Variational principle for contact Hamiltonian systems and its applications

2017/02/15 by Kaizhi Wang, Lin Wang, Wang, Kaizhi +3 · 4 citations
Physics and Astronomy · Engineering · #Quantum chaos and dynamical systems #Control and Stability of Dynamical Systems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1702.04451

Abstract

In \citeWWY, the authors provided an implicit variational principle for the contact Hamilton's equations \ x=(∂ H)/(∂ p)(x,u,p),
p=-(∂ H)/(∂ x)(x,u,p)-(∂ H)/(∂ u)(x,u,p)p, (x,p,u)∈ T^*M\timesR,
u=(∂ H)/(∂ p)(x,u,p)⋅ p-H(x,u,p), . where M is a closed, connected and smooth manifold and H=H(x,u,p) is strictly convex, superlinear in p and Lipschitz in u. In the present paper, we focus on two applications of the variational principle: 1. We provide a representation formula for the solution semigroup of the evolutionary equation w_t(x,t)+H(x,w(x,t),w_x(x,t))=0; 2. We study the ergodic problem of the stationary equation via the solution semigroup. More precisely, we find pairs (u,c) with u∈ C(M,R) and c\inR which, in the viscosity sense, satisfy the stationary partial differential equation H(x,u(x),u_x(x))=c.

Citations

• Action minimizing invariant measures for positive definite Lagrangian systems

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