On continuum dynamics

The theory of continuous dynamical systems is developed with an intrinsic geometric approach based on the action principle formulated in the velocity-time manifold. By endowing the finite dimensional Riemannian ambient manifold with a connection, an induced connection is naturally defined in the infinite dimensional configuration manifold of maps. The motion is shown to be governed, in the Banach configuration manifold, by a generalized Lagrange law and, in the ambient manifold, by a generalized Euler law which is independent of the Banach topology of the configuration manifold. Extended versions of Euler-Poincaŕ law, Euler classical laws and d'Alembert law are also derived as special cases. Stress fields in the body are introduced as Lagrange's multipliers of the rigidity constraint on virtual velocities, dual to the Lie derivative of the metric. No special assumptions are made so that any constitutive behaviors can be modeled.