Extension of Noether's theorem to constrained non-conservative dynamical systems

Abstract

A method based on a differential variational principle is developed in order to extend Noether's theorem to constrained non-conservative dynamical systems. The result is applied to generate constants of the motion for a generic example of a non-linear, dissipative dynamical system with time-varying coefficients represented by the Emden equation. The converse of Noether's theorem, whereby the symmetries of the system are determined from the knowledge of the Lagrangian and a first integral is also considered for both the Emden equation, and that of the damped harmonic oscillator. It is further shown that the presence of ideal constraints (whether holonomic or non-holonomic) does not affect the statement of Noether's theorem. The constraints affect the Jacobi energy integral, however, because they enter into consideration through real work instead of virtual work. It is shown that the Jacobi integral is conserved provided that: (a) the Lagrangian is explicitly independent of time, (b) the real power of the generalized forces not derivable from a potential vanish, (c) the holonomic constraints are explicitly independent of time, (d) the non-holonomic constraints are linear and homogeneous in the generalized velocities.