Two symplectic fixed-point theorems with applications to Hamiltonian dynamics
The authors prove two symplectic fixed point theorems which give information about some finite time perturbations of Hamiltonian systems. They consider a smooth Hamiltonian H in \({\mathbb{R}}^{2n}\) with the standard symplectic structure \(\omega\). They look at solutions of the Hamiltonian system on a fixed energy level \(S=\{H(x)=h\}\). A non autonomous smooth perturbation \(\Delta\) (t,x) of the Hamiltonian is switched on at time \(t=0\) and off at time \(t=1\). The question is whether the perturbation has the property that a trajectory can be found, where the effect of the perturbation is coming back to the original unperturbed solution with a phase shift. Let us call this a phase-shift solution. In the abstract symplectic context, one looks at the symplectic flow \(\psi_ t\), \(0\leq t\leq 1\), generated by the solutions of the perturbed system. Then we need general conditions in order to assert that the time- one map \(\psi_ 1\) takes some point \(x\in S\) into the unperturbed Hamiltonian orbit through x. First of all, they impose the condition that the energy level S is a compact connected \(C^ 2\) hypersurface where the symplectic structure can be restricted to a contact form defined on \({\mathbb{R}}^{2n}.\) The first theorem states that if \(\psi_ t\), \(0\leq t\leq 1\), is an isotopy of the identity with even diffeomorphisms, then the assertion is true. This implies immediately the existence of a phase-shift solution, provided H(x) and \(\Delta\) (t,x) are even functions in x. Besides having a symmetry, the origin is always a fixed point of the flow. In the second theorem they removed the symmetry condition, but had to impose instead a technical condition on the fixed points of \(\psi_ 1\), lying on the bounded component of the complement of S. An immediate corollary is that the assertion is true for linear symplectic flows, extending a previously known result. Another corollary is that if H(t,x) is \(C^ 2\)-close to 0 in a neighborhood \({\mathcal V}\) of [0,1]\(\times S\), there is a phase-shift solution. This improves results of standard perturbation arguments in that \(\psi_ 1(S)\) is only \(C^ 0\)-close to S, rather than \(C^ 1\)-close. As a particular example, take \(H=1/2(| p|^ 2+| q|^ 2)\) the Hamiltonian of a harmonic oscillator in any dimension, with the perturbation a time-dependent potential V(t,q) for \(0\leq t\leq 1\). If V is even in q, the first theorem applies. If \(V'(t,0)=0\) and \(V''(t,q)\) satisfies some conditions besides being positive definite, we also have the existence of a phase-shift solution.
scientific article; zbMATH DE number 4166104