Symmetries, constants of the motion, and reduction of mechanical systems with external forces

Abstract

This paper is devoted to the study of mechanical systems subjected to external forces in the framework of symplectic geometry. We obtain Noether’s theorem for Lagrangian systems with external forces, among other results regarding symmetries and conserved quantities. We particularize our results for the so-called Rayleigh dissipation, i.e., external forces that are derived from a dissipation function, and illustrate them with some examples. Moreover, we present a theory for the reduction in Lagrangian systems subjected to external forces, which are invariant under the action of a Lie group.

Discrete Hamilton-Jacobi theory for systems with external forces

“…Remark 1. It is worth noting that we have changed the sign criteria for discrete forces with respect to our previous papers [6,7,21], in order to be consistent with Lew, Marsden, Ortiz and West's criteria [20,23].…”

“…An external force R is Rayleigh (see Refs. [7,21], see also Ref. [2]) if there exists a function R on T Q such that R = S * (dR),…”

Symplectic and Cosymplectic Reduction for simple hybrid forced mechanical systems with symmetries

“…If H is known to be G-invariant, the subgroup G F of G such that H and F are both G F -invariant can be described as follows. Let ξ ∈ g, then J ξ is a conserved quantity for X H,F if and only if F (ξ c Q ) = 0 (see [18,35]) and ξ leaves F invariant if and only if ι ξ c Q dF = 0. In addition, the vector subspace of g given by…”

“…is the reduced momentum map with µ ∈ g * F a regular value of J F , by denoting (G F ) µ the isotropy subgroup in µ, we have that [18,35] (i)…”

“…Under the assumption of an elastic collision, the reset map (t, x, y, ẋ− , ẏ− ) → (t, x, y, ẋ+ , ẏ+ ) is given by: It is understood that the "minus" square root is taken in ṙ+ (the particle bounces on the boundary after the collision). Note that the momentum map ( JL ) F for θ, The celebrated symplectic reduction of (conservative) mechanical systems with symmetries, due to Marsden and Weinstein [1,38,39], was recently extended for forced autonomous Lagrangian [18,35] as well as Hamiltonian [17] systems. In this paper we have gone a step further and considered simple hybrid forced mechanical systems, both autonomous and non-autonomous.…”