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

Alexei A. Deriglazov

doi:10.1140/epjc/s10052-024-12552-9

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2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2024) 84: 311
https://doi.org/10.1140/epjc/s10052-024-12552-9

Regular Article - Theoretical Physics

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

Alexei A. Deriglazov^a

Depto. de Matemática, ICE, Universidade Federal de Juiz de Fora, Juiz de Fora, MG, Brazil

^a alexei.deriglazov@ufjf.br

Received: 4 September 2023
Accepted: 13 February 2024
Published online: 24 March 2024

Abstract

We consider the Lagrangian dynamical system forced to move on a submanifold $G_{α} (q^{A}) = 0$ $G_\alpha (q^A)=0$ . 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 $p_{i}$ $$p_i$$ , $[i] = [A] - [α]$ $[i]=[A]-[\alpha ]$ . 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.

© The Author(s) 2024

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