Herglotz’ Generalized Variational Principle and Contact Type Hamilton-Jacobi Equations

Abstract

We develop an approach for the analysis of fundamental solutions to Hamilton-Jacobi equations of contact type based on a generalized variational principle proposed by Gustav Herglotz. We also give a quantitative Lipschitz estimate on the associated minimizers.

Vanishing contact structure problem and convergence of the viscosity solutions

“…In recent works [34,36,35], certain implicit variational principle is applied to give representation formula for the viscosity solutions or weak KAM solutions of (HJ λ ). An alternative approach following Herglotz' generalized variational principle is also obtained from the Lagrangian formalism [5], which is later used to obtain a vanishing contact structure result on relavent Cauchy problem for evolutionary equations [39]. But, these results are established under C 2 Tonelli conditions, so they are no longer applicable to our settings in this paper.…”

On the vanishing contact structure for viscosity solutions of contact type Hamilton-Jacobi equations I: Cauchy problem

“…Such a representation formula for Tonelli systems appeared first in [21] and [22] using an implicit variational principle and a fixed point method. In [5], the authors give an alternative approach based on Hergoltz' variational principle. Our following new representation formula for the fundamental solutions is motivated by the multiplier rule (see [10]).…”

“…Sufficient evidences show that the method we use in this paper is also a way to understand the vanishing contact structure limit by developing the Aubry-Mather theory for contact type systems (HJ s ) under suitable conditions. We will use a Langrangian approach of the solutions of (HJ e ) in the viscosity sense developed in [5] using the generalized variational principle proposed by Gustav Herglotz in 1930 (see [5] and the references therein).…”

“…It is clear that (1.2) admits a unique absolutely continuous solution u ξ (s) = u ξ (s; u) on [0, t], by Proposition A.1, if there exists f ∈ L 1 ([0, t], R) such that |L(ξ(s), u,ξ(s))| f (s), a.e., s ∈ [0, t] . In [5], the authors showed that Proposition 1.1. Let L satisfy conditions (L1)-(L3), then for fixed x, y ∈ R, t > 0 and u ∈ R, the problem (1.3) admits a minimizer.…”

Contact variational integrators

“…Another remarkable similarity with standard symplectic Hamiltonian systems is the fact that contact Hamiltonian systems have an associated variational principle, which is due to Herglotz [25,34] (see also [13,55]), and a corresponding theory of generating functions [10].…”