Abstract
With the notion of L-pencil, based on the notion of L-tensor, we construct a new class of Stäckel systems such that the quadratic first integrals associated with the orthogonal separation are computed by a coordinate-independent algebraic process.
Research supported by G.N.F.M. (Gruppo Nazionale di Fisica Matematica) of I.N.d.A.M. (Istituto Nazionale di Alta Matematica, Roma).
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References
Benenti S., Separability in Riemannian manifolds, article submitted (2004) to Royal Society for a special issue on’ The State of the Art of the Separation of Variables’, available on the personal page of the site www.dm.unito.it.
-, Inertia tensors and Stäckel systems in the Euclidean spaces, Rend. Sem. Mat. Univ. Politec. Torino 50 (1992), no. 4, 315–341, Differential geometry (Turin, 1992).
-, Orthogonal separable dynamical systems, Differential geometry and its applications (Opava, 1992), Math. Publ., Vol. 1, Silesian Univ., Opava, 1993, pp. 163–184.
-, Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38 (1997), no. 12, 6578–6602.
-, Special symmetric two-tensors, equivalent dynamical systems, cofactor and bi-cofactor systems, Acta Appl. Math. 87 (2005), no. 1–3, 33–91.
Blaszak M., Separable systems with quadratic in momenta first integrals, J. Phys. A: Math. Gen. 38 (2005), 1667–1685.
Boyer C.P., Kalnins E.G., AND Miller W., Stäckel-equivalent integrable Hamiltonian systems, SIAM J. Math. Anal. 17 (1986), no. 4, 778–797.
Crampin M., Sarlet W., AND Thompson G., Bi-differential calculi, bi-Hamiltonian systems and conformal Killing tensors, J. Phys. A: Math. Gen. 33 (2000), 8755–8770.
Eisenhart R.P., Riemannian Geometry, Princeton University Press, Fifth printing, 1964.
Kalnins E.G., Separation of variables for Riemannian spaces of constant curvature, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 28, Longman Scientific & Technical, Harlow, 1986.
Kalnins E.G. AND Miller W., Separation of variables on n-dimensional Riemannian manifolds. I. The n-sphere \( \mathbb{S}_n \) and Euclidean n-space ℝ 1, J. Math. Phys. 27 (1986), no. 7, 1721–1736.
Okubo S., Integrable dynamical systems with hierarchy. I. Formulation, J. Math. Phys. 30 (1989), 834–843.
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Benenti, S. (2008). Algebraic Construction of the Quadratic First Integrals for a Special Class of Orthogonal Separable Systems. In: Eastwood, M., Miller, W. (eds) Symmetries and Overdetermined Systems of Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 144. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73831-4_13
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DOI: https://doi.org/10.1007/978-0-387-73831-4_13
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