Poisson structure for hyperbolic heat conduction

Abstract

We use the method of extending the space of macroscopic variables (Gambar and Markus, 1994) to construct a Hamilton-Lagrange scheme for hyperbolic transport. Specifically, we propose a Lagrangian density to obtain the telegraphist type equations as the Euler-Lagrange equation of a Hamilton variational principle. Two evolution equations for the components of a conjugated variables space are obtained from the modified Hamilton principle. These equations are particular cases of a more general time evolution equations which contains a Poisson bracket with the Hamiltonian density as the movement generator. The bracket satisfies the Jacobi's identity giving us a Poisson structure for the problem. We discuss some aspects of the time evolution of fluctuations ot the temperature in a rigid heat conductor solid.