A complete solution to the problem of orthogonal separation of variables of the Hamilton–Jacobi equation in three-dimensional Minkowski space is obtained. The solution is based on the underlying ideas of Cartan geometry and ultimately developed into a general new algorithm that can be employed in the study of Hamiltonian systems defined by natural Hamiltonians within the framework of Hamilton–Jacobi theory. To demonstrate its effectiveness, we investigate from this viewpoint the Morosi–Tondo integrable system derived as a stationary reduction of the seventh-order Korteweg–de Vries flow to show explicitly that the system in question is an orthogonally separable Hamiltonian system. The latter result is a new characterization of the Morosi–Tondo system.
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Research Article|
May 05 2009
Hamilton–Jacobi theory in three-dimensional Minkowski space via Cartan geometry
Joshua T. Horwood;
Joshua T. Horwood
a)
1
Numerica Corporation
, 4850 Hahns Peak Drive, Loveland, Colorado 80538, USA
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Raymond G. McLenaghan;
Raymond G. McLenaghan
b)
2Department of Applied Mathematics,
University of Waterloo
, Waterloo, Ontario N2L 3G1, Canada
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Roman G. Smirnov
Roman G. Smirnov
c)
3Department of Mathematics and Statistics,
Dalhousie University
, Halifax, Nova Scotia B3H 3J5, Canada
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J. Math. Phys. 50, 053507 (2009)
Article history
Received:
November 17 2008
Accepted:
February 12 2009
Citation
Joshua T. Horwood, Raymond G. McLenaghan, Roman G. Smirnov; Hamilton–Jacobi theory in three-dimensional Minkowski space via Cartan geometry. J. Math. Phys. 1 May 2009; 50 (5): 053507. https://doi.org/10.1063/1.3094719
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