Operator formulation of Classical mechanics: Levi-Civita map and equivalence of central forces in 2-dimensions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914148433002496 |
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| author | Puranam, Nikhil Kasyap Harikumar, E. |
| author_facet | Puranam, Nikhil Kasyap Harikumar, E. |
| contents | We study the operator formulation of classical mechanics by explicitly applying it to two central potentials in 2 dimensions. After constructing the classical Hamiltonian operators and corresponding Schrödinger like equations, we solve for the corresponding classical wave functions associated with these two potentials, viz; Kepler and harmonic potentials. While satisfying continuity equations, these classical wave functions are shown to be renormalizable only in a finite region of the 2D plane. We also derive the well-known equivalence between these two models within the operator formulation of classical mechanics. This equivalence is shown by relating the Schrödinger-like equations and corresponding classical wave functions of these two systems, using the Levi-Civita map and a reparametrizaton of time(Sundman map). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_08999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Operator formulation of Classical mechanics: Levi-Civita map and equivalence of central forces in 2-dimensions Puranam, Nikhil Kasyap Harikumar, E. Quantum Physics Mathematical Physics Classical Physics We study the operator formulation of classical mechanics by explicitly applying it to two central potentials in 2 dimensions. After constructing the classical Hamiltonian operators and corresponding Schrödinger like equations, we solve for the corresponding classical wave functions associated with these two potentials, viz; Kepler and harmonic potentials. While satisfying continuity equations, these classical wave functions are shown to be renormalizable only in a finite region of the 2D plane. We also derive the well-known equivalence between these two models within the operator formulation of classical mechanics. This equivalence is shown by relating the Schrödinger-like equations and corresponding classical wave functions of these two systems, using the Levi-Civita map and a reparametrizaton of time(Sundman map). |
| title | Operator formulation of Classical mechanics: Levi-Civita map and equivalence of central forces in 2-dimensions |
| topic | Quantum Physics Mathematical Physics Classical Physics |
| url | https://arxiv.org/abs/2412.08999 |