Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909150456315904 |
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| author | Deriglazov, Alexei A. |
| author_facet | Deriglazov, Alexei A. |
| contents | We consider Hamiltonian formulation of a dynamical system forced to move on a submanifold $G_α(q^A)=0$. 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 $p_i$, $[i]=[A]-[α]$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05151 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability Deriglazov, Alexei A. Mathematical Physics Solar and Stellar Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Theory Exactly Solvable and Integrable Systems We consider Hamiltonian formulation of a dynamical system forced to move on a submanifold $G_α(q^A)=0$. 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 $p_i$, $[i]=[A]-[α]$. 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. |
| title | Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability |
| topic | Mathematical Physics Solar and Stellar Astrophysics General Relativity and Quantum Cosmology High Energy Physics - Theory Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2309.05151 |