Extension of many-time Hamiltonian formalism to the theory of deformable bodies

Abstract

For time-independent fields differences between the classical Lagrangian and Himiltonian formalisms vanish. It means that the system of Euler-Lagrange second-order differential equations with respect to time and space coordinates becomes, in Hamiltonian formalism, a system of 2n equations of the first-order, but with respect to time coordinate only. In this note a forgotten Cosserat approach to the construction of a many-time model of deformable continuum is recalled, in particular in the context of formalism of the Hamiltonian field theory. Such formalism leads, in the more natural way, to multi-field variational principles and to the canonical system of governing equations which are of the first order in time and space. The many-time Hamiltonian formalism, when applied to Cosserat and Cauchy continua, allows to explain origins of the Hellinger variational principle.