Constrained systems and analytical mechanics in spaces with torsion Sergei

A system with anholonomic constraints where the trajectories of physical degrees of freedom are autoparallels on a manifold equipped with a general Cartan connection is discussed. A variational principle for the autoparallel trajectories is derived from the d’Alambert-Lagrange principle for anholonomic constrained systems. A geometrical (coordinate-independent) formulation of the variational principle is given. Its relation to Sedov’s anholonomic variational principle for dissipative systems and to Poincaré’s variational principle in anholonomic reference frames is established. A modification of Noether’s theorem due to the torsion force is studied. A non-local action whose extrema contain the autoparallels is proposed. The action can be made local by adding auxiliary degrees of freedom coupled to the original variables in a special way. 1 Anholonomic constrained systems There is no need to explain how important are constrained systems in modern physics (e.g., electrodynamics, Yang-Mills theory, general relativity, etc). Constraints in dynamical systems are usually regarded as a part of the Euler-Lagrange equations of motion which do not involve time derivatives of order higher than one. In other words, both constraints and equations of motion result from the least action principle applied to some Lagrangian. The existence of the Lagrangian formalism is of great importance in constrained systems because it allows one to develop the corresponding Hamiltonian formalism [1] and canonically quantize the system [1]. Yet, the variational principle is a powerful technical tool to find integrals of motion of dynamical systems via symmetries of the Lagrangian. The Hamiltonian (or Lagrangian) constrained systems form a relatively small class of constrained dynamical systems. Given an ”unconstrained” system whose dynamics is governed by a Lagrangian L = L(v, x), v and x being generalized velocities and coordinates, respectively, one can turn it into a constrained system by imposing supplementary conditions Fα(v, x) = 0 (constraints) which has to be fulfilled by the actual motion of the system. There two ways to incorporate the constraints into a dynamical description. First, one can simply modify the Lagrangian L→ L+ λFα with λ α being the Lagrange multipliers and treat the latter as independent dynamical variables. In doing so, we are led to the Lagrangian constrained dynamics. The other way is to supplement the unconstrained Euler-Lagrange equations d/dt(∂vL) − ∂xL = 0 by the constraints Fα = 0. It is well known that if the constraints are not integrable, the two dynamical descriptions DFG fellow; on leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia