\section{Introduction}

Let $1_y$ denote the characteristic function of the integers free of prime factors $\leq y$,
The idea is to work with the   identity
\begin{equation}  \label{eq: TrivId}
\Lambda(n) = \log n - \sum_{\substack{\ell m=n \\ \ell,m>1 }} \Lambda(\ell) ,
\end{equation}
summing it up   over  integers $n\leq x$ for which $1_y(n)=1$, where we might select $y:=\exp(\sqrt{\log x})$ or larger.
In this case the second sum can be written a sum of terms $1_y(\ell) \Lambda(\ell) \cdot 1_y(m)$ which have the bilinear structure that is used in ``Type II sums''. We will see the identity in action in two key results in analytic number theory:

\section{The Bombieri-Vinogradov Theorem}

\begin{theorem}  [The Bombieri-Vinogradov Theorem] \label{thm: 46.3}
If $x^{1/2}/(\log x)^B\leq Q\leq x^{1/2}$ then
  \begin{equation}  \label{eq: 46.6}
\sum_{q\leq Q} \max_{(a,q)=1} \ \left| \pi(x;q,a) - \frac{\pi(x)}{\phi(q)}  \right|  \ll Q x^{1/2}  (\log\log x)^{1/2} .
\end{equation}
\end{theorem}

This is a little stronger than the results in the literature (for example Davenport \cite{Da} has the
$(\log\log x)^{1/2}$ replaced by $(\log x)^{5}$). The reason for this improvement is the simplicity of our identity, and some slight strengthening of the auxiliary results used in the proof.

\begin{proof} Let $y=x^{1/\log\log x}$ . We will instead prove the following result, in which the $\psi$ function replaces $\pi$, and deduce \eqref{eq: 46.6} by partial summation:
\begin{equation}  \label{eq: 46.5}
\sum_{q\leq Q} \max_{(a,q)=1} \ \left| \psi(x;q,a) - \frac{\psi(x)}{\phi(q)}  \right|  \ll Q x^{1/2}  (\log x) (\log\log x)^{1/2} .
\end{equation}

Using \eqref{eq: TrivId} for integers $n$ with $1_y(n)=1$, the quantity on the left-hand side of \eqref{eq: 46.5} is $\leq S_I+S_{II}+E$ where
\[
S_I=\sum_{q\leq Q} \max_{(a,q)=1} \ \left| \sum_{\substack{ n\leq x \\   n\equiv a \pmod q  }} 1_y(n) \log n - \frac {1}{\phi(q)} \sum_{\substack{ n\leq x \\   (n,q)=1  }} 1_y(n) \log n  \right|
\]
which is $\ll xu^{-u+o(u)}\ll_A \frac{x}{(\log x)^A}$ by the small sieve, where $x/Q=y^u$;
and $E$ is the contribution of the powers of primes $\leq y$, which contribute $\leq \pi(y)\log x$ to each sum and therefore $\leq Q \pi(y)\log x\ll_A \frac{x}{(\log x)^A}$ in total. Most interesting is
\[
S_{II}=\sum_{q\leq Q} \ \max_{a:\ (a,q)=1} \ \left| \sum_{\substack{ n\equiv a \pmod q  }} f(n) - \frac {1}{\phi(q)} \sum_{\substack{ (n,q)=1  }} f(n)\right|
\]
where $f(n)= \sum_{\substack{\ell m=n, \ell,m>y }}  \Lambda(\ell)1_y(\ell) \cdot 1_y(m)$. Its bilinearity means that this is a Type II sum, and we can employ the following general result.\footnote{This can be obtained by taking the ideas in proving (5) of chapter 28 of \cite{Da},  along with the method of proof of Theorem 9.16 of \cite{FI}; in any case it is only a minor improvement on either of these results. For full details see chapter 51 of \cite{Gra}.}

\begin{theorem} \label{thm: 46.1} For each integer $n\leq x$ we define
\[
f(n):= \sum_{\substack{\ell m=n }}  \alpha_\ell\beta_m
\]
where $\{ \alpha_\ell\}$ and $\{ \beta_m\}$ are sequences of complex numbers, for which
\begin{itemize}
\item The  $\{ \alpha_\ell\}$ satisfy the Siegel-Walfisz criterion;
\item The  $\{ \alpha_\ell\}$  are only supported in the range $L_0 \leq  \ell  \leq x/y$ ;
\item $\sum_{\ell \leq L} |\alpha_\ell |^2 \leq aL$ and $\sum_{m \leq M} |\beta_m|^2 \leq bM$ for all $L,M\leq x$.
\end{itemize}
For any $B>0$ we have
\begin{equation}  \label{eq: 46.4}
\sum_{q\leq Q} \ \max_{a:\ (a,q)=1} \ \left| \sum_{\substack{ n\equiv a \pmod q  }} f(n) - \frac {1}{\phi(q)} \sum_{\substack{ (n,q)=1  }} f(n)\right| \ll (ab)^{1/2} Q x^{1/2}  \log x,
\end{equation}
where $Q=x^{1/2}/(\log x)^B$,  with $x/y\leq  \frac{Q^2}{(\log x)^2}$ and $L_0\geq y,\exp( (\log x)^\epsilon)$.
\end{theorem}

We deduce that
$S_{II} \ll Q x^{1/2}  (\log x)^{5/4} $ by Theorem \ref{thm: 46.1} since $a\ll \log x$ and $b\ll \frac 1{\log y}=  \frac{\log\log x}{\log x}$.
  \end{proof}

\section{A general bound for a sum over primes}

\begin{proposition} \label{prop: 80.General} For any given function $F(.)$ and $y\leq x$ we have
\[
\left|\sum_{ \substack{ n\leq x \\  p(n)>y}}\Lambda(n)F(n)\right| \ll S_I \log x + (S_{II} \, x (\log x)^5)^{1/2}
\]
where $S_I$ is the Type I sum given by
\[
S_I:= \max_{t\leq x} \left| \sum_{ \substack{   n\leq t \\  p(n)>y}}   F(n) \right|  \leq   \sum_{\substack{   d\geq 1 \\  P(d)\leq y}} \left| \sum_{   m\leq t/d }   F(d  m) \right| ,
\]
and $S_{II}$ is the Type II sum given by
\[
S_{II}:= \max_{\substack{ y<L\leq x/y \\ y< m\leq 2x/L}}   \sum_{m/2<n\leq 2m}  \left| \sum_{\substack{ L<\ell\leq 2L \\ \ell \leq  \frac xm ,\frac xn }} F(\ell m)\overline{F(\ell n)} \right|
\]
\end{proposition}

This simplifies, and slightly improves chapter 24 of \cite{Da}, which is what is used there to bound exponential sums over primes.

\begin{proof} We again use  \eqref{eq: TrivId} so that
\[
\sum_{ \substack{ n\leq x \\  p(n)>y}} \Lambda(n)F(n) =
\sum_{ \substack{ n\leq x \\  p(n)>y}} F(n)    \log n -  \sum_{\substack{ \ell  \\  p(\ell)>y}} \Lambda(\ell)   \sum_{\substack{ m\leq x/\ell \\  p(m)>y}}  F(\ell m).
\]
where $p(n)$ denotes the smallest prime factor of $n$. Now
\[
\sum_{ \substack{ n\leq x \\  p(n)>y}} F(n)    \log n =\sum_{ \substack{ n\leq x \\  p(n)>y}} F(n) \int_1^n \frac {dt}t =  \int_1^x  \sum_{ \substack{ t\leq n\leq x \\  p(n)>y}}    F(n) \frac {dt}t \leq
2\log x \cdot \max_{t\leq x} \left| \sum_{ \substack{   n\leq t \\  p(n)>y}}   F(n) \right|.
\]
Moreover for $P=\prod_{p\leq y}p$,
\[
 \sum_{  \substack{   n\leq t \\  p(n)>y}}   F(n)  =  \sum_{   n\leq t }   F(n)    \sum_{d|P, d|n} \mu(d)
 =  \sum_{d|P } \mu(d) \sum_{   m\leq t/d }   F(d m) .
\]

For the second sum we first split the sums into dyadic intervals ($L<\ell\leq 2L, M<m\leq 2M$) and then Cauchy, so that the square of each subsum is
\begin{align*}
& \leq \sum_{\ell : p(\ell)>y} \Lambda(\ell)^2 \cdot \sum_\ell \left|  \sum_{\substack{ m\leq x/\ell \\  p(m)>y}}    F(\ell m)  \right|^2
\ll L \log L \sum_{\substack{M<m,n\leq 2M  \\  p(m),p(n)>y}}  \sum_{\ell \leq \frac x{\max\{ m,n\} }} F(\ell m)\overline{F(\ell n)} \\
& \ll  x\log x \cdot   \max_{M<m\leq 2M}  \sum_{\substack{m/2<n\leq 2m\\  p(n)>y}}  \left| \sum_{\substack{ L<\ell\leq 2L \\ \ell \leq  \frac xm ,\frac xn }} F(\ell m)\overline{F(\ell n)} \right| \end{align*}
since $m,n\in (M,2M]$, and the result follows.
\end{proof}

\section{Genesis}

The idea for using \eqref{eq: TrivId} germinated from reading the proof of the Bombieri-Vinogradov Theorem (Theorem 9.18)  in \cite{FI}, in which they used Ramar\'e's identity, that if $\sqrt{x}<n\leq x$ and $n$ is squarfree then
\[
  1_{\mathbb P}(n) =  1 - \sum_{\substack{pm=n \\ p \text{ prime} \leq \sqrt{x}}} \frac 1{\omega_{\sqrt{x}}(m)}
\]
where $1_{\mathbb P}$ is the characteristic function for the primes, and  $\omega_{z}(m)=1+\sum_{p|m,\ p\leq z} 1$.
They also had to sum this over all integers free of prime factors $>y$.

\bibliographystyle{plain}


\section{Introduction}
The  galaxy luminosity function (hereafter, LF)  represents  the  number density  of galaxies of a  given luminosity. It is a robust observable, extensively used in the past, to study the properties of galaxy populations \citep[e.g.][and references therein]{blanton2003}. The comparison between the behaviour of the LF and the halo mass function at faint magnitudes has been proposed as a crucial test to understand galaxy formation processes. Indeed, the cold dark matter theory predicts halo mass functions with a slope of $\sim-1.9$ \citep[e.g.][and reference therein]{springel2008}, steeper than the one observed in deep LFs of nearby galaxy clusters, or in the field \citep[$\sim -1.1:-1.5$][]{trenthamt2002,depropris2003,blanton2005}. This is the so-called missing satellite problem, well reported in censuses of galaxies around the Milky Way and M31 \citep[e.g.][]{klypin1999}. Attempts to solve these discrepancies invoke several physical mechanisms that halt the star formation and darken or destroy dwarf galaxies, including high gas cooling times \citep{wr1978}, and suppression of low-mass galaxies due to a combination of feedback, photoionization or/and dynamical processes \citep{benson2002,benson2003,brooks2013}.

Claims of the environmental dependence of the LF are numerous \citep[e.g.][]{tully2002,trentham2002,trentham2005,infante2003}, but the exact shape and significance of this dependence is still a matter of debate. Many previous studies suggest that the most striking differences between low-density and high-density environments concern the faint-end slope of the LF, with cluster galaxies showing a higher abundance of low-luminosity galaxies than the field \citep[e.g.][and references therein]{blanton2005, popesso2006}. This has important implications, as it suggests that, whatever the mechanisms preventing the formation of galaxies within low-mass satellites are, they depend on the host halo mass. \cite{tully2002} suggest that perhaps reionization inhibited the collapse of gas in late-forming dwarfs in low-density environments, whereas a larger fraction of low-mass haloes formed before the photoinization of the intergalactic medium in overdensities that ultimately become clusters. Interestingly, this mechanism could be at the heart of the discrepancies existing in the literature about the faint-end of the LF in clusters. While some authors observe a marked upturn at faint magnitudes \citep[][]{yagi2002,popesso2006,barkhouse2007}, others find a more regular behaviour \citep[e.g.][]{sanchez2005,andreon2006}. When observed, this upturn is usually due to early-type dwarfs located in the outer regions of clusters, suggesting that cluster-related mechanisms may be responsible for the paucity of dwarfs observed in the denser inner regions \citep{pfeffer2013}. Moreover, the slopes derived in the upturn region are rather steep, as much as to be fully consistent with that of the halo mass function. Unfortunately, there is no evidence for the existence of such a dramatic steeping in the few cluster LFs where spectroscopic or visual membership has been determined \citep[][]{hilker2003,rines2003,mieske2007,misgeld2008,misgeld2009}. More extensive spectroscopic cluster surveys are needed to derive robust results on the faint-end slope of the LF, and in this way put constraints on the role played by the environment on the formation of baryonic structures within low-mass dark matter halos.

We have undertaken a project in order to obtain spectroscopy of galaxies in nearby clusters down to the dwarf regime ($M > M^*+6$). This dataset will allow us to infer accurate cluster membership and analyze several properties of the dwarf galaxy population in nearby clusters, minimizing the contribution of background sources. In the present work, we present a study of the spectroscopic LF of  (A85), a nearby ($z = 0.055$) and massive cluster \citep[$M_{200} = 2.5 \times 10^{14} \; M_{\odot}$, and $R_{200} = 1.02 \, h^{-1} \,$Mpc][]{rines2006}. This cluster is not completely virialized, since several substructures and infalling groups have been identified \citep{durret1999,aguerri2010,cava2010}. A85 is an ideal target for a deep study of the LF, as spectroscopy within the virial radius is almost complete down to m$_r \sim$ 18, resulting in 273 confirmed members \citep[][]{aguerri2007}. The new dataset presented here reaches three mag fainter, and almost doubles the number cluster members.

Throughout this work we have used the cosmological parameters $H_0 = 75 \; \mathrm{km} \, \mathrm{s}^{-1} \mathrm{Mpc}^{-1}$, $\Omega _m = 0.3$ and $\Omega _{\Lambda} = 0.7$.

\section{The observational data on A85}
\subsection{Deep VLT/VIMOS Spectroscopy}
Our parent photometric catalogue contains all galaxies brighter than $m_r = 22$ mag\footnote{The apparent magnitudes used are the dered SDSS-DR6 $r$-band magnitudes.} from the SDSS-DR6 \citep[][]{adelman2008}, and within $R_{200}$\footnote{The cluster center is assumed to be at the brightest cluster galaxy (BCG, $\alpha$(J2000): $00^h \, 41^m \, 50.448^s$ $\delta$(J2000): $-9^{\circ} \, 18'  \, 11.45''$). This is a sensible assumption because the peak of X-ray emission lies at only 7 kpc from the BCG \citep{popesso2004}.}. Figure \ref{cmd} shows the colour-magnitude diagram of A85 for the galaxies included in this catalogue. The target galaxies for our spectroscopic observations were selected among those with no existing redshift in the literature and bluer that $g-r = 1.0$ (see Fig. \ref{cmd}). This is the colour of a 12 Gyr old stellar population with [Fe/H] = +0.25 supersolar metallicity \citep{worthey1994}, typical of very luminous early-type galaxies. As a result, this colour selection should minimize the contamination by background sources, while matching at the same time the colour distribution of galaxies in the nearby Universe \citep[e.g.][]{hogg2004}.

The observations were carried out using the multi-object-spectroscopy (MOS) mode of VLT/VIMOS, in combination with the LR-blue+OS-blue grisms and filters (Program 083.A-0962(B), PI R. S\'anchez-Janssen). To maximize the number of targets, and avoid the gaps between the instrument CCDs, we designed 25 masks with large overlaps covering an area of 3.0 $\times$ 2.6 Mpc$^2$ around the central galaxy in A85 -- i.e. extending out to more than 1\,R$_{200}$. This observational strategy allowed us to obtain 2861 low-resolution spectra (R=180) of galaxies down to $m_r= 22$ mag. We exposed during 1000 s to reach a signal-to-noise ($S/N$) in the range $6 - 10$ down to the limiting magnitude. The data were reduced using GASGANO and the provided pipeline \citep{izzo2004}. The spectra were wavelength calibrated using the HeNe lamp, which yield a wavelength accuracy  of $\sim 0.5$ $\rm \AA$ pixel$^{-1}$ in the full spectral range ($3700 - 6700 \, \rm \AA$).

\subsection{Redshift Determination and Cluster Membership}
The recessional velocity of the observed galaxies were determined using the \textit{rvsao.xcsao} \textit{IRAF} task \citep{kurtz1992}. This task cross correlates a template spectrum \citep[in this work][]{kennicutt1992} with the observed galaxy spectrum. This technique allowed us to determine the recessional velocity in 2070 spectra. The remaining spectra had too low $S/N$ to estimate reliable redshifts. \textit{xcsao} formal errors are smaller than true intrinsic errors \citep[e.g.][]{bardelli1994}, and reliable errors can only be estimated by observing galaxies more than once. Our observational strategy allowed us to obtain 676 repeated spectra result in a one-sigma velocity uncertainty of $\sim 500$ km s$^{-1}$. The redshifts from the literature (SDSS-DR6 and NED catalogues), together with our new data, result in a total number of 1593 galaxy velocities in the A85 direction within R$_{200}$ and $14  < m_r < 22$.

\begin{figure}
\centering
  \includegraphics[width=1\linewidth]{cmd}
  \caption{Lower panel: colour-magnitude diagram of the galaxies in the direction of A85. Grey points are the target galaxies. Red and blue symbols show red and blue cluster members, respectively. The solid line represents the red sequence of the cluster. Upper panel: spectroscopic completeness  ($C$, diamonds) and cluster member function ($f_m$, black triangles) as a function of $r$-band magnitude. The dashed vertical line represents our limit magnitude for the spectroscopic LF.}
\label{cmd}
\end{figure}

The caustic method \citep{diaferio1997,diaferio1999,serra2011} estimates the escape velocity and the mass profile of galaxy clusters in both their virial and infall regions, where the assumption of dynamical equilibrium does not necessarily hold. A by-product of the caustic technique is the identification of the members of the cluster with an interloper contamination of only 2\% within R$_{200}$, on average \citep{serra2013}. The application of the caustic technique to our spectroscopic catalogue resulted in a sample of  $434$ cluster members within $R_{200}$, 284 of which are new data.

We define the completeness of our data as $C = N_z / N_{phot}$, with $N_z$ being the number of measured redshift and $N_{phot}$ the number of photometric targets. Figure \ref{cmd} shows that $C$ is higher than 90 $\%$ for galaxies with M$_r < -19$ and it decreases around $40 \, \%$ at $M_r \sim -16$. We also defined the member fraction as $f_m = N_m / N_z$, being $N_m$ the number of members. The member fraction also strongly depends on luminosity. Thus, $f_m$ is higher than 80 $\%$ down to M$_r < -19$ and then rapidly decreases down to $\sim$ 20 $\%$ at   M$_r = -16$ (see Fig. \ref{cmd}).

\section{The  spectroscopic  LF  of  A85}
The A85 LF is computed using all cluster members with m$_r \leq 21$ mag and $\langle \mu_{e,r} \rangle \leq 24$ mag arcsec$^{-2}$. These limits correspond to the values where the galaxy counts stop increasing, and thus determine our completeness limits. We note that our uniform spectroscopic selection function (cf. Sect. 2.1) does not introduce any bias in magnitude, $\langle \mu_{e,r} \rangle $, or colour.

Figure \ref{LF} shows the $r$-band spectroscopic LF of A85. It is computed as $\phi(M_r) = N_{phot}(M_r) \times  f_m(M_r) / (0.5 \times A)$, where $A$ is the observed area and 0.5 is the magnitude bin.
The Pearson test shows that the observed LF cannot be modelled by a single Schechter at 99$\%$ confidence level, due to the presence of a statistically significant upturn at $M_r > -18$ (see Fig. \ref{LF}). The observed LF is better parameterized using two Schechter functions. Because of the degeneracy in fitting a double Schechter profile, we followed a two-steps process \citep[see][]{barkhouse2007}. First, we fitted the bright part ($b: -22.5<$ $M_{r}<-19.0$) of the LF obtaining $M^*_b$ and $\alpha_b$. Then, these parameters were fixed when a double Schechter fit was performed to the total LF. Table \ref{tabella_data} shows the M$_r^*$ and $\alpha$ values of the faint ($f: -19.0<$ $M_{r}<-16.0$) and bright parts of the best-fit LF.

\begin{figure}
\includegraphics[width=1\linewidth]{lf}
\caption{Black points are the observed spectroscopic LF of A85. The solid blue line shows the fit of the LF by a double Schechter. The bright ($b$) and faint ($f$) components of the fit are represented by a dashed and dotted lines, respectively. The 68$\%$ and 99$\%$ c.l. for the fitted parameters are shown in the insets.}
\label{LF}
\end{figure}

In order to better understand the nature of the galaxies responsible for the upturn at the faint end of the LF, we classified galaxies in blue and red ones according to their $(g-r)$ colours. Thus, blue galaxies are those with $(g-r) < (g-r)_{RS} - 3 \sigma $ and red ones are the remaining cluster members; where $(g-r)_{RS}$ is the colour of the red sequence of A85, and $\sigma_{RS}$ represents its dispersion\footnote{The red sequence of the cluster and its dispersion were measured in the magnitude range $-22.5 < $ M$_r < -19.0$.} (see Fig. \ref{cmd}). Figure \ref{LF_rb} shows the spectroscopic LF of the blue and red populations of A85. Naturally, red galaxies completely dominate in number. The red LF departs from the single Schechter shape, showing the characteristic flattening at intermediate luminosities followed by a (mild) upturn at the faint end. The blue LF, however, is well fitted by a single Schechter function. This is in qualitative agreement with previous work \citep[e.g.][P06 hereafter]{popesso2006}. Nevertheless, it is remarkable that the faint-end slopes of both the red and blue populations are virtually identical (see Table \ref{tabella_data}).

\begin{table}
\begin{center}
\caption{Schechter Function Parameters.\label{tabella_data}}
\begin{tabular}{cccc}
\hline\hline
 mag interval &$M_{r}^* $ [mag] &$\alpha $\\
\hline
-22.5 $<$ M$_{r}$ $<$ -19.0  &$-20.85\; ^{+0.14}_{-0.14}$  &$-0.79\; ^{+0.08}_{-0.09}$ \\
-19.0 $<$ M$_{r}$ $<$ -16.0   &$-18.36\; ^{+0.41}_{-0.40}$  &$-1.58\; ^{+0.19}_{-0.15}$ \\
\hline
&     red &\\
\hline
-22.5 $<$ M$_{r}$ $<$ -19.0  &$-20.71\; ^{+0.13}_{-0.15}$  &$-0.63\; ^{+0.09}_{-0.08}$ \\
-19.0 $<$ M$_{r}$ $<$ -16.0    &$-17.90\; ^{+0.35}_{-0.19}$  &$-1.46\; ^{+0.18}_{-0.17}$ \\
\hline
&      blue &\\
\hline
-22.5 $<$ M$_{r}$ $<$ -16.0   &$-21.29 \; ^{+0.36}_{-0.44}$  &$-1.43 \; ^{+0.06}_{-0.05}$ \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}
\includegraphics[width=1\linewidth]{lf_b_r}
\caption{The spectroscopic LF of A85 (black points), blue and red diamonds show the LF of blue and red galaxies of A85. The full lines correspond to the Schecter function fits. The histograms are the LF of field red and blue galaxies (Blanton et al. 2005).}
\label{LF_rb}
\end{figure}

\section{Discussion}
In Fig.3 we additionally show the LFs of blue and red field populations from \cite{blanton2005} \footnote{Throughout this work, all the literature LFs are normalized so that they have the same number of counts as our A85 LF in the magnitude range $-22 < M_{r} < -19$.}. Their field LF  is best described by a double Schechter function with faint-end slope -1.5.

Figure \ref{LF_cfr} shows the comparison between our LF and other cluster and field LFs from the literature. We compare with the LF of the Virgo and Abell\,2199 clusters \citep{rines2008} because they are the two deepest spectroscopic LFs of nearby clusters covering a significant fraction of their respective virial radii (out to $\sim 0.7\,R_{200}$, very similar to our coverage). In addition, we include the LF from P06 because it is a photometric LF obtained by stacking a large sample of 69 clusters, and it exhibits the most statistically meaningful example of an upturn. Finally, we compare with the field LF from \cite{blanton2005}. We note that all LFs in Fig.4 have been derived using SDSS photometry. The very steep upturn present by the stacked photometric LF  ($\alpha \sim -2.2$) is not observed in any of the nearby clusters. While A85 shows a slightly steeper slope than Virgo and A2199, it is not nearly as steep as the photometric LF.

The discrepancy between the LF of A85 and the composite cluster LF from P06 can be traced back to the different methodologies applied to derive them. First, we introduced a stringent colour cut in the selection of our spectroscopic targets (see Sect.2.1). Second, our spectroscopic LF is based on accurate cluster membership using the galaxy recessional velocity, while photometric LFs necessarily rely on a statistical background subtraction. The open circles in Fig. \ref{LF_cfr} show the photometric LF of A85, derived from the number counts of candidate cluster galaxies with $(g-r) < 1.0$, and after performing a statistical background subtraction using galaxies (with the same colour cut) from 50 SDSS random fields of the same area as our A85 survey. It is clear that the photometric LF of A85 closely matches the spectroscopic LF, except for a steeper faint-end slope, which moves towards the P06 value. We quantify this difference by fitting a power law function to these LFs in the $ -18 \leq M_{r} \leq -16$ magnitude range. We prefer this simple approach to minimize the degeneracies present when fitting a double Schechter function (see Sect.3). We derive power-law faint-end slopes $\alpha_{f} = -1.5, -1.8,$ and  $-2.1$ for the spectroscopic LF of A85, the photometric LF of A85, and the stacked photometric LF from  P06, respectively. We note that cosmic variance of the cluster LF, or any mass dependence, can not explain the discrepancy between the composite cluster LF and that of A85. P06 show that 90 per cent of their nearby cluster sample have individual LFs consistent  with the composite one shown in Fig.\ref{LF_cfr}. Moreover, the photometric LF for A85 itself from P06 is totally consistent with having a very steep faint end (see their Figure 5). This all suggests that the composite photometric LF from  P06 suffers from contamination at the faint end, resulting in much steeper slopes than what is found using spectroscopic data.

\begin{figure}
\includegraphics[width=1\linewidth]{lf_cfr}
\caption{Comparison between our LF and others present in literature: the stacked photometric LF of 69 clusters from Popesso et al. (2006), the spectroscopic LFs of A2199 and Virgo from Rines \& Geller (2008), and the field galaxy LF from Blanton et al. (2005). The photometric LF of A85 is also shown with open circles.}
\label{LF_cfr}
\end{figure}

The field spectroscopic LF presented by \citep[][]{blanton2005} shows an upturn and a faint-end slope ($\alpha=-1.5$) similar to the one of A85. Contrary to some claims from photometrically derived cluster LFs, Fig. \ref{LF_cfr} shows that the faint-end slope in A85 and the field are consistent with each other -- i.e. clusters of these masses do not seem to contain a significant excess of dwarf galaxies with respect to the field. This, in turn, suggests that the environment may not play a major role in determining the abundance of low-mass galaxies \citep[cf.][]{tully2002}, but only acts to modify their star formation activity. While the overall abundance of dwarfs in the field and in clusters like A85 is the same, blue dwarfs dominate in the former environments, and red dwarfs in the latter.

Environmental processes involving the loss of the galaxy gas reservoirs \citep[e.g.][]{quilis2000, bekki2002}, followed by the subsequent halt of the star formation, and the reddening of their stellar populations are the obvious mechanisms invoked to explain the colour transformation of cluster dwarfs.
It is however not yet clear whether the efficiency of these processes depends on the halo mass or not. \cite{lisker2013} propose that quiescent cluster dwarfs originate as the result of early \citep[see also][] {sanchez2012} \emph{and} prolonged environmental influence in \emph{both} group- and cluster-size haloes. Along these lines, \cite{wetzel2013} find that group preprocessing is responsible for up to half of the quenched satellite population in massive clusters. On the other hand, P06 suggest that the excess of red dwarfs in clusters is a threshold process that occurs within the cluster virial radius \citep[see also][]{sanchez2008}, but their exact abundance is nevertheless a function of clustercentric radius: the upturn becomes steeper as the distance from the centre increases.
Our dataset calls for a study of the LF as a function of radial distance from the cluster centre, and an investigation of the properties of A85 dwarfs in substructures and infalling groups. This will be the subject of future  papers in this series.

\section{Conclusions}
We obtained 2861 low resolution spectra for galaxies down to $m_{r}=22$ mag within the virial radius of the nearby ($z=0.055$) galaxy cluster A85. This unique dataset allowed us to identify 438 galaxy cluster members, and build the spectroscopic LF of A85 down to M* + 6. The resulting LF is best modelled by a double Schechter function due to the presence of a statistically significant upturn at the faint-end. The amplitude of this upturn ($\alpha = -1.58^{+0.19}_{-0.15}$), however, is much smaller than that of most photometric cluster LFs previously reported in the literature. The faint-end slope of the LF in A85 is consistent, within the uncertainties, with that of the field. We investigate the nature of the galaxy population dominating the faint end of the LF of A85 by  dividing the galaxies according to their $(g-r)$ colour. The red population dominates at low luminosities, and is the main responsible for the upturn. This is different from the field LF: while a relatively steep upturn is also present in the red population, blue galaxies are the prevalent population. The fact that the slopes of the spectroscopic LFs in the field and in a cluster as massive as A85 are similar suggests that the cluster environment does not play a major role in determining the abundance of low-mass galaxies, but it does influence the star formation history of dwarfs.

\textbf{\textit{Acknowledgements.}}
IA, AD and ALS acknowledge partial support from the
INFN grant InDark and from the grant Progetti di Ateneo TO Call 2012 0011 ‘Marco Polo’
of the University of Torino.


\section{Numerical and asymptotic solutions of equation for pressure}

Eq.(11) of our main paper has the following form:
\begin{gather}
-\mu U=\frac{\partial }{\partial r}\left( H  \langle  k \rangle \frac{\partial p}{%
\partial r}\right) +\frac{H \langle k\rangle}{r}\frac{\partial p}{\partial r}+%
\frac{H \langle k\rangle}{r^{2}}\frac{\partial ^{2}p}{\partial \varphi ^{2}}
\label{a1} \\
+\cos 2\varphi \left[ \frac{\partial }{\partial r}\left(H \Delta k \frac{%
\partial p}{\partial r}\right) -\frac{H\Delta k }{r}\frac{\partial p}{%
\partial r}-\frac{H\Delta k }{r^{2}}\frac{\partial ^{2}p}{\partial
\varphi ^{2}}\right]   \notag \\
-\frac{\sin 2\varphi }{r}\left[ 2H \Delta k \frac{\partial ^{2}p}{%
\partial r\partial \varphi }+\left( \frac{d\left(H \Delta k \right) }{dr}%
-\frac{2H\Delta k }{r}\right) \frac{\partial p}{\partial \varphi }\right]
,  \notag
\end{gather}%
and its solution can be presented in
terms of cosine series:
\begin{equation} \label{Gener_sol_p}
p=p_{0}\left( r\right) +p_{1}\left( r\right) \cos 2\varphi +p_{2}\left(
r\right) \cos 4\varphi +...
\end{equation}%

Here we derive equations for functions $p_{n}\left( r\right) $
in series Eq.(\ref{Gener_sol_p}) and describe their numerical solution. When $\Delta k$ is small, we construct an asymptotic solution.

To solve Eq.(\ref{a1}) we substitute (%
\ref{Gener_sol_p}) into (\ref{a1}) and collect terms proportional to $\cos
2n\varphi $. To eliminate singularities, we then use logarithmic
substitution for variable $r$ and introduce the following dimensionless
quantities:
\begin{eqnarray*}
\xi  &=&\ln \left( \frac{r}{\sqrt{hR}}\right) ,\quad \eta =\frac{H}{h}=1+%
\frac{\exp \left( 2\xi \right) }{2}, \\
P_{n} &=&\frac{p_{n}h^{2}}{3\mu UR}, \\
G\left( \eta \right)  &=&-\frac{\eta ^{3} \langle k \rangle}{\langle k%
\rangle\left( 0\right) },\quad D\left( \eta \right) =-\frac{\eta ^{3}\
\Delta k}{\Delta k\left( 0\right) }
\end{eqnarray*}%
This leads to a system of ordinary differential
equations
\begin{equation}
\mathrm{L}_{0}P_{0}+\varepsilon \mathrm{L}_{0}^{+}P_{1}=\frac{4 k }{\langle k%
\rangle\left( 0\right) }\exp \left( 2\xi \right) ,  \label{a5}
\end{equation}%
\begin{equation}
\mathrm{L}_{n}P_{n}+\varepsilon \mathrm{L}_{n}^{+}P_{n+1}+\varepsilon
\mathrm{L}_{n}^{-}P_{n-1}=0\quad \text{for}\quad n>0,  \label{a6}
\end{equation}%
which is expressed in a more compact form by using differential operators
\begin{equation*}
\mathrm{L}_{n}=\frac{d}{d\xi }\left( G\frac{d}{d\xi }\right) -4n^{2}G,
\end{equation*}%
\begin{eqnarray*}
\mathrm{L}_{n}^{+} &=&\frac{d}{d\xi }\left( \frac{D}{2}\frac{d}{d\xi }%
\right) +\left( 2n+1\right) D\frac{d}{d\xi } \\
&&+\left( n+1\right) \frac{dD}{d\xi }+2n\left( n+1\right) D,
\end{eqnarray*}%
\begin{eqnarray*}
\mathrm{L}_{n}^{-} &=&\frac{d}{d\xi }\left( \frac{D}{2}\frac{d}{d\xi }%
\right) -\left( 2n-1\right) D\frac{d}{d\xi } \\
&&-\left( n-1\right) \frac{dD}{d\xi }+2n\left( n-1\right) D,
\end{eqnarray*}%

To solve a finite-difference version of ODE system (\ref{a5}), (\ref{a6}) we
resolve the truncated system with respect to $d^{2}P_{n}/d\xi ^{2},$ $%
n=1,...,\ N,$ by using the Gauss routine, and then integrate numerically the obtained system by
using the fourth-order Runge-Kutta method. This system has a boundary condition
$P_{n}\rightarrow 0$ both at small and large $r,$ or at $\xi \rightarrow -\infty
$ and $\xi \rightarrow +\infty$. Therefore, the dimensionless gap and
the permeabilities take a form%
\begin{equation*}
\eta \rightarrow 1,\quad dG/d\xi =dD/d\xi \rightarrow 0\quad \text{as}\quad
\xi \rightarrow -\infty ,
\end{equation*}%
\begin{equation*}
\eta \simeq \frac{\exp \left( 2\xi \right) }{2},\quad G,\ D\simeq \eta
^{3}\simeq \exp \left( 6\xi \right) \quad \text{as}\quad \xi \rightarrow
+\infty .
\end{equation*}%
Asymptotic linearly independent solutions of (\ref{a5}), (\ref%
{a6}) can be then found in the form: $P_{ni}=a_{ni}^{0}\exp \left( \lambda
_{i}^{0}\xi \right) ,$ $i=1,...,\ 2N$ as $\xi \rightarrow -\infty $ and $%
P_{ni}=a_{ni}^{\infty }\exp \left( \lambda _{i}^{\infty }\xi \right) $ as $%
\xi \rightarrow +\infty .$ The eigenvalues, $\lambda _{i}^{0}$ and $\lambda
_{i}^{\infty },$ and eigenvectors, $\mathbf{a}_{i}^{0}$ and $\mathbf{a}%
_{i}^{\infty },$ were found using IMSL routine DEVCRG. The system admits
both exponentially growing and decaying solutions $P_{ni}$ whereas the
boundary conditions require the decaying one. Thus, we choose solutions with
$\lambda _{i}^{0}>0$ and $\lambda _{i}^{\infty }<0$. To resolve all linearly
independent solutions properly at $\left\vert \xi \right\vert \gg 1$ the
orthonormalization method~\cite{godunov1961nsb,conte1966nsl} is applied.
The boundaries of the integration domain are set at $r_{\min }=0.01$ and $%
r_{\max }=20,$ the number of taken into account harmonics  is $N=4.$ The
calculations show that the expansion converges fast with $N,$ since the
ratio $\left\vert P_{n+1}/P_{n}\right\vert $ is usually small for all $n$, which is illustrated in Fig.~\ref{pn}.

\begin{figure}[tbp]
\includegraphics[width=7 cm]{pn.eps}\newline
\caption{ Functions $P_{n}\left( r\right) =p_{n}h^{2}/3\protect\mu UR$ in
series (\protect\ref{Gener_sol_p}) for $h\ll \min \{ b_2,L \},$ $%
b_{1}/h=0.5$, $\protect\phi _{2}=0.8$. Solid line corresponds to an analytical
solution for $\tilde{P}=\tilde{p} h^{2}/3\protect\mu UR$, circles - to
a numerical evaluation of $P_0$, dashed line - to $10^3P_1$ and dash-dotted
line - to $10^5P_2$.}
\label{pn}
\end{figure}

The solution of Eq. (\ref{a1}) is simple if ratio $k_{\parallel }/k_{\perp }$ (and hence $\Delta k/ \langle k \rangle$) is
constant, i.e., when the permeabilities depend on $r$ similarly, $%
k_{\parallel }^{\ast },\ k_{\perp }^{\ast }\sim g\left( r\right) .$ This is the case in a thin channel with $%
b_{1}=0$, when all the permeabilities are proportional to $H^{2}$~\cite%
{feuillebois.f:2009}. Then the solution includes only axisymmetric term $%
p_{0}$. It can be verified directly that solution
\begin{equation}
\frac{d\tilde p}{dr}=-\frac{\mu Ur}{2 H \langle k \rangle}=-\frac{12\mu Ur}{%
H^{3}\left(k_\parallel^{\ast }(H)+k_\perp^{\ast }(H)\right)},  \label{derivP}
\end{equation}%
\begin{equation}
\frac{\tilde p(H)}{12 \mu UR}=\int\limits_{H}^{\infty }{\frac{ dH^{\prime }}{%
H^{\prime }{}^{3} \left(k_\parallel^{\ast }+k_\perp^{\ast }\right)}}  \label{Pressure}
\end{equation}
satisfies Eq. (\ref{a1}) since the boundary conditions%
are homogeneous, $\partial \widetilde{p}/\partial \varphi =0$, and
terms
\begin{equation*}
\frac{\partial }{\partial r}\left(H \Delta k \frac{\partial \widetilde{p}%
}{\partial r}\right) =\frac{\Delta k}{2 \langle k \rangle}\mu
U,\quad \frac{H\Delta k }{r}\frac{\partial \widetilde{p}}{\partial r}%
=\frac{\Delta k}{2 \langle k \rangle}\mu U,
\end{equation*}%
cancel out. The analytical expressions for $\widetilde{P}=\widetilde{p}%
h^{2}(3\mu UR)^{-1}$ and corresponding resistance force in the case when
$h\ll \min \{ b_{2},L \} $ are obtained in the Appendix section of the main paper.

The first three harmonics in series (\ref{Gener_sol_p}), plotted as
functions of $r/\sqrt{hR}$, are presented in Fig.~\ref{pn}. It shows that
the axisymmetric part of pressure distribution is very close to the
approximate solution $\widetilde{P}$ while the non-axisymmetric part is
several orders less.

The permeability difference is typically small, $|\Delta k|\ll |\langle k\rangle|$, for example,
for large distances between surfaces, $H\gg L.$ The asymptotic solution of
Eqs. (\ref{a5}), (\ref{a6}) can be found in this important case as power
series expansion $P_{n}=\sum\nolimits_{j=0}\varepsilon ^{j}P_{n}^{(j)}$ with
respect to small parameter $\varepsilon $. We do not construct the full
asymptotic solution, but show that the isotropic part $P_{0}$, which only
contributes to the drag force, differs from the approximate solution $%
\widetilde{P}$ given by Eq. (\ref{Pressure}) by the value $O\left(
\varepsilon ^{2}\right) .$ Substituting the expansions into the system (\ref%
{a5}), (\ref{a6}) and collecting the terms proportional to $\varepsilon ^{j}$
one obtains equations governing the first three terms of expansion:
\begin{equation}
\mathrm{L}_{0}P_{0}^{(0)}=\frac{4 k}{\langle k\rangle\left( 0\right) }\exp
\left( 2\xi \right) ,  \label{a7}
\end{equation}%
\begin{equation*}
P_{n}^{(0)}=0\quad \text{as}\quad n>0,
\end{equation*}%
\begin{equation*}
\mathrm{L}_{1}P_{1}^{(1)}=-\mathrm{L}_{1}^{-}P_{0}^{(0)},
\end{equation*}%

\begin{equation*}
P_{n}^{(1)}=0\quad \text{as}\quad n\neq 1,
\end{equation*}%
\begin{equation*}
\mathrm{L}_{0}P_{0}^{(2)}=-\mathrm{L}_{0}^{+}P_{1}^{(1)},
\end{equation*}%
\begin{equation*}
\mathrm{L}_{2}P_{2}^{(2)}=-\mathrm{L}_{2}^{-}P_{1}^{(1)},
\end{equation*}%
\begin{equation*}
P_{1}^{(2)}=0,\ P_{n}^{(2)}=0\quad \text{as}\quad n>2.
\end{equation*}%
The equation (\ref{a7}) is the dimensionless form of Eq. (\ref{derivP}). For
further terms, one can deduce that
\begin{equation*}
P_{n}=\varepsilon ^{n}P_{n}^{(n)}+\varepsilon
^{n+2}P_{n}^{(n+2)}+\varepsilon ^{n+4}P_{n}^{(n+4)}+...
\end{equation*}%
Thus the first-order correction to the pressure field is $\varepsilon
P_{1}^{(1)},$ while the correction to the isotropic part $\varepsilon
^{2}P_{0}^{(2)}$ is much smaller.

\iffals
In the most practical situations the permeability ratio $\left\vert \Delta
k/\langle k\rangle\right\vert $ is small and/or weakly dependent
on $r$. As a result of both factors, numerical solution coincides well with
approximate solution $\widetilde{P}_{0}$ (see Fig.~\ref{pn}). \f

\bibliographystyle{rsc}


\section{Introduction}
Westerlund (1961) discovered that the 41day Cepheid RS Pup was surrounded
by a remarkable nebulosity. This is in the shape of rudimentary rings but
with much distorted structure and condensations. Havlen (1972) showed that
portions of the nebulosity varied in the period of the Cepheid but with
with various phase lags. A very beautiful set of measurements of phase lags
at various points in the nebula has recently been obtained by
Kervella et al. (2008) (= Kervella et al.). In general, the expected phase lag
at a point $i$ may be written:
\begin{equation}
(N_{i} +\Delta \phi_{i}) =
5.7755 10^{-3} D\theta_{i} (1+ \sin \alpha_{i})/P\cos \alpha_{i}
\end{equation}

Here $\Delta \phi_{i}$ is the fractional phase lag, $N_{i}$ the whole
number of pulsation periods elapsed, $D$ is the distance to RS Pup in parsecs,
$\theta_{i}$ is the angular distance of $i$ from the star in arcsec,
$P$ is the pulsation period in days and $\alpha_{i}$ is the angle between the
line joining the star to $i$ and the plane of the sky (positive if $i$
is further away than the star, negative if it is nearer). The measured
quantities are $\Delta \phi_{i}$ and $\theta_{i}$. $P$ is assumed known and
here it is taken as 41.4389 days ( Kervella et al.).
In an attempt to determine $D$, Kervella et al. assume
$\alpha_{i} = \rm {constant} =0$. That is they assume that
all the features measured by them lie in the plane of the sky
and the values of $N_{i}$ are then chosen to obtain the
best fit to this model. The
justification for this assumption is that if the nebulosity consisted of
a series of thin, uniform, spherical shells centred on the star,
then the deviation of all measured points from the plane of the sky would
be small. However an examination of the structure of the nebulosity
(for instance from the figures in Kervella et al.) shows that it is far
from corresponding to this idealized model. There is much distortion and
density variation in the rudimentary rings. Kervella et al. place
special emphasis on the ten condensations or blobs shown in their fig 7.
The existence of such blobs is not consistent with the idealized model
and leaves open the question of whether they or other features are actually in,
or near,
the plane of the sky.
In view of these
uncertainties it cannot
be claimed that a definitive distance to RS Pup can be found based on
the ``in-the-plane" assumption. In the next section this assumption is dropped
and it is shown that a simple and astrophysically interesting model for
the nebulosity is found
if a distance for RS Pup is adopted from a period-luminosity
relation.
\section{An equatorial disc model}
van Leeuwen et al. (2007) established a reddening-free period-luminosity
relation in $V,I$ based on HST (Benedict et al. 2007) and revised Hipparcos
parallaxes. This together with the data in table A1 of that paper leads
to a predicted distance of 1728pc for RS Pup
\footnote{The distance, $1830^{+109}_{-94}$pc derived from the pulsational
parallax by Fouqu\'{e} et al. (2007) is not significantly different from
this.}.
Adopting this distance it is
possible to
use eq. 1 to study the three dimensional structure of the nebulosity.
In principle the values of $N_{i}$ can be arbitrarily assigned. However they
should obviously be chosen to account for apparent continuities in the
structure and to conform to some simple, physically reasonable model.
It was quickly found by trial and error that there is a consistent
set of values of $N_{i}$
in which the points measured by Kervella et al. are further away than
the star on the south side and nearer on the north, i.e. an inclined disc
model is indicated. This is indeed the simplest model, if the
uniform spherical shell model is rejected. In such a model the values of
$N_{i}$ have to chosen such that $(N_{i} +\Delta \phi_{i})/\theta_{i}$
values are as near constant as possible in a given direction from the
star and vary smoothly with direction.
The details are given
in Table 1. This contains data on the 31 points observed by Kervella et al.
and I am greatly indebted to them for supplying detail of their observations
which were not given in the original paper. The Table lists:\\
1. Position number, $i$.\\
2. Angular distance from the star, $\theta_{i}$, in arcsec.\\
3.Azimuth of the point relative to the star, $\beta_{i}$, measured from north
through east, in degrees.\\
4. $\Delta \phi_{i}$ and its standard error.\\
5. The value of $N_{i}$ ($=N_{i}^{K}$) adopted by Kervella et al. to fit their
model
assumptions.\\
6. The distance, $d_{y}^{K}$, behind (positive) or in front (negative) of the
plane of the
sky through the star.
This is found by using eq.1 together with adopted values
of $N_{i}$ and $D$ to derive $\alpha_{i}$ in each case. Then,
\begin{equation}
 d_{y} = 4.848.10^{-2} \theta_{i} D \tan\alpha_{i}
\end{equation}
where $d_{y}$ is in units of $10^{-4}$pc. For $d_{y}^{K}$
the value of $D$ estimated by Kervella et al. (1992 pc) was combined
with their $N_{i}^{K}$ values.\\
7. The value of $N_{i}$ adopted in the present paper\\
8. The distance, $d_{y}$, behind or in front of the plane
of the sky through the star assumed to be at its PL distance (1728pc)
and adopting the revised values of $N_{i}$. The units are also in
$10^{-4}$pc.\\
9. The perpendicular distance $d_{x}$ of the point from the intersection of the
disc with the plane of the sky and projected onto the plane of the sky.
In the same units as $d_{y}$.
This is given by:
\begin{equation}
d_{x} = 83.77\theta_{i} \sin(\beta_{i} -\gamma)
\end{equation}
where $\gamma$ is the angle (azimuth) at which the plane of the disc
cuts the plane of the sky.
In this test of the model this
is take as close to the line from the star to point 9
(i.e. $\gamma =  80^{\circ}$).\\
\begin{table*}
\centering
\caption{The phase lag observations of Kervella et al. with derived linear
positions}
\begin{tabular}{rrrrrrrrr}
\hline
$i$ & $\theta_{i}$ & $\beta_{i}$ & $\Delta \phi_{i} \pm s.e.\;\;\;\;$ &
$N_{i}^{K}$ & $d_{y}^{K}$ & $N_{i}$ & $d_{y}$ & $d_{x}$\\
\hline
 1 & 21.10 & 139 & $0.983\pm 0.020$ & 5 &   43 & 5 & 290 & 1515\\
 2 & 21.16 & 170 & $0.809\pm 0.012$ & 5 & --24 & 5 & 233 & 1773\\
 3 & 21.65 & 196 & $0.989\pm 0.013$ & 5 & --7  & 5 & 252 & 1630\\
 4 & 16.58 & 186 & $0.576\pm 0.009$ & 4 & --10 & 4 & 190 & 1335\\
 5 & 16.03 & 167 & $0.504\pm 0.083$ & 4 &   19 & 4 & 210 & 1341\\
 6 & 12.89 & 213 & $0.102\pm 0.023$ & 3 & --179& 3 &--1  & 790\\
 7 & 10.96 & 249 & $0.098\pm 0.023$ & 3 &   19 & 3 & 148 & 175\\
 8 & 29.28 & 312 & $0.207\pm 0.036$ & 8 &   25 & 6 &--313 & --1933\\
 9 & 19.42 &  80 & $0.697\pm 0.016$ & 5 &  103 & 4 & 7   & 0 \\
10 & 17.28 & 149 & $0.602\pm 0.014$ & 4 & --70 & 4 & 146 & 1351\\
11 & 11.03 & 252 & $0.108\pm 0.004$ & 3 &   16 & 3 & 146 &  129\\
12 & 11.84 & 201 & $0.692\pm 0.010$ & 3 &  133 & 3 & 259 & 850\\
13 & 16.26 & 166 & $0.462\pm 0.001$ & 4 & --17 & 4 & 179 & 1359\\
14 & 16.45 &  67 & $0.526\pm 0.001$ & 4 & --14 & 3 & --161 & --310\\
15 & 16.98 & 148 & $0.572\pm 0.033$ & 4 & --49 & 4 & 159 & 1319\\
16 & 17.04 & 187 & $0.589\pm 0.006$ & 4 & --49 & 4 & 160 & 1365\\
17 & 17.58 &  86 & $0.394\pm 0.009$ & 4 & --178& 4 &  55 & 154\\
18 & 18.84 & 247 & $0.543\pm 0.005$ & 5 &  108 & 4 &   2 & 355\\
19 & 20.59 & 170 & $0.770\pm 0.015$ & 5 &   20 & 5 &  263& 1725\\
20 & 20.99 & 140 & $0.965\pm 0.021$ & 5 &   48 & 5 & 293 & 1523\\
21 & 24.43 & 255 & $0.925\pm 0.002$ & 6 &   50 & 5 &  15 &  178\\
22 & 24.75 &  84 & $0.182\pm 0.025$ & 6 & --252& 6 &  76 &  145\\
23 & 26.76 & 200 & $0.460\pm 0.003$ & 7 &    12& 7 & 329 & 1941\\
24 & 28.35 & 273 & $0.081\pm 0.004$ & 7 & --289& 7 &  87 & --534\\
25 & 28.74 & 156 & $0.717\pm 0.010$ & 8 &   247& 7 & 263 & 2336\\
26 & 29.61 & 312 & $0.138\pm 0.001$ & 8 & --27 & 6 & -373 & --1955\\
27 & 30.06 &  70 & $0.158\pm 0.015$ & 8 &  --65& 7 & --28 & --437\\
28 & 33.25 & 214 & $0.570\pm 0.055$ & 9 &  117 & 8 & 190 & 2004\\
29 & 34.81 &  83 & $0.454\pm 0.006$ & 9 & --72 & 8 &  25 &  153\\
30 & 37.76 & 275 & $0.956\pm 0.004$ &10 &  163 & 8 & --48 & -819\\
31 & 39.14 & 283 & $0.938\pm 0.008$ &10 &   26 & 8 & --174 & --1281\\

\hline
\end{tabular}
\end{table*}
Fig. 1 shows a plot of $d_{x}$ against $d_{y}$. This indicates
a clear, apparently linear relation, between the two quantities.
That is, the points lie
in a tilted plane, presumably an equatorial disc.
The line shown is a least squares
fit through the origin and is given by:
\begin{equation}
 d_{y} = 0.143(\pm 0.010)d_{x}
\end{equation}
 The tilt of the disc to the plane of the sky is $\tan^{-1} 0.143 =
8^{\circ}.1\pm0^{\circ}.6$. The rms scatter about the line in $d_{y}$ is only
$73.10^{-4}$pc,
much smaller than the diameter
of the disc, which is
$\sim 6800.10^{-4}$pc out to the limits of the phase-lag survey.
This rms scatter may be compared to the rms
scatter of $d_{y}^{K}$ which is $111.10^{-4}$pc.
No attempt has been made to optimize the disc model
by, for instance, varying $\gamma$ to find a better fit. This might reduce
the rms scatter slightly.
However this is already small compared
to the diameter of the part of the disc surveyed
indicating a relatively thin disc.
A realistic disc will have some significant depth perpendicular to its axis
and, indeed,  Kervella et al. note that their observations at some positions
suggest smoothing attributed to
a non-zero
depth in the line of sight.
An inclined disc model broadly similar the one just discussed could
probably be derived for other distances, if there were
good evidence for these. However it should be noted that to take
the distance as a free parameter in an attempt to reduce the rms scatter
in the model is to assume that the disc must conform as closely as possible to
an idealized model which is perfectly flat and of negligible thickness.
There is no a priori justification for such an assumption.
\begin{figure}
\includegraphics[width=8.5cm]{RSnewfig4.ps}
\caption{A plot of the distances, $d_{y}$,from the plane of the sky through
the star against $d_{x}$ the perpendicular distance, in the plane of the sky,
to a line of azimuth $\gamma =80^{\circ}$. The units are $10^{-4}$pc. Note the
expanded $d_{y}$ scale.}

\end{figure}

An equatorial disc model for RS Pup is particularly interesting from the
astrophysical point of view. Whilst it has seemed possible, for instance,
that Cepheids might have ejected shells in a previous evolutionary phase,
it has been puzzling that only for RS Pup is such a prominent structure
found. The interpretation of the nebulosity as a disc at a small angle
to the plane of the sky opens up the possibility of a deeper understanding
of this phenomenon and its rarity. Obvious possibilities are loss
of mass in the equatorial plane by unusually rapid rotation and/or
binary interaction at an earlier evolutionary stage.
\section{Conclusion}
The structure seen in the RS Pup nebulosity makes questionable the assumption
that phase-lag observations all refer to points close to the plane of the sky
and this makes distance estimates based on this assumption questionable.
An inclined disc model at a distance
predicted by a period-luminosity relation gives a good fit to the data
and opens new possibilities for understanding the system, including a
possible interacting binary model.
\section*{Acknowledgments}
 I am grateful to Dr Pierre Kervella for a very helpful exchange of
correspondence and to him and his colleagues for sending me detail of their
beautiful
work not given in their paper. I am also grateful to Dr Kurt van der Heyden
for his help and to the referee for suggestions.


\section{Introduction}

 Hamilton's principle is one of the basic principles of Physics.
 Anthony \cite{Anthony} states: "In theoretical physics, a theory
 is often considered to be complete if its variational principle
 in the sense of Hamilton is known".
 When a Hamilton's principle is known, the whole information
 concerning the processes of a particular system is included into
 its Lagrangian.
 When known, the Hamilton's principe could be used in many
 different ways. For example, it could serve as the basis for
 obtaining first integrals via N\"{o}ther's theory, or to generate
 approximate solutions to the relevant system of equations by the
 use of Ritz procedure (cf.\ \cite{VujanovicAtanackovic}).

 In this paper we investigate necessary conditions
 for solutions of fractional variational problems (Euler-Lagrange
 equations). Such investigations has been initiated in
 \cite{Riewe96, Riewe97},
 and continued in \cite{Agrawal02} - \cite{AKP-fracEL} and
 \cite{BalTru08}; see also \cite{Rekh} for
 the importance of introducing fractional derivatives into
 the Lagrangian density of Hamilton's principle.
 In general, we refer to
 \cite{Dacorogna} - \cite{Jumarie07},
 \cite{MillRoss, OldSpa, Podlubny},
 \cite{SamkoKM} - \cite{WestBolognaGrigolini},
 for different aspects of the calculus of variations and fractional
 calculus, motivations and applications.

 When a fractional variational problem is studied,
 a natural question arises how one can choose $\alpha$, the order
 of fractional derivative, in order to
 achieve the minimal value of a functional under consideration.
 More generally, one can address the same question for any problem
 involving fractional operators.
 Usually, in application, a good choice of $\alpha$
 is obtained by experiments, numerical methods or
 computational simulations. However, experimental results give
 different values for $\alpha$ within certain interval.
 In this paper we
 propose a method which a priori gives values for $\alpha$
 which optimize the considered variational problem following the
 fundamental minimization principle of Hamilton's action.
 In fact, we address the question of finding stationary
 points for Hamilton's action integral with fractional Lagrangian
 in a more general setting. Namely, we allow the stationarity
 of the action integral with respect to a set of admissible
 functions \textit{and} with respect to the order of fractional
 derivatives, appearing in the Lagrangian.
 Up to our knowledge the problem when both $y$
 and $\alpha$ are varied has
 not been analyzed so far. It leads to stationarity conditions
 as a basis for generalized Hamilton's principle
 for the action integral
 \begin{equation} \label{eq:fvp-alpha}
 I[y,\alpha]: = \int_{0}^{b}L( t,y( t),{}_{0}D_{t}^{\alpha }y,\alpha)\,
 dt,\quad y\in \cU,\alpha\in A:=[0,\alpha_0],\alpha_0\leq 1,
 \end{equation}
 where $\cU$ is a set of admissible functions: Find
 \begin{equation}\label{eq:min}
 \min_{(y,\alpha)\in \cU\times A} I[y, \alpha]
 \qquad \mbox{or}
 \end{equation}
 \begin{equation}\label{eq:minmin}
 \min_{\alpha\in A}(\min_{y\in\cU} I[y, \alpha])
 \qquad \mbox{or}
 \end{equation}
 \begin{equation}\label{eq:minimin}
 \min_{y\in\cU}( \min_{\alpha\in A} I[y, \alpha]).
 \end{equation}
 In this paper we analyze stationarity conditions for
 (\ref{eq:min}) and (\ref{eq:minmin}).
 Stationarity conditions with respect to $\alpha$
 in (\ref{eq:minimin}) are more difficult,
 and, contrary to (\ref{eq:minmin}), this case is less natural in
 applications. Note that in (\ref{eq:min}), (\ref{eq:minmin}) and
 (\ref{eq:minimin}) one can look for maximums instead of minimums.
 So, our general problem is determination of stationary points.

 So far, parameter $\alpha$, the order of fractional derivative,
 has been determined experimentally (cf.\ e.g. \cite{Rogers83}).
 This approach offers a rational way for choosing the precise
 $\alpha$.

 The paper is organized as follows. To the end of Introduction, we
 recall definitions and properties of fractional derivatives.
 In Section \ref{sec:Hamilton} we present a framework in which we
 shall study variational problems (\ref{eq:min}) and
 (\ref{eq:minmin}). Then in Section \ref{SecOptCond}
 we derive stationarity conditions for (\ref{eq:min}).
 Section \ref{SecEqProbl} is devoted to additional
 assumptions which provide equivalence of problems
 (\ref{eq:min}) and (\ref{eq:minmin}).
 Results which are obtained in previous sections are illustrated
 by several examples in Section \ref{SecEx}.
 Moreover, examples of this section give further motivation for
 our investigation.
 In the last remark of
 Section \ref{SecEx} we propose a new formulation of a
 fractional variational problem.

 Throughout this paper we shall use the following notation. The
 mapping $(t,\alpha)\mapsto {}_0D_t^\alpha(y)$, which defines
 the left Riemann-Liouville fractional derivative of
 order $\alpha$, will be denoted by ${}_0D_t^\alpha y$, or
 by ${}_0D_t^\alpha y(t)$. Recall,
 $$
 {}_{0}D_{t}^{\alpha}y:=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}
 \int_{0}^{t}\frac{y(\tau)}{(t-\tau)^{\alpha}} \, d\tau,
 \quad t\in[0,b],\, 0\leq\alpha <1,
 $$
 where $\Gamma$ is the Euler gamma function,
 and its existence is provided whenever
 \begin{equation} \label{eq:acfd}
 [0,b]\ni t\mapsto
 \int_0^t \frac{y(\tau)}{(t-\tau)^{\alpha}}\, d\tau
 \end{equation}
 is an absolutely continuous function. Recall, the space of
 absolutely continuous functions is denoted by
 $AC([0,b])$ and it is supplied with the norm
 $||f||=\sup_{x\in[0,b]} |f(x)|$
 (clearly, it is not a Banach space).
 For example, (\ref{eq:acfd}) is absolutely continuous if
 $y\in AC([0,b])$.
 However, there are some cases when with less regularity in $y$ we
 still have a well-defined operator of fractional differentiation (cf.\
 \cite{SamkoKM}). For instance, ${}_{0}D_{t}^{\alpha}y$ exists for
 functions with integrable singularities
 (a continuous and locally integrable function $f$ in $(0,b]$ has
 an integrable singularity
 at $\tau=0$ of order $r<1$ if
 $\lim_{\tau\to 0}\tau^r f(\tau)\not= 0$).
 In particular we can take $y(t)=t^{-\mu}$,
 $t\in(0,b]$ (for any $b>0$), $0<\mu <1$.
 Then we obtain the so-called Euler formula
 (cf.\ \cite[(2.26)]{SamkoKM})
 $$
 {}_0D_t^\alpha t^{-\mu}
 =\frac{\Gamma(1-\mu)}{\Gamma(1-\mu-\alpha)}
 \frac{1}{t^{\mu+\alpha}}, \quad t\in(0,b].
 $$

 The right Riemann-Liouville fractional derivative of order
 $\alpha$ is defined as
 $$
 {}_{t}D_{b}^{\alpha}y:=\frac{1}{\Gamma(1-\alpha)}
 \left(-\frac{d}{dt}\right)
 \int_{t}^{b}\frac{y(\tau)}{(\tau-t)^{\alpha}} \, d\tau,
 \quad t\in[0,b],\, 0\leq\alpha <1.
 $$
 The conditions for its existence are similar as in the case of
 the left fractional derivative.

 In the sequel we shall consider cases involving both fractional
 derivatives and work with integrable functions for which these
 derivatives (or one of them) exist. In such cases notation
 ${}_{0}D_{t}^{\alpha}y$, resp.\ ${}_{t}D_{b}^{\alpha}y$, $t\in[0,b]$,
 means that $y$ and ${}_{0}D_{t}^{\alpha}y$, resp.\
 ${}_{t}D_{b}^{\alpha}y$, are considered as integrable functions
 which can take values $+\infty$ or $-\infty$ at some points.

 Recall (\cite{Nakhushev}), that
 ${}_0D_t^\alpha y\to y'$ and ${}_tD_b^\alpha y\to -y'$ in $\cC([0,b])$,
 as $\alpha\to 1^{-}$, whenever $y\in \cC^1([0,b])$.

 Also, we shall make use of Caputo fractional derivatives.
 The left, resp.\ right, Caputo fractional derivative
 of order $\alpha\in[0,1)$ is defined as
 $$
 {}_{0}^{C}D_{t}^{\alpha}y: = \frac{1}{\Gamma(1-\alpha)}
 \int_{0}^{t} \frac{y'(\tau)}{(t-\tau)^{\alpha}}\, d\tau
 \quad \mbox{resp.} \quad
 {}_{t}^{C}D_{b}^{\alpha}y: = \frac{1}{\Gamma(1-\alpha)}
 \int_{t}^{b} \frac{-y'(\tau)}{(\tau-t)^{\alpha}}\, d\tau.
 $$
 One can show that for $y\in AC([0,b])$ and $t\in [0,b]$,
 $$
 {}_0D_t^\alpha y= {}_{0}^{C}D_{t}^{\alpha}y
 + \frac{1}{\Gamma (1-\alpha)}
 \frac{y(0)}{(t-a)^\alpha},
 \quad
 {}_tD_b^\alpha y= {}_{t}^{C}D_{b}^{\alpha}y
 + \frac{1}{\Gamma (1-\alpha)}
 \frac{y(b)}{(b-t)^\alpha},
 $$
 (cf.\ e.g. \cite{KilSriTru}).
 Therefore, ${}_{0}D_{t}^{\alpha}y={}_{0}^{C}D_{t}^{\alpha}y$,
 resp.\ ${}_{t}D_{b}^{\alpha}y={}_{t}^{C}D_{b}^{\alpha}y$,
 whenever $y(0)=0$, resp.\ $y(b)=0$.

 \section{Formulation of the problem}
 \label{sec:Hamilton}

 We investigate stationary points of (\ref{eq:fvp-alpha})
 for $\alpha\in[0,\alpha_0]$ and
 all admissible functions $y$, whose properties will be specified
 in the sequel.
 We shall distinguish two cases: $\alpha_0$ strictly less than $1$
 and $\alpha_0=1$.
 In the case $\alpha_0<1$, set
 $$
 \mathcal{U}_l:=\{y\in L^1([0,b])\,|\,
 {}_0D_t^\alpha y\in L^1([0,b])\}.
 $$
 Obviously, $AC([0,b])$ is a subset of $\cU_l$.
 In the case $\alpha_0=1$ we assume that
 $y\in\mathcal{U}_l$ and that, in addition,
 ${}_0D_t^1 y$ exists, and ${}_0D_t^1 y=y'$
 is an integrable function.
 Let us note that one can consider $\mathcal{U}_l$ defined with
 $L^p([0,b])$ (or their subspaces) instead of $L^1([0,b])$
 (see Remark \ref{rem:lin ops}).

 In general, we shall use notation
 \beq \label{admisset}
 \mathcal{U}:=\{y\in \cU_l\,|\,
 y \mbox{ satisfies specified boundary conditions}\}.
 \eeq
 We shall sometimes write $\mathcal{U}$ also for $\mathcal{U}_l$
 (then the set of specified boundary conditions is empty).

 In the sequel, Lagrangian $L(t,y(t),{}_{0}D_{t}^{\alpha}y,\alpha)$
 (Lagrangian density, in Physics) satisfies:
 \beq \label{eq:conditions on L}
 \left.\ba{c}
 L\in\cC^1([0,b]\times \R\times \R\times [0,1])\\
 \mbox{and}\\
 t\mapsto\pd_3 L(t,y(t),{}_0D_t^\alpha y,\alpha)
 \in \cU_r, \mbox{ for every } y\in \cU_l
 \ea\right\}
 \eeq
 where $\mathcal{U}_r:=\{y\in L^1([0,b])\,|\,
 {}_tD_b^\alpha y\in L^1([0,b])\}$.

 Recall, our generalization of Hamilton's principle is realized
 through the determination of
 $(y^{\ast},\alpha^{\ast})\in\mathcal{U}\times A$ such that
 \begin{equation} \label{eq:fvp-alpha-min}
 \underset{(y,\alpha) \in \mathcal{U}\times A}{\min}
 \int_{0}^{b} L(t,y(t),{}_{0}D_{t}^{\alpha}y,\alpha)\, dt
 =
 \int_{0}^{b} L( t,y^{\ast}(t),{}_{0}D_{t}^{\alpha^{\ast}}y^{\ast},
 \alpha^{\ast})\,dt.
 \end{equation}

 There are two special cases of (\ref{eq:fvp-alpha-min}).
 The first one is obtained when $A=\{1\}$.
 Then, since $\left.{}_0D_t^\alpha y\right\vert_{\alpha=1}
 =y'(t)$ for $y\in\cC^1([0,b])$,
 the solution $y^{\ast}$ of (\ref{eq:fvp-alpha-min})
 satisfies the classical Euler-Lagrange equation
 $$
 \frac{d}{dt}\frac{\partial L}{\partial y'}-
 \frac{\partial L}{\partial y} =0.
 $$

 If $A$ has a single element $A=\{\alpha\}$, $0< \alpha < 1$, then
 $\underset{(y,\alpha)\in\mathcal{U}\times \{\alpha\}}{\min}
 I[y,\alpha]$ leads to the fractional Euler-Lagrange equation
 (cf.\ (\cite{Agrawal02, AKP-fracEL}))
 $$
 {}_{t}D_{b}^{\alpha}\frac{\partial L}{\partial {}_{0}D_{t}^{\alpha}y}
 +\frac{\partial L}{\partial y}=0.
 $$

 We proceed with finding stationary points related to
 (\ref{eq:fvp-alpha}).

 \section{Optimality conditions}
 \label{SecOptCond}

 A necessary condition for the existence of solutions to variational
 problem (\ref{eq:fvp-alpha-min}) is given in the following
 proposition.

 \bprop \label{prop:ELeqs}
 Let $L$ satisfy (\ref{eq:conditions on L}).
 Then a necessary condition that functional
 (\ref{eq:fvp-alpha}) has an extremal point at
 $(y^{\ast},\alpha^{\ast})\in\cU\times A$
 is that $(y^{\ast},\alpha^{\ast})$ is
 a solution of the system of equations
 \begin{eqnarray}
 \frac{\partial L}{\partial y}+{}_{t}D_{b}^{\alpha}
 \frac{\partial L}{\partial{}_{0}D_{t}^{\alpha}y}
 &=& 0,  \label{eq:ELeqs-y} \\
 \int_{0}^{b} \left(\frac{\partial L}{\partial{}_{0}D_{t}^{\alpha }y}G( y,\alpha)
 +\frac{\partial L}{\partial \alpha }\right)\, dt
 &=& 0,  \label{eq:ELeqs-alpha}
 \end{eqnarray}
 where
 $$
 G(y,\alpha)=\frac{\partial{}_{0}D_{t}^{\alpha}y}{\partial\alpha}
 =\frac{d}{dt}(f_{1}\ast_t y)(t,\alpha), \,
 f_{1}(t,\alpha)=\frac{1}{t^{\alpha}\Gamma(1-\alpha)}
 [\psi(1-\alpha)-\ln t], \, t>0,
 $$
 with the Euler function $\psi(z)=\frac{d}{dz}\ln\Gamma(z)$,
 and
 $(f_{1}\ast_t y)(t,\alpha)
 =\int_{0}^{t}f_{1}(\tau,\alpha)y(t-\tau)\, d\tau$.
 \eprop

 \pr
 Let $(y^{\ast},\alpha^{\ast})$ be an element of
 $\mathcal{U}\times A$
 for which $I[y,\alpha]$ has an extremal value. Let
 $y(t) =y^{\ast}(t) +\varepsilon_{1}f(t)$,
 $\alpha =\alpha^{\ast}+\varepsilon_{2}$,
 $\varepsilon_1,\varepsilon_2\in\R$,
 with $f\in\cU_l$, and the boundary conditions
 on $f$ are specified so that the varied path
 $y^{\ast}+\varepsilon_{1}f$ is an element of $\mathcal{U}$.
 Then $I[y,\alpha]=I[y^{\ast}+\varepsilon_{1}f,
 \alpha^{\ast}+\varepsilon_{2}]=:I(\varepsilon_{1},
 \varepsilon_{2})$.
 A necessary condition for an extremal value of $I[y,\alpha]$ is
 $$
 \left.\frac{\partial I(\varepsilon_{1},\varepsilon_{2})}{\partial\varepsilon_{1}}
 \right\vert_{\varepsilon_{1}=0,\varepsilon_{2}=0}=0,
 \qquad
 \left.\frac{\partial I(\varepsilon_{1},\varepsilon_{2})}{\partial\varepsilon_{2}}
 \right\vert_{\varepsilon_{1}=0,\varepsilon_{2}=0}=0.
 $$
 Therefore we obtain
 \begin{eqnarray}
 \int_{a}^{b}\left(\frac{\partial L}{\partial y}f(t)
 +\frac{\partial L}{\partial{}_{0}D_{t}^{\alpha}y}\,
 {}_{0}D_{t}^{\alpha }f\right)\, dt &=&0, \label{eq:11-1} \\
 \int_{0}^{b}\left(\frac{\partial L}{\partial {}_{0}D_{t}^{\alpha}y}\,
 \frac{\partial {}_{0}D_{t}^{\alpha}y}{\partial\alpha}
 +\frac{\partial L}{\partial \alpha }\right)\, dt &=&0.  \label{eq:11-2}
 \end{eqnarray}
 Applying the fractional integration by parts formula
 (cf.\ \cite{KilSriTru}):
 \begin{equation} \label{eq:frac int by parts}
 \int_{0}^{b} g(t)\cdot {}_{0}D_{t}^{\alpha }f(t)\, dt
 = \int_{0}^{b}\, f(t)\cdot {}_{t}D_{b}^{\alpha }g(t)\, dt,
 \end{equation}
 to (\ref{eq:11-1}), (\ref{eq:11-1}) is transformed
 to
 $$
 \int_{0}^{b} \left(\frac{\partial L}{\partial y}
 +{}_{t}D_{b}^{\alpha }\frac{\partial L}{\partial{}_{0}D_{t}^{\alpha}y}
 \right)f(t)\, dt=0.
 $$
 From this equation, using the fundamental lemma of
 the calculus of variations
 (see \cite[p.\ 115]{Dacorogna}), we conclude that
 condition (\ref{eq:ELeqs-y}) holds for the
 optimal values $y^{\ast}$ and $\alpha^{\ast}$.
 In (\ref{eq:ELeqs-alpha}) the term
 $\frac{\partial {}_{0}D_{t}^{\alpha}y}{\partial\alpha}$
 is transformed by the use of expression
 \begin{eqnarray}
 \frac{\partial {}_{0}D_{t}^{\alpha}y}{\partial\alpha}
 &=&\psi(1-\alpha)\, {}_{0}D_{t}^{\alpha}y -
 \frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}
 \int_{0}^{t}
 \frac{\ln(t-\tau) y(\tau)}{(t-\tau)^{\alpha}}\, d\tau \notag \\
 &=&\frac{d}{dt}(f_{1}\ast_t y) (t,\alpha) \notag \\
 &=& G(y,\alpha), \quad (y,\alpha)\in\cU\times A \label{eq:14}
 \end{eqnarray}
 (cf.\ \cite[p.\ 592]{AOP07}).
 We obtain (\ref{eq:ELeqs-alpha}) by substituting (\ref{eq:14})
 into (\ref{eq:11-2}).
 \ep

 \brem
 In general, in  solving  equations
 (\ref{eq:ELeqs-y}) and (\ref{eq:ELeqs-alpha}), the most delicate
 task is the calculation of expression
 $\frac{\partial{}_{0}D_{t}^{\alpha}y}{\partial \alpha}$.
 Although its general form (\ref{eq:14}) has been derived
 in \cite[p.\ 592]{AOP07}, various difficulties can still appear.
 We illustrate this by examples in Section \ref{SecEx}.
 However, the simplified form of
 $\frac{\partial{}_{0}D_{t}^{\alpha}y}{\partial \alpha}$,
 in some special cases, is important.

 Already in
 \cite{AOP07} it has been shown that for $y\in AC([0,b])$
 \begin{eqnarray}
 \left.\frac{\partial {}_{0}D_{t}^{\alpha}y}{\partial\alpha}
 \right\vert_{\alpha =0^{+}}
 &=& -(\gamma +\ln t) y(0) -\int_{0}^{t}(\gamma +\ln \tau)
 y( t-\tau)\,d\tau \notag \\
 &=& -(\gamma +\ln t) y(t)
 + \int_{0}^{t}\frac{y(t)-y(t-\tau)}{\tau}\, d\tau, \label{16bbb}
 \end{eqnarray}
 where $\gamma =0.5772156...$ is the Euler constant. (Another form
 of $\left.\frac{\partial {}_{0}D_{t}^{\alpha}y}{\partial\alpha}
 \right\vert_{\alpha =0^{+}}$ is also given in
 \cite[p.\ 111]{WestBolognaGrigolini}.)

 Let us obtain a simplified form
 of $\frac{\partial {}_{0}D_{t}^{\alpha}y}{\partial\alpha}$ at
 $\alpha=1^{-}$. In order to do that we use the method
 proposed in \cite{TarasovZaslavsky06}.
 We recall the expansion of $(t-\tau)^{\eps}/\Gamma(1+\eps)$ with
 respect to $\eps$, at $\eps=0$, with $\tau<t$ (cf.\
 \cite[p.\ 401]{TarasovZaslavsky06}), which will be used in the
 sequel:
 \begin{equation} \label{eq:expansion}
 \frac{(t-\tau)^{\eps}}{\Gamma(1+\eps)}
 = \frac{e^{\eps\ln(t-\tau)}}{\Gamma(1+\eps)}
 = 1+\eps(\gamma+\ln(t-\tau))+ o(\eps).
 \end{equation}
 Assume now that $y\in C^{2}([0,b])$.
 Then, as in \cite{Podlubny}, for $t\in[0,b]$,
 \begin{eqnarray*}
 {}_{0}D_{t}^{\alpha }y &=& \frac{1}{\Gamma(1-\alpha)}
 \frac{d}{dt} \int_{0}^{t}\frac{y(\tau)}{(t-\tau)^{\alpha }}\, d\tau \\
 &=& \frac{y(0)}{\Gamma(1-\alpha) t^{\alpha}}
 +\frac{1}{\Gamma(1-\alpha)}
 \int_{0}^{t}\frac{y^{(1)}(\tau)}{(t-\tau)^{\alpha}}\, d\tau \\
 &=& \frac{y(0)}{\Gamma(1-\alpha) t^{\alpha }}+
 \frac{y^{(1)}(0)}{\Gamma(2-\alpha)t^{\alpha -1}}
 +\frac{1}{\Gamma(2-\alpha)}\int_{0}^{t}
 (t-\tau)^{1-\alpha}y^{(2)}(\tau)\, d\tau.
 \end{eqnarray*}
 With $\alpha=1-\varepsilon$ we obtain
 \begin{equation} \label{eq:16ab}
 {}_{0}D_{t}^{1-\varepsilon }y =
 \frac{y(0)}{\Gamma(\varepsilon) t^{1-\varepsilon}}
 +\frac{y^{(1)}(0) t^{\varepsilon}}{\Gamma(1+\varepsilon)}
 +\frac{1}{\Gamma(1+\varepsilon)}
 \int_{0}^{t}(t-\tau)^{\varepsilon}y^{(2)}(\tau)\, d\tau.
 \end{equation}
 From (\ref{eq:16ab}) and (\ref{eq:expansion}) it follows that
 \begin{eqnarray}
 \left.\frac{\partial {}_{0}D_{t}^{\alpha }y}{\partial \alpha}
 \right\vert_{\alpha =1^{-}}
 &\!\!\!\!\!\!\!\!\!=\!\!\!\!& \left.
 -\frac{\partial {}_{0}D_{t}^{1-\varepsilon}y}{\partial \varepsilon}
 \right\vert_{\varepsilon =0^{+}} \notag \\
 &\!\!\!\!\!\!\!\!\!=\!\!\!\!&
 -\frac{y(0)}{t}-y^{(1)}(0) \ln t- \gamma y^{(1)}(t)
 - \int_{0}^{t}y^{(2)}(\tau) \ln(t-\tau)\, d\tau. \label{eq:16bbbb}
 \end{eqnarray}
 Assuming $y(0)=0$ in (\ref{eq:16bbbb}), we recover the results
 presented in \cite{TarasovZaslavsky06} and
 \cite{TofighiGolestani} for the Caputo fractional derivative.
 Since ${}_{0}D_{t}^{\alpha}y={}_{0}^{C}D_{t}^{\alpha}y$
 when $y(0)=0$, it follows that
 with $y(0)=0$ (\ref{eq:16bbbb}) becomes
 $$
 \left.\frac{\partial {}_{0}D_{t}^{\alpha}y}{\partial\alpha}
 \right\vert_{\alpha =1^{-}}
 =
 \left.\frac{\partial\, {}_{0}^{C}D_{t}^{\alpha}y}{\partial\alpha}
 \right\vert_{\alpha =1^{-}}
 = -y^{(1)}(0) \ln t-\gamma y^{(1)}(t)
 -\int_{0}^{t} y^{(2)}(\tau) \ln (t-\tau)\, d\tau.
 $$
 \erem

 \brem \label{rem:lin ops}
 Functional (\ref{eq:fvp-alpha})
 is a special case of the class of functionals with Lagrangians
 depending on linear operators,
 see \cite[p.\ 51]{Filippov}. Indeed, suppose that Lagrangian $L$
 depends on $t$,
 $y$ and $\cL y$, where $\cL:M\to L^p([0,b])$, $p\in[1,+\infty)$,
 is a linear operator acting on a set of admissible
 functions $M$, which is linear, open and dense in $L^p([0,b])$
 (i.e. $\cL$ belongs to $Lin(M,L^p([0,b]))$, the space of continuous,
 linear functions with the uniform norm).
 Suppose that $L$ is continuously differentiable
 with respect to $t$ and $y$ and twice continuously
 differentiable with respect to $\cL y$.
 Moreover, assume that
 function $t\mapsto L(t,y(t), \cL y(t))$, $t\in [0,b]$,
 is continuous, for all $y\in M$.
 Then the Euler-Lagrange equation reads
 $$
 \frac{\pd L}{\pd y} + \cL^\ast \frac{\pd L}{\pd (\cL y)}=0,
 $$
 where $\cL^\ast$ denotes the adjoint operator of $\cL$.
 In case when $\cL$ is the left Riemann-Liouville operator
 ${}_0D_t^\alpha$, with the adjoint ${}_tD_{b}^\alpha$, the
 latter equation coincides with (\ref{eq:ELeqs-y}).

 If instead of $\cL$ one considers a family $\{\cL_\alpha, \alpha\in
 A\}$, where $A=[0,1]$ (or some other interval), and the mapping
 $A\to Lin(M,L^p([0,b]))$, $\alpha\mapsto \cL_\alpha$, is differentiable,
 then a more general problem of finding stationary points with respect
 to $y$ and $\alpha$ can be formulated. In that case,
 one can derive the second stationarity condition similar to
 (\ref{eq:ELeqs-alpha}):
 $$
 \int_0^b \left( \frac{\d L}{\d\cL}\frac{\d \cL}{\d\al} + \frac{\d
 L}{\d\al}\right)dt=0.
 $$
 \erem

 \section{Equivalent problems}
 \label{SecEqProbl}

 In this section we shall give conditions which provide that problems
 (\ref{eq:min}) and (\ref{eq:minmin}) coincide.

 \bprop \label{EqvivPobl}
 Let the Lagrangian $L$ satisfy (\ref{eq:conditions on L}).
 Assume that for every $\alpha\in [0,1]$ there is a unique
 $y^\ast(t,\alpha)\in\cU$, a solution to (\ref{eq:ELeqs-y}), and
 that the mapping $\alpha\mapsto y^\ast(t,\alpha)$ is
 differentiable as a mapping from $[0,1]$ to $\cU$.
 Then
 the problem $\min_{(y,\alpha)\in\cU\times A}I[y,\alpha]$
 is equivalent to
 the problem $\min_{\alpha\in A}(\min_{y\in\cU}I[y,\alpha])$.
 \eprop

 \pr
 As we have shown in Proposition \ref{prop:ELeqs}, any solution to the
 problem $\min_{(y,\alpha)\in\cU\times A}I[y,\alpha]$ satisfies
 system
 (\ref{eq:ELeqs-y})-(\ref{eq:ELeqs-alpha}). It can be
 solved as follows.
 We first solve (\ref{eq:ELeqs-y}) and the corresponding
 boundary conditions to obtain $y^{\ast}=y^{\ast}(t,\alpha)$.
 According to the assumption, the solution $y^\ast$ is unique.
 Then we insert $y^\ast$ in (\ref{eq:ELeqs-alpha}) to obtain
 $\alpha^{\ast}$.
 In this case, functional $I[y,\alpha]$
 becomes a functional depending only on $\alpha$,
 $\alpha\mapsto I[y^\ast(t,\alpha),\alpha]=I[\alpha]$, and
 therefore (\ref{eq:ELeqs-alpha}) transforms to the total
 derivative of $I[\alpha]$ since
 \begin{eqnarray}
 \frac{dI[\alpha]}{d\alpha} &=&
 \left.\frac{dI[\alpha+\varepsilon]}{d\varepsilon}\right\vert_{\varepsilon=0}
 \nonumber\\
 &=& \int_0^1\left[
 \frac{\pd L}{\pd y}\frac{\pd y}{\pd \alpha} +
 \frac{\pd L}{\pd {}_{0}D_{t}^{\alpha}y} \left(
 {}_{0}D_{t}^{\alpha}\left(\frac{\pd y}{\pd\alpha}\right)
 + \left(\frac{\pd}{\pd\alpha}{}_{0}D_{t}^\alpha\right)y
 \right)
 +\frac{\pd L}{\pd\alpha}
 \right]\, dt \nonumber\\
 &=& \int_0^1\left[
 \frac{\pd y}{\pd \alpha} \left(
 \frac{\pd L}{\pd y} + {}_{t}D_{b}^{\alpha}
 \frac{\partial L}{\partial{}_{0}D_{t}^{\alpha}y}\right)
 +
 \frac{\pd L}{\pd {}_{0}D_{t}^{\alpha}y}
 \left(\frac{\pd}{\pd\alpha}{}_{0}D_{t}^\alpha\right)y
 +\frac{\pd L}{\pd\alpha}
 \right]\, dt \nonumber\\
 &=& \int_0^1 \left(
 \frac{\pd L}{\pd {}_{0}D_{t}^{\alpha}y}
 \left(\frac{\pd}{\pd\alpha}{}_{0}D_{t}^\alpha\right)y
 +\frac{\pd L}{\pd\alpha}
 \right)\, dt, \label{eq:oio}
 \end{eqnarray}
 where we used fractional integration by parts formula
 (\ref{eq:frac int by parts}) in the third,
 and equation (\ref{eq:ELeqs-y}) in the last equality.
 This proves the claim.
 \ep

 The following simple assertion is of particular interest:

 \bprop
 Let $L$ satisfy (\ref{eq:conditions on L}).
 Assume that for every $\alpha\in[0,1]$ there exists a unique
 $y_{\alpha}\in\cU$, a solution to the
 fractional variational problem (\ref{eq:fvp-alpha-min}), and that
 $I[y_{\alpha},\alpha]$ is the corresponding minimal value of
 the functional $I$. Assume additionally that
 $$
 \frac{dI}{d\alpha}(y,\alpha)|_{y=y_\alpha}>0,\quad
 \forall y_\alpha\in\cU.
 $$
 Then the minimal, resp.\ maximal
 value of the functional $I[y,\alpha]$ is attained at $\alpha=0$,
 resp.\ at $\alpha=1$.
 \eprop

 \pr
 Under the above assumptions we have that
 $$
 I[y_{0},0]\leq I[y_{\alpha},0]\leq
 I[y_{\alpha},\alpha]\leq I[y_1,\alpha]
 \leq I[y_1,1],
 \quad \forall\alpha\in[0,1],
 $$
 which proves the claim.
 \ep

 \brem
 The same argument can be applied to
 the case when $dI/d\alpha<0$, for any
 fixed $y_\alpha\in\cU$, i.e. when $I$ is an decreasing function
 of $\alpha$, for any fixed $y_\alpha\in\cU$. In that case the
 minimal, resp.\ maximal value of $I$ is at $\alpha=1$, resp.\
 $\alpha=0$.
 \erem

 \section{Examples}\label{SecEx}

 \subsection{Examples with Lagrangians linear in $y$}

 \bex
 Consider the action integral for the
 inertial motion (no force acting) of a material point of the
 form
 \begin{equation} \label{eq:16aa}
 I[y,\alpha]: = \int_{0}^{1} ({}_{0}D_{t}^{\alpha}y)^{2}\,dt,
 \quad (y,\alpha)\in\cU\times A,
 \end{equation}
 where $\cU:=\{y\in \cU_l\,|\, y(0)=0,\, y(1) =1\}$ and $A=[0,1]$.

 Obviously, the minimal value of $I[y,\alpha]$ is zero, and it is
 attained whenever ${}_0D_t^\alpha y=0$. Solutions to
 equation ${}_0D_t^\alpha y=0$ are of the form $y(t)=C\cdot
 t^{1-\alpha}$, $t\in[0,1]$, $C\in\R$ (cf.\ \cite{SamkoKM}).
 All solutions satisfy the Euler-Lagrange equation
 $$
 {}_{t}D_{1}^{\alpha}({}_{0}D_{t}^{\alpha}y)=0.
 $$
 Stationarity condition (\ref{eq:ELeqs-alpha}) reads:
 $$
 \int_{0}^{1} {}_{0}D_{t}^{\alpha}y
 \left(\psi(1-\alpha) {}_{0}D_{t}^{\alpha}y
 -\frac{1}{\Gamma(1-\alpha)}
 \frac{d}{dt} \int_{0}^{t}
 \frac{\ln (t-\tau)y(\tau)}{(t-\tau)^{\alpha}}\,d\tau\right)\,
 dt=0
 $$
 and is automatically satisfied.

 Note that $C\cdot t^{1-\alpha}\in \cU_l$, for all $C\in\R$,
 but only $t^{1-\alpha}\in \cU$. Hence, we conclude that
 $(y^\ast,\alpha^\ast)=(t^{1-\alpha},\alpha)$, $\alpha\in[0,1]$,
 are solutions to the variational problem $I[y,\alpha]\to\min$,
 for $I$ defined by (\ref{eq:16aa}).
 \eex

 \brem
 If $L(t,y(t),{}_{0}D_{t}^{\alpha}y,\alpha)=
 ({}_{0}D_{t}^{\alpha}y)^{2}+(\alpha-\alpha_0)^2$, for a fixed
 $\alpha_0\in(0,1)$, then the problem
 $\int_0^1 L(t,y(t),{}_{0}D_{t}^{\alpha}y,\alpha)\, dt \to\min$
 has a unique minimizer $(y^\ast,\alpha^\ast)=
 (t^{1-\alpha_0},\alpha_0)$.
 \erem

 \bex
 Let the Lagrangian $L$ be of the form
 $$
 L(t,y(t),{}_0D_t^\alpha y,\alpha):= ({}_0D_t^\alpha y)^2- c\cdot y,
 \quad c\in\R,
 $$
 and let $\cU=\{y\in \cU_l\,|\, y(0)=0\}$, $A=[0,1]$,
 for the variational problem:
 $$
 \min_{(y,\alpha)\in\cU\times A}I[y,\alpha]=
 \min_{(y,\alpha)\in\cU\times A} \int_0^1
 (({}_0D_t^\alpha y)^2- c\cdot y(t))\, dt.
 $$
 Equations (\ref{eq:ELeqs-y}) and (\ref{eq:ELeqs-alpha}) become:
 \bea
 {}_tD_1^\alpha {}_0D_t^\al y & = & c, \label{eq:ELyEx2}\\
 \int_0^1 {}_0D_t^\alpha y\cdot \frac{\pd {}_0D_t^\alpha y}{\pd\alpha}\,dt
 & = & 0.\nonumber
 \eea
 Equation (\ref{eq:ELyEx2}) could be solved as follows.
 First, one introduces a substitution $z(t)={}_0D_t^\alpha y$,
 $t\in[0,1]$, and solve ${}_tD_1^\alpha z = c$:
 $$
 z(t)=c \cdot \frac{(1-t)^\al}{\Gamma(1-\al)},
 \quad t\in[0,1],\alpha\in A.
 $$
 Therefore,
 \beq \label{eq:fde}
 {}_0D_t^\al y (t)= c \cdot \frac{(1-t)^\alpha}{\Gamma(1-\al)},
 \quad t\in[0,1],\alpha\in A.
 \eeq
 Recall,
 $$
 {}_0I_t^\alpha y := \frac{1}{\Gamma(\alpha)} \int_0^t
 (t-\tau)^{\alpha} y(\tau)\, d\tau,
 \quad t\in[0,1],\alpha\in A
 $$
 and apply it on the both sides of (\ref{eq:fde}).
 Using \cite[Th.\ 2.4]{SamkoKM}, i.e.
 ${}_0I_t^\alpha ({}_0D_t^\alpha y)= y(t)$,
 one obtains
 \beas
 y(t,\al) &=& \frac{c}{\Gamma(\al)\Gamma(1+\al)} \int_0^t (t-\tau)^{\al-1}
 (1-\tau)^{\al}\,d\tau\\
 &=& \frac{c}{\Gamma (1+\al)}\sum_{p=0}^{\infty}
 \frac{\Gamma(p-\al)\Gamma(1+p)}{\Gamma(-\al)p\,!\Gamma
 (1+p+\al)}t^{p+\al},
 \quad t\in[0,1],\alpha\in A.
 \eeas
 This solution is unique and belongs to $\cU$.
 Since $\al\mapsto y(t,\al)$ is differentiable,
 Proposition \ref{EqvivPobl} holds.

 We substitute obtained $y(t,\al)$ into $I[y,\al]$ which yields
 \begin{eqnarray*}
 I[\al] &=&
 \int_0^1 \left(\left(c \cdot \frac{(1-t)^\al}{\Gamma(1-\al)}\right)^2-
 \frac{c^2}{\Gamma(\al)\Gamma(1+\al)}\int_0^t(t-\tau)^{\al-1}
 (1-\tau)^{\al}d\tau\right)\,dt \\
 &=&
 \int_0^1 \left(\left(c \cdot \frac{(1-t)^\al}{\Gamma(1-\al)}\right))^2 -
 \frac{c^2}{\Gamma (1+\al)}\sum_{p=0}^{\infty}
 \frac{\Gamma(p-\al)\Gamma(1+p)}{\Gamma(-\al)p\,!\Gamma
 (1+p+\al)}t^{p+\al}\right)\,dt.
 \end{eqnarray*}
 Simple numerical calculations show that $I[\alpha]$ is an increasing
 function and attains extremal values at the boundaries.

 \brem
 Equation (\ref{eq:ELyEx2}) represents a fractional generalization of
 the equation of motion for a material point (with unit mass)
 under the action of constant force equal to $c$. Our result
 shows that an optimal value of Hamilton's action is attained for
 $\al=1$, that is for integer order dynamics. We note that
 different generalizations of classical equation of motion can be
 found in \cite{Kwok}, where the problem
 ${}_0D_t^\al y = c$, $1<\alpha\leq 2$, was analyzed.
 \erem
 \eex

 \subsection{Examples with Lagrangians linear in ${}_0D_t^\alpha y$}

 \bex
 Let
 \beq \label{eq:ex3-L}
 L(t,y(t),{}_0D_t^\alpha y,\alpha):=
 \Gamma(1-\alpha){}_0D_t^\alpha y - \frac{1}{2} c y^2,
 \quad c>0, c\not=1,
 \eeq
 and consider the problem of finding stationary points for the
 functional (\ref{eq:fvp-alpha}),
 where $\cU:=\{y\in\cU_l\,|\,y(0)=\frac{1}{c}\}$
 and $\alpha_0<\frac12$.
 Note that $L$ satisfies the so called primary constraint in Dirac's
 classification of systems with constraints
 (cf.\ \cite{HenneauxTeitelboim}).
 In the setting of fractional derivatives such
 Lagrangians have been recently treated in \cite{Bal04} and
 \cite{MuslihBaleanu}.

 Equations (\ref{eq:ELeqs-y}) and (\ref{eq:ELeqs-alpha}) become
 \beq \label{eq:ex3-ELy}
 \Gamma(1-\alpha){}_tD_1^\alpha 1 - c y=0
 \eeq
 and
 \beq \label{eq:ex3-ELalph}
 \int_0^1 \left(\Gamma(1-\alpha)\frac{\pd{}_0D_t^\alpha y}{\pd\alpha}
 + \frac{\pd \Gamma(1-\alpha)}{\pd\alpha}\right)\, dt=0.
 \eeq
 Equation (\ref{eq:ex3-ELy}) has a unique solution
 $y^\ast=\frac{1}{c(1-t)^\alpha}$, $t\in[0,1]$,
 $\alpha\in[0,\alpha_0]$. This implies
 \beas
 I[y^\ast,\alpha] &=& \int_0^1 \left[\frac{d}{dt} \int_0^t
 \frac{d\tau}{c(1-\tau)^\alpha(t-\tau)^\alpha}
 -\frac{1}{2c(1-t)^{2\alpha}}\right] \, dt \\
 &=& \left. \int_0^t
 \frac{d\tau}{c(1-\tau)^\alpha(t-\tau)^\alpha}
 \right\vert_{0}^{1}
 - \int_0^1
 \frac{1}{2c(1-t)^{2\alpha}} \, dt \\
 &=& \int_0^1
 \frac{d\tau}{c(1-\tau)^{2\alpha}}
 - \int_0^1
 \frac{1}{2c(1-t)^{2\alpha}} \, dt \\
 &=&
 \frac{1}{2c} \int_0^1
 \frac{1}{(1-t)^{2\alpha}} \, dt.
 \eeas
 Since $\alpha_0<1/2$ we have that $I[y^\ast,\alpha]$ exists and
 is an increasing function with respect to $\alpha$. Hence,
 $I[y^\ast,\alpha]$ attains its
 minimal value at $\alpha=0$, and it equals $1/(2c)$.
 We also have that the maximal value of $I[y^\ast,\alpha]$ is
 attained at $\alpha_0$.
 \eex

 \bex
 Let $\cU:=\{y\in \cU_l\,|\, y(0)=0\}$,
 $c\neq 0$ and let $L$ be of the form
 \begin{equation}\label{eq:exL}
 L(t,y(t),{}_0D_t^\alpha y,\alpha): =
 c\cdot{}_0D_t^\alpha y + f(y(t)),\quad t\in[0,1],
 \end{equation}
 where the properties of $f$ are going to be specified.

 In this example we are dealing with integrable functions which
 can take values $+\infty$ or $-\infty$ at some points.
 We are going to analyze stationary points of
 $$
 I[y,\alpha]= \int_0^1
 (c\cdot{}_0D_t^\alpha y(t) + f(y(t)))\, dt.
 $$

 Equations (\ref{eq:ELeqs-y}) and (\ref{eq:ELeqs-alpha}) become
 \bea
 {}_tD_1^\alpha c + \frac{\pd f}{\pd y} & = & 0 \label{eq:ELyEx}\\
 c \cdot \int_0^1 \frac{\pd {}_0D_t^\alpha y}{\pd\alpha}\,dt
 & = & 0. \label{eq:ELalEx}
 \eea
 Since ${}_tD_1^\alpha c =
 \frac{c}{\Gamma (1-\alpha) (1-t)^{\alpha}}$, $t\in[0,1]$,
 we see that in order to solve (\ref{eq:ELyEx})-(\ref{eq:ELalEx})
 we have to assume that $f\in\cC^1(\R)$, and that $f'$ is
 invertible so that $t\mapsto (f')^{-1}
 (\frac{c}{\Gamma (1-\alpha) (1-t)^{\alpha}})\in\cU_l$.
 Then equation (\ref{eq:ELyEx}) is solvable
 with  respect to $y$:
 \begin{equation}\label{eq:Ex_y_od_c}
 y_c(t,\alpha) = \left(\frac{\d f}{\d y}\right)^{-1}
 \left( \frac{c}{\Gamma (1-\alpha) (1-t)^{\alpha}} \right), \quad t\in [0,1].
 \end{equation}
 Since $c\neq 0$, (\ref{eq:ELalEx}) implies
 $\int_0^1 \frac{\pd {}_0D_t^\alpha y}{\pd\alpha}\,dt=0$. Thus
 \bea
 0 & = &
 \int_0^1 \frac{\pd {}_0D_t^\alpha y}{\pd\alpha}\,dt
 = \int_0^1 G(y, \alpha)(t)\,dt
 = \int_0^1 \frac{d}{dt}(f_1\ast_t y)(t,\alpha)\,dt \nonumber \\
 & = &  f_1\ast_t y(t,\alpha)|_{t=1} - f_1\ast_t y(t,\alpha)|_{t=0} =
 f_1\ast_t y(t,\alpha)|_{t=1},\label{integral}
 \eea
 where we have used that  $f_1 \in L^1([0,1])$ and that $y\in \cU$.
 Substitution of (\ref{eq:Ex_y_od_c}) into (\ref{integral}) gives
 $(f_1\ast y_c(t,\alpha))(t)|_{t=1}=0$ or
 \begin{equation} \label{eq:exc}
 \int_0^1 \frac{\psi(1-\alpha)-\ln (1-\tau)}{\Gamma (1-\alpha)(1-\tau)^{\alpha}}
 \left(\frac{\d f}{\d y}\right)^{-1}\left(\frac{c}{\Gamma (1-\alpha)(\tau-1)^{\alpha}}\right)
 \,d\tau=0.
 \end{equation}
 Solving this equation is obviously difficult.
 Hence, we consider some special cases.

 a)\ $f(y(t)):= d \cdot \frac{y(t)^2}{2}$, $t\in[0,1]$, $d\in\R$.
 Then the Lagrangian is
 $$
 L(t,y(t),{}_0D_t^\alpha y,\alpha) =
 c\cdot{}_0D_t^\alpha y(t) + d \cdot \frac{y(t)^2}{2},
 $$
 and
 $$
 y_c(t,\alpha)=-\frac{1}{d}\frac{c}{\Gamma(1-\al)(1-t)^\al},
 \quad t\in[0,1].
 $$
 Also, (\ref{eq:exc}) becomes
 $$
 \int_0^1 \frac{\psi(1-\alpha)-\ln (1-\tau)}
 {\Gamma (1-\alpha)^2(1-\tau)^{2\alpha}}\,d\tau =0.
 $$
 By a simple numerical calculation one shows that this equation
 does not have any solution for $\al\in (0,1)$. Hence, in this
 case there does not exist any point $(y,\alpha)$ which is an
 extremal of functional $I[y,\alpha]$.

 b)\ $f(y(t)): = \ln y(t)$, $t\in[0,1]$. Then (\ref{eq:ELyEx})
 becomes
 $$
 \frac{c}{\Gamma(1-\al)(1-t)^\al}=\frac{1}{y},
 $$
 and therefore
 $$
 y=\frac{\Gamma(1-\al)}{c}(1-t)^\al\in AC([0,1]).
 $$
 In this particular case we take the set of admissible functions
 to be $\cU:=\{y\in\cU_l\,|\,y(1)=0\}$.
 Using (\ref{integral}), equation (\ref{eq:ELalEx}) reads
 $$
 \int_0^1 \left(\psi (1-\al)-\ln (1-\tau)\right) \,d\tau = 0,
 $$
 which, after integration, yields $\psi (\al-1)=1$.
 A unique solution of this equation in $(0,1)$ is
 $\alpha=0.604...$.

 Therefore, a unique stationary point of
 $$
 I[y,\alpha]=\int_0^1 (c\cdot{}_0D_t^\alpha y
 + \ln y(t))\, dt
 $$
 is the point $(y,\alpha)=
 (\frac{\Gamma(0,396)}{c}(1-t)^\al;0,604)$.
 \eex

 \brem
 So far, we have considered variational problems defined via functionals
 of type (\ref{eq:fvp-alpha}). In fact, we have allowed fractional
 derivatives of functions to appear in Lagrangians. The natural
 generalization of such problems consists of replacing the
 Lebesgue integral in (\ref{eq:fvp-alpha}) by the Riemann-Liouville
 fractional integral.
 More precisely, for $\beta>0$ set
 \beas
 I_\beta[y,\alpha] :&=& {}_0I_b^\beta
 L(t,y(t),{}_{0}D_{t}^{\alpha}y,\alpha) \\
 &=& \frac{1}{\Gamma (\beta)}
 \int_{0}^{b}
 (b-t)^{\beta -1}L(t,y( t),{}_{0}D_{t}^{\alpha}y,\alpha)\, dt,
 \quad t\in (0,b).
 \eeas
 Then the fractional variational problem consists of finding extremal
 values of the functional $I_\beta[y,\alpha]$.

 In the above construction we have used the left Riemann-Liouville
 fractional integral of order $\beta$ (which, in general, differs
 from the order of fractional differentiation $\alpha$), evaluated
 at $t=b$. The choice $\beta=1$ turns us back to the problem
 (\ref{eq:fvp-alpha}).

 The study of such fractional variational problems is reduced
 to the case we have already considered in the following way.
 It suffices to redefine the Lagrangian as
 $$
 L_1(t,y(t),{}_{0}D_{t}^{\alpha}y,\alpha,\beta):=
 \frac{1}{\Gamma (\beta)} (b-t)^{\beta -1}
 L(t,y( t),{}_{0}D_{t}^{\alpha}y,\alpha).
 $$
 Then we have to consider the functional
 \beq \label{eq:fracfrac-vp}
 I_\beta[y,\alpha]=\int_{0}^{b}
 L_1(t,y(t),{}_{0}D_{t}^{\alpha}y,\alpha,\beta)\, dt.
 \eeq
 In case $\beta>1$, $L_1$ is of the same regularity as $L$, so
 the straightforward application of the results derived in
 previous sections to the Lagrangian $L_1$ leads to
 the optimality conditions for the variational problem
 defined via the functional (\ref{eq:fracfrac-vp}).
 However, when $0<\beta<1$, continuity, as well as
 differentiability of $L_1$ with respect to $t$ may be violated
 (which depends of the explicit form of $L$), and hence it may be
 not possible to use the theory developed so far.
 \erem

 \section{Conclusion}

 We formulated Hamilton's principle so that the order of derivative
 in the Lagrangian is also subject to variation. The stationarity
 conditions are derived in (\ref{eq:ELeqs-y}) and
 (\ref{eq:ELeqs-alpha}). We introduced additional assumptions which
 resulted in equivalent problems, simpler for solving.
 Several examples are given in order to illustrate the theory
 presented in the paper.
 We concluded our work with a consideration
 of Hamilton's principle defined in terms of Riemann-Liouville
 fractional integrals.

 \subsection*{Acknowledgement}

 This work is supported by Projects $144016$ and $144019$ of the
 Serbian Ministry of Science and START-project Y-237 of the
 Austrian Science Fund.