Symplectic Homogenization
Copyright policy )Symplectic Homogenization
Let H(q,p) be a Hamiltonian on T * T n . Under suitable assumptions on H, we show that the sequence (H k ) k≥1 defined by H k (q,p)=H(kq,p) converges in the γ-topology—defined in [Vit92]—to an integrable continuous Hamiltonian H ¯(p). This is extended to the case of non-autonomous Hamiltonians, and the more general setting in which only some of the variables are homogenized: we consider the sequence H(kx,y,q,p) and prove it has a γ-limit H ¯(y,q,p), thus yielding an “effective Hamiltonian”. The goal of this paper is to prove convergence of the above sequences, state the first properties of the homogenization operator, and give some applications to solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that when H is convex in p, the function H ¯ coincides with Mather’s α function defined in [Mat91] and associated to the Legendre dual of H. This gives a new proof—in the torus case—of its symplectic invariance first discovered by P. Bernard in [Ber07].
- New York University United States
- Institute for Advanced Study United States
- École Normale Supérieure Burundi
variational solutions, 37J05, 53D35 (Primary) 35F20, 49L25, 37J40, 37J50 (Secondary), Symplectic and contact topology in high or arbitrary dimension, homogenization, Dynamical Systems (math.DS), Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.), Hamiltonian flow on tori, Global theory of symplectic and contact manifolds, Mathematics - Analysis of PDEs, FOS: Mathematics, Mathematics - Dynamical Systems, Mathematics - Optimization and Control, Symplectic structures in 4 dimensions, Nonautonomous Hamiltonian dynamical systems (Painlevé equations, etc.), General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants, Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games, Hamilton-Jacobi equation, symplectic topology, Mathematics - Symplectic Geometry, Optimization and Control (math.OC), Symplectic Geometry (math.SG), symplectic invariance, Hamilton-Jacobi equations, nonautonomous Hamiltonian, Mather's function, Analysis of PDEs (math.AP)
variational solutions, 37J05, 53D35 (Primary) 35F20, 49L25, 37J40, 37J50 (Secondary), Symplectic and contact topology in high or arbitrary dimension, homogenization, Dynamical Systems (math.DS), Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.), Hamiltonian flow on tori, Global theory of symplectic and contact manifolds, Mathematics - Analysis of PDEs, FOS: Mathematics, Mathematics - Dynamical Systems, Mathematics - Optimization and Control, Symplectic structures in 4 dimensions, Nonautonomous Hamiltonian dynamical systems (Painlevé equations, etc.), General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants, Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games, Hamilton-Jacobi equation, symplectic topology, Mathematics - Symplectic Geometry, Optimization and Control (math.OC), Symplectic Geometry (math.SG), symplectic invariance, Hamilton-Jacobi equations, nonautonomous Hamiltonian, Mather's function, Analysis of PDEs (math.AP)
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