Special Symmetric Two-tensors, Equivalent Dynamical Systems, Cofactor and Bi-cofactor Systems
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Special Symmetric Two-tensors, Equivalent Dynamical Systems, Cofactor and Bi-cofactor Systems SERGIO BENENTI Department of Mathematics, University of Turin, Italy. e-mail: [email protected] Abstract. A general analysis of special classes of symmetric two-tensor on Riemannian manifolds is provided. These tensors arise in connection with special topics in differential geometry and analytical mechanics: geodesic equivalence and separation of variables. It is shown that they play an important role in the theory of correspondent (or equivalent) dynamical systems of Levi-Civita. By applying some new developments of this theory, it is shown that the recent notions of cofactor and cofactorpair systems arise in a natural way, as non-Lagrangian systems having a Lagrangian equivalent. This circumstance extends the Hamiltonian methods, including the separation of variables of the Hamilton–Jacobi equation, to a special class of nonconservative systems. In this extension the case of indefinite metrics, may occur. Hence, it is shown that also pseudo-Riemannian geometry plays an important role also in classical mechanics. Mathematics Subject Classifications (2000): Key words: Riemannian geometry, symmetric two-tensors, geodesic equivalence, dynamical equivalence, cofactor and cofactor-pair systems, separation of variable, special Stäckel systems.
1. Introduction The aim of this tutorial paper is twofold: (i) To propose a unified and general introduction to the symmetric two-tensors of special kind which, in recent years, have been employed in the variable separation theory of the Hamilton–Jacobi and Schrödinger equations, in the geodesic equivalence theory, in the study of factor and bi-cofactor (or cofactor-pair) systems, bi-Hamiltonian systems, etc. (ii) To show how the notions of cofactor system and bi-cofactor system arise in a simple and natural way from the theory of equivalent (or correspondent) dynamical systems, whose foundation is due to Levi-Civita (1896a, 1896b) and in which special symmetric two-tensors play a significant fundamental role. The leading ideas are explained in the following items. (I) A holonomic mechanical system with time-independent ideal constraints and with n degrees of freedom is characterized by a triple (Q, g, F), where Q is the n-dimensional configuration manifold covered by local Lagrangian coordinate Research sponsored by the Dept. of Mathematics, University of Turin, and by INDAM-GNFM.
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systems q = (q i ) and endowed with a metric tensor g = (gij ), representing the kinetic energy K = 12 gij q˙ i q˙ j , and where F = (F i ) is a vector field on Q representing the Lagrangian force, i.e., all the active forces acting on the system (we consider the case of time-independent and velocity-independent forces). In the present paper by dynamical system we mean a triple of this kind. As shown by Lagrange, the motions are locally represented by the solutions q(t) of the second-order differential equations d ∂K ∂K − i = Fi , i dt ∂ q˙ ∂q
. Fi = gij F j ,
(
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