dorsal/arxiv
View SchemaIllustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited
| Authors | Ross C. O'Connell, Kannan Jagannathan |
|---|---|
| Categories | |
| ArXiv ID | physics/0212043 |
| URL | https://arxiv.org/abs/physics/0212043 |
| DOI | 10.1119/1.1524165 |
Abstract
The inverse square force law admits a conserved vector that lies in the plane of motion. This vector has been associated with the names of Laplace, Runge, and Lenz, among others. Many workers have explored aspects of the symmetry and degeneracy associated with this vector and with analogous dynamical symmetris. We define a conserved dynamical variable $\alpha$ that characterizes the orientation of the orbit in two-dimensional configuration space for the Kepler problem and an analogous variable $\beta$ for the isotropic harmonics oscillator. This orbit orientation variable is canonically conjugate to the angular momentum component normal to the plane of motion. We explore the canoncial one-parameter group of transformations generated by $\alpha (\beta).$ Because we have an obvious pair of conserved canonically conjugate variables, it is desirable to us them as a coordinate-momentum pair. In terms of these phase space coordinates, the form of the Hamiltonian is nearly trivial because neither member of the pair can occur explicitly in the Hamiltonian. From these considerations we gain a simple picture of the dynamics in phase space. The procedure we use is in the spirit of the Hamilton-Jacobi method.
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"abstract": "The inverse square force law admits a conserved vector that lies in the plane\nof motion. This vector has been associated with the names of Laplace, Runge,\nand Lenz, among others. Many workers have explored aspects of the symmetry and\ndegeneracy associated with this vector and with analogous dynamical symmetris.\nWe define a conserved dynamical variable $\\alpha$ that characterizes the\norientation of the orbit in two-dimensional configuration space for the Kepler\nproblem and an analogous variable $\\beta$ for the isotropic harmonics\noscillator. This orbit orientation variable is canonically conjugate to the\nangular momentum component normal to the plane of motion. We explore the\ncanoncial one-parameter group of transformations generated by $\\alpha (\\beta).$\nBecause we have an obvious pair of conserved canonically conjugate variables,\nit is desirable to us them as a coordinate-momentum pair. In terms of these\nphase space coordinates, the form of the Hamiltonian is nearly trivial because\nneither member of the pair can occur explicitly in the Hamiltonian. From these\nconsiderations we gain a simple picture of the dynamics in phase space. The\nprocedure we use is in the spirit of the Hamilton-Jacobi method.",
"arxiv_id": "physics/0212043",
"authors": [
"Ross C. O\u0027Connell",
"Kannan Jagannathan"
],
"categories": [
"physics.class-ph",
"physics.ed-ph"
],
"doi": "10.1119/1.1524165",
"title": "Illustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited",
"url": "https://arxiv.org/abs/physics/0212043"
},
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