Conformal Hamiltonian systems

Abstract

Vector fields whose flow preserves a symplectic form up to a constant, such as simple mechanical systems with friction, are called "conformal". We develop a reduction theory for symmetric conformal Hamiltonian systems, analogous to symplectic reduction theory. This entire theory extends naturally to Poisson systems: given a symmetric conformal Poisson vector field, we show that it induces two reduced conformal Poisson vector fields, again analogous to the dual pair construction for symplectic manifolds. Conformal Poisson systems form an interesting infinite-dimensional Lie algebra of foliate vector fields. Manifolds supporting such conformal vector fields include cotangent bundles, Lie-Poisson manifolds, and their natural quotients.

On time-dependent Hamiltonian realizations of planar and nonplanar systems

“…It is now a matter of direct calculation to show that the reduced system (38) is Hamiltonian generating by the Hamiltonian function (39) and the symplectic two-form Ω in (40). There remains to add the term cy to the right hand side of the second equation in (38).…”

Conformal conservation laws and geometric integration for damped Hamiltonian PDEs

“…Then, the non-Hamiltonian part of the flow, given by z t ¼ Dz, may be solved exactly to give the flow map U t ðzÞ ¼ expðDtÞz. Thus, the system (28) may be solved by composing the flow maps, so that U Dt W Dt denotes the (approximate) flow map for (28).…”

Optimal error estimate of a conformal Fourier pseudo‐spectral method for the damped nonlinear Schrödinger equation

“…When , the DNLS equation reduces to the classical NLS equation. According to McLachlan and Perlmutter , the DNLS equation admits the conformal mass conservation law and the conformal momentum conservation law where represents the complex conjugate of .…”