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Aubry-Mather theory for contact Hamiltonian systems. (English) Zbl 1429.37034

The goal of this paper is to formulate an Aubry-Mather theory for contact Hamiltonian systems. The authors develop criteria for action-minimizing methods for such systems. Their work involves taking basic results for Aubry-Mather systems together with prior results in weak KAM theory for conventional Hamiltonian systems.
The authors look at the contact manifold as the manifold of 1-jets on a connected, closed and smooth Riemannian manifold \(M\) with a \(C^\infty\) metric. They note that such a manifold is canonically diffeomorphic to T\(^*M\times\mathbb{R}\). In local coordinates the equations of the contact flow generated by the Hamiltonian H are: \(\dot{x}=\frac{\partial H}{\partial p}(x, u, p)\); \(\dot{p}=\frac{\partial H}{\partial x}\) \((x,u,p)-\frac{\partial H}{\partial u}(x, u, p)\) \(p\); and \(\dot{u}=\frac{\partial H}{\partial p}\) \((x,u,p)\cdot p-H(x,u,p)\) for \((x,u,p) \in T^*M\times\mathbb{R}\).
The authors make the following assumptions: (1) For every \((x,u,p) \in T^*M\times\mathbb{R}\), the second derivative \(\frac{\partial ^2 H}{\partial p^2}\) is positive definite; (2) \(H(x,u,p)\) is superlinear in \(p\) for every \((x,u) \in M \times\mathbb{R}\); and (3) There is a positive constant \(\lambda\) such that \(0<\frac{\partial H}{\partial p}(x, u, p)\leq\lambda\) for every \((x,u,p) \in T^*M\times\mathbb{R}\).
For their main results, the authors also need an admissability assumption: there exists an \(a \in\mathbb{R}\) such that the Mañé critical value \(c(a)\) of \(H(x,a,p)\) is zero.
The authors’ main theorems characterize the Aubry and Mather sets of contact Hamiltonians as well as the global minimizing curves.

MSC:

37J55 Contact systems
37J51 Action-minimizing orbits and measures for finite-dimensional Hamiltonian and Lagrangian systems; variational principles; degree-theoretic methods
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
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