Multiple orbits for hamiltonian systems on starshaped surfaces with symmetries
We prove the existence of infinitely many geometrically distinct homoclinic orbits for a class of asymptotically periodic second order Hamiltonian systems.
A proof of the Chekanov theorem is discussed from a geometric point of view. Similar results in the context of projectivized cotangent bundles are proved. Some applications are given.
See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case , and satisfy conditions (11.7) when is a polynomial in , conditions (C)-i.e. (11.8) and (11.7) with -are proved to be necessary for treating satisfactorily 's hyper-impulsive motions (in which positions can suffer first order discontinuities)....
In [1] I and II various equivalence theorems are proved; e.g. an ODE with a scalar control is linear w.r.t. iff its solution with given initial conditions (chosen arbitrarily) is continuous w.r.t. in a certain sense, or iff