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in this paper we study the applicability of nekhoroshev s theorem @xcite ( see also @xcite ) to the central motion . the main point is that nekhoroshev s theorem applies to perturbations of integrable systems whose hamiltonian is a _ steep _ function of the actions . even if such a property is known to be generic , it is very difficult ( and not at all explicit ) to verify it . here we prove that , under an explicit condition on the potential ( see eq . ) , the hamiltonian of the central motion is a quasi - convex function of the actions and thus it is steep , so that nekhoroshev s theorem applies . actually , the form of nekhoroshev s theorem used here is not the original one , but that for degenerate systems proved by fass in @xcite . this is due to the fact that the hamiltonian of the central motion is a function of two actions only , namely , the modulus of the total angular momentum and the action of the effective one dimensional hamiltonian describing the motion of the radial variable .    on the one hand , as pointed out in @xcite , this feature creates some problems for the proof of nekhoroshev s theorem , but these problems were solved in @xcite . on the other hand , degeneracy reduces the difficulty for the verification of steepness or of the strongest property of being quasi - convex , since , in the two - dimensional case , quasi - convexity is generic and equivalent to the nonvanishing of the arnold determinant    @xmath0    a property that it is not too hard to verify .    indeed , since is an analytic function of the actions , it is enough to show that it is different from zero at one point in order to ensure that it is different from zero almost everywhere . here we explicitly compute the expansion of @xmath1 at a circular orbit and we show that , provided the central potential @xmath2 does not satisfy identically a fourth order differential equation that we explicitly write , the hamiltonian @xmath1 is quasi - convex on an open dense domain ( whose complementary is possibly empty ) . the rest of the paper is organized as follows : in sect . [ statement ] we introduce the central motion problem and state the main results . [ proof ] contains all the proofs . in the appendix we prove that in the two dimensional case quasi - convexity is equivalent to arnold isoenergetic nondegeneracy condition . _ acknowledgements . _ we thank f. fass for a detailed discussion on action angle variables in the central motion problem , m. guzzo , l. niederman and g. pinzari for pointing to our attention some relevant references and a. maspero for interesting discussions . we first recall the structure of the action angle variables for the central motion . introducing polar coordinates , the hamiltonian takes the form @xmath3 and the actions on which @xmath4 depends are @xmath5 and the action @xmath6 of the effective one dimensional hamiltonian system @xmath7 by construction @xmath4 turns out to be a function of the two actions only . we still write @xmath8    according to fass s theorem , if @xmath4 depends on @xmath9 in a steep way , then nekhoroshev s estimate applies . we recall that steepness is actually implied by quasi - convexity , the property that we are now going to verify .    a function @xmath4 of the actions is said to be _ quasi - convex _ at a point @xmath10 if the system @xmath11 admits only trivial solutions . here we denoted by @xmath12 the second differential of @xmath4 at @xmath10 applied to the two vectors @xmath13 .    to define the set @xmath14 in which the actions vary we first assume that there exists an interval @xmath15 such that , for @xmath16 one has @xmath17 then we define @xmath18 and , in order to fix ideas , we assume that @xmath19 . then for @xmath20 , the effective potential @xmath21 has a non degenerate minimum at some @xmath22 . then , there exists a curve @xmath23 such that for @xmath24 , all the orbits of the hamiltonian are periodic . correspondingly , their action @xmath25 vary in some interval @xmath26 . thus , the domain @xmath27 of the actions @xmath28 has the form @xmath29 we remark that @xmath27 is simply connected , a property that will play an important role in the following . our main result is the following . [ teorema1 ] consider the hamiltonian @xmath30 with @xmath2 analytic on @xmath31 . assume that there exists a value @xmath32 of the radius such that the following fourth order equation @xmath33 is _ not _ satisfied . then , there exists a subset @xmath34 of the action space , with the property that its intersection with any compact set is composed by at most a finite number of lines , and such that @xmath4 restricted to @xmath35 is quasi - convex . the fourth order equation can be rewritten as a second order ordinary differential equation in terms of the variable @xmath36 , namely , @xmath37    it is interesting to see what are the homogeneous potentials which do not give rise to steep hamiltonians . thus take @xmath38 with @xmath39 , then the assumptions of theorem [ teorema1 ] are fulfilled if @xmath40 thus the excluded cases are the kepler and the harmonic potentials . we also remark that the equation has also the solution @xmath41 , which however is excluded by the second of . fix a positive parameter @xmath42 and denote by @xmath43 the open ball of radius @xmath42 and center @xmath28 and define @xmath44    we now consider a small perturbation @xmath45 of @xmath4 , with @xmath46 a function of the original cartesian coordinates @xmath47 in @xmath48 which is analytic . [ teorema2 ] fix a positive small parameter @xmath42 and consider the hamiltonian @xmath49 . then , for every compact set @xmath50 , there exist positive constants @xmath51 such that if the initial value @xmath52 of the actions fulfills @xmath53 and @xmath54 one has @xmath55 for all times @xmath56 satisfying @xmath57    the main dynamical consequence is that , as in the central motion , for any initial datum as above , there exist @xmath58 such that @xmath59 for the times .    actually , one can weaker the analyticity requirement on @xmath46 , since it would be enough to have that it is analytic on the set of @xmath60 and @xmath61 s such that the action is close to @xmath52 . the theorem holds also if one couples the system to an other system with a dynamic taking place over a much faster time scale . for example , this occurs in the case of a soliton interacting with radiation in the nls equation as in @xcite . as anticipated above , in the case of _ two _ actions , quasi - convexity is equivalent to the nonvanishing of the arnold determinant @xmath62 ( cf . ) .    in order to compute @xmath62 , we have to compute @xmath63 . to do this we proceed as follows .    first , as explained in sect . [ statement ] , it is easy to introduce the action @xmath64 which coincides with the modulus of the total angular momentum . then , @xmath6 is the action of the effective one dimensional system in which @xmath64 plays the role of a parameter . in order to have an explicit formula for the hamiltonian as a function of the actions , we work at circular orbits . precisely , we exploit the remark that in one dimensional systems birkhoff normal form converges in a neighborhood of a nondegenerate minimum . indeed , birkhoff normal form allows to construct an analytic canonical transformation which , in a neighborhood of the critical point , conjugates the hamiltonian @xmath4 to a function of the form @xmath65 which moreover is explicitly constructed as a power series in @xmath66 . thus , one can define the first action @xmath6 by @xmath67 . remark that , since , as a function of @xmath68 is analytic in a whole _ complex _ neighborhood of zero , then @xmath63 is analytic for @xmath6 in a whole neighborhood of zero . then , from uniqueness of the actions in one dimensional systems , one has that the expression one gets is actually the expression of @xmath4 as a function of the actions as defined in sect . [ statement ] . it also follows that @xmath69 as a function of @xmath6 extends to a complex analytic function in whole neighborhood of zero , and such a function can be computed using the expression of @xmath4 obtained from . we use this remark in order to compute the arnold determinant at the equilibrium point @xmath70 of the effective one dimensional system described by @xmath4 , which coincides with a circular orbit of the original system . this can be done by computing explicitly the second order taylor expansion of @xmath4 which coincides with the fourth order birkhoff normal form of @xmath71 at @xmath70 . actually this was already done in @xcite getting @xmath72 where @xmath73    fix a point @xmath74 and let @xmath75 be the corresponding critical point of the effective potential .    inserting into the arnold determinant @xmath62 , the first and second derivatives of the hamiltonian evaluated at the point @xmath76 , one gets @xmath77 where we denoted by @xmath78 thus , @xmath79 is equivalent to @xmath80 isolating the fourth order derivative of @xmath81 one gets @xmath82 using that @xmath70 is a critical point of @xmath83 , one can express @xmath74 in terms of @xmath70 , namely , @xmath84 and computing @xmath85 one can rewrite the equation in terms of the radius @xmath70 , obtaining the equation . thus , if there exists @xmath70 such that is not satisfied , then at this point @xmath86 . this concludes the proof of the theorem . @xmath87    * proof of theorem [ teorema2 ] * from the previous result , one gets that the hamiltonian is quasi - convex on the subset @xmath88 with uniform bounds on the quasi - convexity constants . thus , if we choose an initial datum in such a set and a sufficiently small parameter @xmath89 , then fass s version of the nekhoroshev theorem for degenerate hamiltonians applies ( see @xcite for details ) . we remark that , in the case of quasi - convex hamiltonians , one gets optimal nekhoroshev s exponents which for two dimensional systems are @xmath90 .    @xmath87 in this appendix we show that , in the two dimensional case , quasi - convexity is equivalent to the nonvanishing of the arnold determinant @xmath62 . we start by recalling the arnold condition . let @xmath4 be a complete integrable hamiltonian with @xmath91 degrees of freedom and frequency @xmath92 . then , @xmath4 is said to satisfy the _ arnold condition _ at @xmath93 if the following map @xmath94 has maximal rank at @xmath95 . thus , we conclude that , in the case @xmath105 , the arnold condition is equivalent to the request of the second differential @xmath106 to be different from zero on the hyperplane generated by the vector @xmath107 normal to the gradient , namely , quasi - convexity .
in this paper we prove a nekhoroshev type theorem for perturbations of hamiltonians describing a particle subject to the force due to a central potential . precisely , we prove that under an explicit condition on the potential , the hamiltonian of the central motion is quasi - convex . 
 thus , when it is perturbed , two actions ( the modulus of the total angular momentum and the action of the reduced radial system ) are approximately conserved for times which are exponentially long with the inverse of the perturbation parameter .