Symmetry preserving deformations of the Kepler problem. (English) Zbl 0683.70016
Certainly one of the most studied problems in mechanics is the Kepler problem - that which describes the motion of two particles governed by a force varying inversely as the square of the distance between the two particles. One of the more delightful aspects of this model is the great number of remarkable properties that it possesses. For example, it has Kepler’s three laws, a Lenz vector, Delaunay variables and a fifteen- dimensional symmetry group.
The plan of this paper is to realize the differential equations for the Kepler problem as a member of a one-parameter family of differential equations, which are Hamiltonian at each value of the parameter. If we think of the parameter as representing a deformation, then this deformation is Hamiltonian. Next we show that there is a false-time that linearizes the differential equations, and provides their integration. Then we show that many of the wonderful properties of the Kepler problem have suitable parametrized analogues. For instance, Kepler’s tree laws have a parametrized analogue, and the fifteen-dimensional symmetry group of the Kepler problem is preserved under the deformation as well. Looking at the Kepler problem as a member of a whole family of problems allows us to view it as a limit, and so gain a new insight on almost all aspects of the Kepler problem, especially some of the recent work on regularization.
The plan of this paper is to realize the differential equations for the Kepler problem as a member of a one-parameter family of differential equations, which are Hamiltonian at each value of the parameter. If we think of the parameter as representing a deformation, then this deformation is Hamiltonian. Next we show that there is a false-time that linearizes the differential equations, and provides their integration. Then we show that many of the wonderful properties of the Kepler problem have suitable parametrized analogues. For instance, Kepler’s tree laws have a parametrized analogue, and the fifteen-dimensional symmetry group of the Kepler problem is preserved under the deformation as well. Looking at the Kepler problem as a member of a whole family of problems allows us to view it as a limit, and so gain a new insight on almost all aspects of the Kepler problem, especially some of the recent work on regularization.
MSC:
| 70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |
| 70H15 | Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
Kepler problem; Lenz vector; Delaunay variables; fifteen-dimensional symmetry group; one-parameter family of differential equationsReferences:
| [1] | Abraham, R.; Marsden, J., Foundations of Mechanics (1978), Benjamin-Cummings: Benjamin-Cummings Reading, Mass · Zbl 0393.70001 |
| [2] | Arnold, V. I., Mathematical Methods of Classical Mechanics (1978), Springer: Springer New York · Zbl 0386.70001 |
| [3] | Bacry, H., Nuovo Cimento, 41A, 222-234 (1966) · Zbl 0138.44701 |
| [4] | Bates L.: Hidden Symmetry and Complete Lifts; Bates L.: Hidden Symmetry and Complete Lifts |
| [5] | Bates L.: Ph.D. Thesis, University of Calgary.; Bates L.: Ph.D. Thesis, University of Calgary. |
| [6] | Burgoyne, N.; Cushman, R., Journal of Algebra, 44, 339-362 (1977) · Zbl 0347.20026 |
| [7] | Cordani, B., Comm. Math. Phys., 103, 403-413 (1986) · Zbl 0599.70014 |
| [8] | Cushman, R.; Knörrer, H., The Energy Momentum Mapping of the Lagrange Top, (Differential-Geometric Methods in Mathematics. Differential-Geometric Methods in Mathematics, Springer Lecture Notes in Mathematics, vol. 1139 (1985), Springer: Springer New York) · Zbl 0615.70002 |
| [9] | Iosifescu, M.; Scutaru, H., J. Math. Phys., 21, 2033-2045 (1980) · Zbl 0463.22018 |
| [10] | Kirillov, A., Elements of the Theory of Representations (1978), Springer: Springer New York |
| [11] | Kummer, M., Indiana Math., 30, 2, 281-291 (1981) · Zbl 0425.70019 |
| [12] | Ligon, T.; Schaaf, M., Rep. on Math. Phys., 9, 281-300 (1976) · Zbl 0347.58005 |
| [13] | Moser, J., Comm. Pure Appl. Math., 23, 609-636 (1970) · Zbl 0193.53803 |
| [14] | Schonfeld, J., J. Math. Phys., 10, 2528-2533 (1980) |
| [15] | Souriau, J. M., Atti Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur., 117, suppl. 1, 369-418 (1983), See also Math. Reviews 86d:70004 |
| [16] | Souriau, J. M., Structure des Systemes Dynamiques (1970), Dunod: Dunod Paris · Zbl 0186.58001 |
| [17] | Kamp, P.van de, Elements of Astromechaics (1964), W.H. Freeman: W.H. Freeman San Francisco, Ca · Zbl 0121.46001 |
| [18] | Weyl, H., The Classical Groups (1946), Princeton Univ. Press: Princeton Univ. Press Princeton · JFM 65.0058.02 |
| [19] | Wolf, J., Representations associated to Minimal Co-adjoint Orbits, (Differential-Geometric Methods in Mathematical Physics II. Differential-Geometric Methods in Mathematical Physics II, Springer Lecture Notes in Mathematics, vol. 676 (1978), Springer: Springer New York), (proceedings, Bonn 1977) · Zbl 0388.22008 |
| [20] | Woodhouse, N., Geometric Quantization (1980), Oxford University Press: Oxford University Press Oxford · Zbl 0458.58003 |
| [21] | Zwanziger, D., Phys. Rev., 5, 1480-1488 (1968) |
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