This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in mechanics ranges from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many

The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the wellknown three-wave equations that also apply, for example, in laser-matter interaction in a cavity. We use Hamiltonian reduction and pattern evocation techniques to derive a formula that describes the characteristic feature of this system's dynamics, namely, the stepwise precession of its azimuthal angle. PACS numbers: 02.40.-k, 05.45.-a, 45.10.Db, 45.20.Jj Keywords: Classical mechanics, Variational principles, Averaged Lagrangian, Elastic Pendulum, Nonlinear Resonance. email: dholm@lanl.gov y email: Peter.Lynch@met.ie 1 D. D. Holm & P. Lynch Precession of the Swinging Spring 2 Contents 1

In this paper we discuss asymptotic stability in energy-preserving systems which have an almost Poisson structure. In particular we consider a class of Poisson systems which includes the Toda lattice. In standard Hamiltonian systems one of course does not expect asymptotic stability.

The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the wellknown three-wave equations that also apply, for example, in laser-matter interaction in a cavity. We use Hamiltonian reduction and pattern evocation techniques to derive a formula that describes the characteristic feature of this system’s dynamics, namely, the stepwise precession of its azimuthal angle.