Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability

TL;DR

This paper compares two Hamiltonian formalisms for systems constrained to submanifolds, illustrating their application by deriving equations for a spinning body's dynamics and analyzing their Poisson structures.

Contribution

It introduces and compares Hamiltonian reduction of Dirac brackets and an intermediate formalism for constrained systems, with applications to spinning bodies.

Findings

Derived Euler-Poisson equations using intermediate formalism

Established Poisson structure for spinning body dynamics

Provided general solutions via Hamiltonian vector fields

Abstract

We consider Hamiltonian formulation of a dynamical system forced to move on a submanifold $G_\alpha(q^A)=0$. If for some reasons we are interested in knowing the dynamics of all original variables $q^A(t)$, the most economical would be a Hamiltonian formulation on the intermediate phase-space submanifold spanned by reducible variables $q^A$ and an irreducible set of momenta $p_i$, $[i]=[A]-[\alpha]$. We describe and compare two different possibilities for establishing the Poisson structure and Hamiltonian dynamics on an intermediate submanifold: Hamiltonian reduction of the Dirac bracket and intermediate formalism. As an example of the application of intermediate formalism, we deduce on this basis the Euler-Poisson equations of a spinning body, establish the underlying Poisson structure, and write their general solution in terms of the exponential of the Hamiltonian vector field.