Tensor invariants of natural mechanical systems on compact surfaces and the corresponding integrals (original) (raw)

Commutative conservation laws for geodesic flows of metrics admitting projective symmetry

Mathematical Research Letters, 2002

We prove that the geodesic flow of a pseudo-Riemannian metric g that admits a "nontrivial" projective symmetry X is completely integrable. Nontriviality condition of the projective symmetry is expressed in the terms of the invariants of the pair forms g and L X g, where L X denotes the Lie derivative with respect to the vector field X. The theorem we propose can be considered as a "commutative" analog of the Noether theorem.

A class of integrable geodesic flows on the symplectic group and the symmetric matrices

2005

This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Frobenius metric is an integrable system that is not contained in the Mishchenko-Fomenko class of rigid body metrics. This system may be expressed as a flow on symmetric matrices and is bi-Hamiltonian. This analysis is extended to cover flows on symmetric matrices when an isomorphism * Research partially supported by the NSF. † Research partially supported by the California Institute of Technology and NSF. ‡ Research partially supported by the Swiss NSF. 1 Introduction 2 with the symplectic Lie algebra does not hold. The two Poisson structures associated with this system, including an analysis of its Casimirs, are completely analyzed. Since the system integrals are not generated by its Casimirs it is shown that the nature of integrability is fundamentally different from that exhibited in the Mischenko-Fomenko setting.

A Generalization of The Poincare-Cartan Integral Invariant

Based on the d'Alembert-Lagrange-Poincaré variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincaré-Hamilton equations, and study a version of corresponding Poincaré-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincaré variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain line integral under the motion of a mechanical system of the type considered characterizes the Poincaré-Hamilton equations as underlying equations of the motion. As a special case, an invariant analogous to Poincaré linear integral invariant is obtained.

Integral invariants for non-barotropic flows in a four dimensional space time manifold

Physics Letters A, 2019

Properties of non-barotropic flows are described using Lie derivatives of differential forms in a Euclidean four dimensional space-time manifold. Vanishing of the Lie derivative implies that the corresponding physical quantity remains invariant along the integral curves of the flow. Integral invariants of nonbarotropic perfect and viscous flows are studied using the concepts of relative and absolute invariance of forms. The four dimensional expressions for the rate of change of the generalized circulation, generalized vorticity flux, generalized helicity and generalized parity in the case of ideal and viscous non-barotropic flows are thereby obtained.

Symmetries and conserved quantities in geodesic motion

Journal of Mathematical Physics, 1986

Recently obtained results linking several constants of motion to one (non-Noetherian) symmetry to the problem of geodesic motion in Riemannian space-times are applied. The construction of conserved quantities in geodesic motion as well as the deduction of geometrical statements about Riemannian space-times are achieved.

Geodesic equivalence and integrability

Geometriae Dedicata

We suggest a construction that, given a trajectorial diffeomorphism between two Hamiltonian systems, produces integrals of them. As the main example we treat geodesic equivalence of metrics. We show that the existence of a non-trivially geodesically equivalent metric leads to Liouville integrability, and present explicit formulae for integrals. 2 3ḡ (ξ, ξ) is an integral of the geodesic flow of the metric g.

On the generally invariant lagrangians for the metric field and other tensor fields

International Journal of Theoretical Physics, 1978

The Krupka and Trautman method for the description of all generally invariant functions of the components of geometrical object fields is applied to the invariants of second degree of the metrical field and other tensor fields. The complete system of differential identities fulfilled by the invariants mentioned is found and it is proved that these invariants depend on the tensor quantities only.

Double Bracket Equations and Geodesic Flows on Symmetric Spaces

Communications in Mathematical Physics, 1997

In this paper we consider the geometry of Hamiltonian flows on the cotangent bundle of coadjoint orbits of compact Lie groups and on symmetric spaces. A key idea here is the use of the normal metric to define the kinetic energy. This leads to Hamiltonian flows of the double bracket type. We analyze the integrability of geodesic flows according to the method of Thimm. We obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity. We demonstrate for example integrability of the geodesic flow on the real and complex Grassmannians. We also consider right invariant systems and the generalized rigid body equations in this setting.

Jacobi vector fields of integrable geodesic flows

1997

We show that an invariant surface allows to construct the Jacobi vector field along a geodesic and construct the formula for the normal component of the Jacobi field. If a geodesic is the transversal intersection of two invariant surfaces (such situation we have, for example, if the geodesic is hyperbolic), then we can construct a fundamental solution of the the

Geometry and Dynamical Systems

2013

For a compact real analytic symplectic manifold (M, ω) we describe an approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and corresponding geodesics on the space of Kähler metrics. In this approach, motivated by recent work on quantization, the complexified Hamiltonian flows act, through the Gröbner theory of Lie series, on the sheaf of complex valued real analytic functions, changing the sheaves of holomorphic functions. This defines an action on the space of (equivalent) complex structures on M and also a direct action on M. This description is related to the approach of [BLU] where one has an action on a complexification M C of M followed by projection to M. Our approach allows for the study of some Hamiltonian functions which are not real analytic. It also leads naturally to the consideration of continuous degenerations of diffeomorphisms and of Kähler structures of M. Hence, one can link continuously (geometric quantization) real, and more general non-Kähler, polarizations with Kähler polarizations. This corresponds to the extension of the geodesics to the boundary of the space of Kähler metrics. Three illustrative examples are considered. We find an explicit formula for the complex time evolution of the Kähler potential under the flow. For integral symplectic forms, this formula corresponds to the complexification of the prequantization of Hamiltonian symplectomorphisms. We verify that certain families of Kähler structures, which have been studied in geometric quantization, are geodesic families.