Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy
Brake orbits of a conservative system \(\ddot q=-V'(q)\) \((q(t)\in {\mathbb{R}}^ N)\) are non-constant solutions, satisfying \(\dot q(0)=\dot q(T)=0\) for \(T>0\). They give rise to a 2T-periodic solution. The author presents a constructive formulation of brake orbits with prescribed energy \(E>0\) via an analytical mini-max procedure. The potential V is assumed to be strictly convex, with global minimum at zero. The starting point is a kind of modified principle of least action for fixed energy orbits: after a rescaling of the equations of motion, the author constructs a function \(J_ E\) on the Sobolev space \(X=W_{1,2}([0,1],{\mathbb{R}}^ N)\), whose critical points are the brake orbits of energy E. It is argued that X is the union of halfspaces \(M_{\xi}\) in \({\mathbb{R}}^{N+1}\), where \(\xi\) runs over the space of normalized functions with zero mean value. Restricted to each \(M_{\xi}\), \(J_ E\) is shown to have a unique critical point (a positive maximum) in the interior. Letting \(N_ E\) denote the union of these maxima, the problem is posed to minimize \(J_ E\) over \(N_ E\). It is shown that this problem has at least one solution x, with \(J_ E(x)>0\) and that the corresponding brake orbit is one with minimal period. In the final section, the author calls on Lyusternik-Schnirelmann theory to derive a multiplicity result concerning the number of distinct brake orbits with energy E.
scientific article; zbMATH DE number 4089002
scientific article