On the general form of the law of dynamics

Giovanni Romano , Raffaele Barretta , Marina Diaco

The dynamics of lagrangian systems is formulated with a differential geometric approach and according to a new paradigm of the calculus of variations. Discontinuities in the trajectory, non-potential force systems and linear constraints are taken into account with a coordinate-free treatment. The law of dynamics, characterizing the trajectory in a general nonlinear configuration manifold, is expressed in terms of a variational principle and of differential and jump conditions. By endowing the configuration manifold with a connection, the general law is shown to be tensorial in the velocity of virtual flows and to depend on the torsion of the connection. This result provides a general expression of the Euler-Lagrange operator. Poincare and Lagrange forms of the law are recovered as special cases corresponding respectively to the connection induced by natural and mobile reference frames. For free motions, the geodesic property of the trajectory is directly inferred by adopting the LeviCivita connection induced by the kinetic energy.

Mots clés: Dynamics; Connection; Torsion; PoincarŽ law; Lagrange law

Giovanni Romano, Raffaele Barretta, Marina Diaco. On the general form of the law of dynamics. International Journal of Non-Linear Mechanics, 2009, 44 (6), pp.689. ⟨10.1016/j.ijnonlinmec.2009.02.011⟩. ⟨hal-00538619⟩