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

Abstract

AbstractWe consider the Lagrangian dynamical system forced to move on a submanifold $$G_\alpha (q^A)=0$$ G α ( q A ) = 0 . If for some reason we are interested in knowing the dynamics of all original variables $$q^A(t)$$ q A ( t ) , the most economical would be a Hamiltonian formulation on the intermediate phase-space submanifold spanned by reducible variables $$q^A$$ q A and an irreducible set of momenta $$p_i$$ p i , $$[i]=[A]-[\alpha ]$$ [ i ] = [ A ] - [ α ] . 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.