I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre Transformation is degenerate. I could not find the meaning of this on googling but I guessed that there could be multiple Hamiltonians which could be generated from the Legendre transformation of the Lagrangian. I am not sure how is that possible. So in brief, what is the meaning of the 'degenerate Legendre Transformation' and it it possible for only special cases of Lagrangians describing the system (which seems easy to figure out) or in general for a particular kind of system where such a problem usually appears, such as electrical circuits (which seems difficult to figure out). Or could you provide me a reference that describes this ? Thanks. 
Also, suggestions for better tags is much appreciated. 



What is a degenerate Legendre Transformation?
I) We give here a possible definition for a $\mathbb{R}$-vector space $V$, and leave it to the reader to generalize the construction e.g. to a tangent bundle.


Let $C\subseteq V$ be a convex subset. Let there be given a function $L:C\to\mathbb{R}$. Define a new function $L\_H: V^{\ast} \times C \to \mathbb{R}$ as 


$$ L\_H(p,v)~:=~\langle p,v\rangle - L(v).$$


Define the [Legendre](http://en.wikipedia.org/wiki/Legendre_transformation) [transform](http://en.wikipedia.org/wiki/Convex_conjugate) $H: V^{\ast} \to ~]-\infty,\infty]$ as


$$ H(p)~:=~\sup\_{v\in C} L\_H(p,v).$$


Define the [domain](http://en.wikipedia.org/wiki/Legendre_transformation) $$C^{\ast}~:=~ \{p\in V^{\ast}\mid H(p) <\infty\}.$$


Define for all $p\in V^{\ast}$ the set


$$C(p)~:=~\{ v\in C \mid L\_H(p,v)=H(p) \} . $$


Call the Legendre transformation *regular* or *non-degenerate* if 


$$C^{\ast}=V^{\ast}\qquad \text{and} \qquad
\forall p\in V^{\ast}:C(p)\text{ is a singleton}.$$ 


Call it *singular* or *degenerate* otherwise.


II) In the physics literature, one often downplays the role of convexity, cf. [this](https://physics.stackexchange.com/q/103997/2451) Phys.SE post. In practice, physicists often use another definition of Legendre transform that doesn't rely on convexity, see e.g. [this](https://physics.stackexchange.com/q/105912/2451) Phys.SE post and Ref. 1. Degenerate Legendre transformations are vital to the topics of gauge theory and constraint dynamics, see e.g. Ref. 2.


References:


1. Herbert Goldstein, *Classical Mechanics.*
2. M. Henneaux and C. Teitelboim, *Quantization of Gauge Systems,* 1994.


