Abstract
In this work, an efficient method for constructing a complete integral of the geodesic Hamilton–Jacobi equation on pseudo-Riemannian manifolds with simply transitive groups of motions is suggested. The method is based on using a special transition to canonical coordinates on coadjoint orbits of the group of motion. As a non-trivial example, we consider the problem of constructing a complete integral of the geodesic Hamilton–Jacobi equation in the McLenaghan–Tariq–Tupper spacetime. An essential feature of this example is that the Hamilton-Jacobi equation is not separable in the corresponding configuration space.
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References
Eisenhart, L.: Riemannian Geometry. Princeton University Press, Princeton (1966)
Goldstein, H., Poole, C., Safko, J.: Classical Mechanics. Addison-Wesley, Boston (2001)
Arnol’d, V.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)
Woodhouse, N.: Killing tensors and the separation of the Hamilton–Jacobi equation. Commun. Math. Phys. 44, 9–38 (1975)
Shapovalov, V.N.: Stäckel spaces. Sib. Math. J. 20, 790–800 (1979)
Kalnins, E.G., Miller, W. Jr: Killing tensors and variable separation for Hamilton–Jacobi and Helmholtz equations. SIAM J. Math. Anal. 11, 1011–1026 (1980)
Bagrov, V.G., Obukhov, V.V.: Complete separation of variables in the free Hamilton–Jacobi equation. Theor. Math. Phys. 97, 1275–1289 (1993)
Benenti, S., Chanu, C., Rastelli, G.: Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds. J. Math. Phys. 42, 2065–2091 (2001)
Benenti, S.: Separability in Riemannian manifolds. SIGMA 12, 013 (2016)
Eisenhart, L.P.: Separable systems of stäckel. Ann. Math. 284–305 (1934)
Kalnins, E.G., Miller, W. Jr: Killing tensors and nonorthogonal variable separation for Hamilton–Jacobi equations. SIAM J. Math. Anal. 12, 617–629 (1981)
Klishevich, V.V.: On the existence of the second Dirac operator in Riemannian space. Class. Quantum Gravity. 17, 305–318 (2000)
Babelon, O., Bernard, D., Talon, M.: Introduction to Classical Integrable Systems. Cambridge University Press, Cambridge (2003)
Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121–130 (1974)
Marsden, J.E., Abraham, R.: Foundations of Mechanics. Addison-wesley Reading, MA (1978)
Wang, H.: Hamilton–jacobi theorems for regular reducible Hamiltonian systems on a cotangent bundle. J. Geom. Phys. 119, 82–102 (2017)
de León, M., de Diego, D.M., Vaquero, M.: Hamilton–jacobi theory, symmetries and coisotropic reduction. J. Math. Pures Appl. 107, 591–614 (2017)
McLenaghan, R.G., Tariq, N.: A new solution of the Einstein–Maxwell equations. J. Math. Phys. 16, 2306–2312 (1975)
Tariq, N., Tupper, B.A.: Class of algebraically general solutions of the Einstein–Maxwell equations for non-null electromagnetic fields. Gen. Relativ. Gravit. 6, 345–360 (1975)
Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of the Einstein Field Equations. Cambridge University Press, Cambridge (2003)
Shirokov, I.: Darboux coordinates on K-orbits and the spectra of Casimir operators on Lie groups. Theor. Math. Phys. 123, 754–767 (2000)
Kamalin, S.A., Perelomov, A.M.: Construction of canonical coordinates on polarized coadjoint orbits of Lie groups. Commun. Math. Phys. 97, 553–568 (1985)
Adams, M.R., Harnad, J., Hurtubise, J.: Darboux coordinates and Liouville–Arnold integration in loop algebras. Commun. Math. Phys. 155, 385–413 (1993)
Kirillov, A.A.: Elements of the Theory of Representations. Springer, Berlin (1976)
Kostant, B.: Quantization and Unitary Representations. Lectures in Modern Analysis and Applications III, pp 87–208. Springer, New York (1970)
Dixmier, J.: Enveloping Algebras. North-Holland Publishing Co, Amsterdam (1977)
Magazev, A.A., Mikheyev, V.V., Shirokov, I.V.: Computation of composition functions and invariant vector fields in terms of structure constants of associated Lie algebras. SIGMA 11, 066 (2015)
Baranovskii, S., Shirokov, I.: Deformations of vector fields and canonical coordinates on coadjoint orbits. Sib. Math. J. 50, 580–586 (2009)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Academic Press, Cambridge (2014)
Acknowledgements
The author expresses his deep gratitude to Igor V. Shirokov for his constant and fruitful discussions and support. Dr. S. V. Danilova is gratefully acknowledged for careful reading of the manuscript.
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Magazev, A.A. Constructing a Complete Integral of the Hamilton–Jacobi Equation on Pseudo-Riemannian Spaces with Simply Transitive Groups of Motions. Math Phys Anal Geom 24, 11 (2021). https://doi.org/10.1007/s11040-021-09385-3
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DOI: https://doi.org/10.1007/s11040-021-09385-3
Keywords
- Geodesic Hamilton–Jacobi equation
- Simply transitive action
- Lie group
- Coadjoint orbit
- Darboux coordinates