On Lagrangian and Hamiltonian systems with homogeneous trajectories

arXiv:1003.1495v3 [math-ph] 19 Aug 2010 On Lagrangian and Hamiltonian systems with homogeneous trajectories G´abor Zsolt T´oth KFKI Research Institute for Particle and Nuclear Physics, Hungarian Academy of Sciences, P.O.B. 49, 1525 Budapest, Hungary email: tgzs@rmki.kfki.hu, tgzs@cs.elte.hu Abstract Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homoge- neous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of La- grangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solution of the Euler-Lagrange or Hamiltonian equations. In particular, we generalize the ‘geodesic lemma’ known in Riemannian geometry to Lagrangian and Hamiltonian systems. We present results on the existence of homogeneous trajectories of Lagrangian systems. We study Hamiltonian and Lagrangian g.o. spaces, i.e. homogeneous spaces G/H with G-invariant Lagrangian or Hamiltonian functions on which every solution of the equa- tions of motion is homogeneous. We show that the Hamiltonian g.o. spaces are related to the functions that are invariant under the coadjoint action of G. Riemannian g.o. spaces thus corre- spond to special Ad∗(G)-invariant functions. An Ad∗(G)-invariant function that is related to a g.o. space also serves as a potential for the mapping called ‘geodesic graph’. As illustration we discuss the Riemannian g.o. metrics on SU(3)/SU(2). Keywords: g.o. space, homogeneous space, relative equilibrium, momentum map, La- grangian and Hamiltonian systems with symmetry PACS numbers: 45.20.Jj, 02.40.Ky, 02.40.Ma 1 Introduction Let M be a Riemannian manifold. A geodesic in M is called homogeneous if it is the orbit of a one-parameter group of isometries of M. A homogeneous Riemannian manifold M = G/K, where G is a connected Lie group and K is a closed subgroup, is a geodesic orbit (g.o.) space with respect to G, if every geodesic in it is the orbit of a one-parameter subgroup of G. The homogeneous space M = G/K is called a reductive space, if there exists a direct sum decomposition (called reductive decomposition) g = m⊕k of the Lie algebra of G, where m is an ad(K)-invariant linear subspace of g and k is the Lie algebra of K. It is known that all Riemannian homogeneous spaces are reductive. If M = G/K is Riemannian and there exists a reductive decomposition g = m ⊕k such that each geodesic in M starting at the origin o ∈M is an orbit of a one-parameter subgroup of G generated by some element of m, then M is called a naturally reductive space with respect to G, and m is called a natural complement. The origin o is the image of K by the canonical projection G →G/K. Obviously, every naturally reductive space is a g.o. space as well. It was believed some decades ago that the converse is also true, i.e. every g.o. space is isometric to some naturally reductive space. A counter example, however, was found by A. Kaplan [1], initiating the extensive study of g.o. spaces [3]-[21]. Pseudo-Riemannian g.o. spaces were also investigated recently [22, 23, 24]. Before Kaplan’s example appeared, J. Szenthe discovered a geometrical background for the situation when a g.o. space is not naturally reductive [2], not knowing whether such a situation can be realized or not. This result had considerable influence on the later studies. In general, it is possible that a homogeneous Riemannian space M = G/K is not naturally reductive with respect to G, but one can take other groups G′ and K′ so that M = G′/K′ and M is naturally reductive with respect to G′. The same situation can occur for g.o. spaces as well. It is also possible in some cases that a g.o. space can be made naturally reductive by taking a different symmetry group G′, but there also exist g.o. spaces for which this is not possible, i.e. which are in no way naturally reductive. Kaplan’s example is of the latter type. Since Riemannian (and pseudo-Riemannian) manifolds can be viewed as a special class of the manifolds with a Lagrangian or Hamiltonian function, it is interesting to consider the generalization of the g.o. property to homogeneous spaces with invariant Lagrangian and Hamiltonian functions and to ask whether the known results for the Riemannian spaces can be generalized, and whether the techniques of Lagrangian or Hamiltonian dynamics can be used for the study of Riemannian g.o. spaces. In this paper we present the results that we obtained in relation to these questions. A subject closely related to the study of g.o. spaces is the characterization of the homo- geneous geodesics in Riemannian manifolds. Homogeneous geodesics are of interest also in Finsler geometry, pseudo-Riemannian geometry and in dynamics. We refer the reader to [25]-[41] and further references therein. The present paper is also concerned with the characterization of homogeneous trajectories in Lagrangian and Hamiltonian dynamical systems, partly because this is necessary for

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