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

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.

17 pages, typos corrected, notation improved, discussion expanded, matches with published version

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Cited by in corpus (2)

• Has the problem of the motion of a heavy symmetric top been solved in quadratures?
• General solution to Euler-Poisson equations of a free symmetric body by direct summation of power series