Intrinsic variational structure of higher-derivative formulations of classical mechanics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909124189487104 |
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| author | Sanders, John W. |
| author_facet | Sanders, John W. |
| contents | This paper investigates the geometric structure of higher-derivative formulations of classical mechanics. It is shown that every even-order formulation of classical mechanics higher than the second order is intrinsically variational, in the sense that the equations of motion are always derivable from a minimum action principle, even when the system is non-Hamiltonian. Particular emphasis is placed on the fourth-order formulation, as that is shown to be the lowest order for which the governing equations are intrinsically variational. The Noether symmetries and associated conservation laws of the fourth-order formulation, including its Hamiltonian, are derived along with the natural auxiliary conditions. The intrinsic variational structure of higher-derivative formulations makes it possible to treat non-Hamiltonian systems as if they were Hamiltonian, with immediate classical applications. A case study of the classical damped harmonic oscillator is presented for illustration, and an action is formulated for a higher-order Navier-Stokes equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_10175 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Intrinsic variational structure of higher-derivative formulations of classical mechanics Sanders, John W. Classical Physics This paper investigates the geometric structure of higher-derivative formulations of classical mechanics. It is shown that every even-order formulation of classical mechanics higher than the second order is intrinsically variational, in the sense that the equations of motion are always derivable from a minimum action principle, even when the system is non-Hamiltonian. Particular emphasis is placed on the fourth-order formulation, as that is shown to be the lowest order for which the governing equations are intrinsically variational. The Noether symmetries and associated conservation laws of the fourth-order formulation, including its Hamiltonian, are derived along with the natural auxiliary conditions. The intrinsic variational structure of higher-derivative formulations makes it possible to treat non-Hamiltonian systems as if they were Hamiltonian, with immediate classical applications. A case study of the classical damped harmonic oscillator is presented for illustration, and an action is formulated for a higher-order Navier-Stokes equation. |
| title | Intrinsic variational structure of higher-derivative formulations of classical mechanics |
| topic | Classical Physics |
| url | https://arxiv.org/abs/2301.10175 |