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

Mathematical Notes

https://doi.org/10.1007/BF02676443

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Abstract

The classification problem for tensor invariants of geodesic flows on compact surfaces is considered. Such invariants include first integrals, multivalued integrals, and symmetry fields. The problem is solved for tensor invariants of arbitrary degrees. It is shown that the invariants under consideration can vanish at some point only in the presence of two-valued symmetry fields and multivalued integrals.

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.

A note on the relation between joint and differential invariants

Lobachevskii Journal of Mathematics, 2016

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allow to compute differential invariants from joint invariants.

Time-dependent Lagrangians invariant by a vector field

Letters in Mathematical Physics, 2001

We study the reduction of non-autonomous regular Lagrangian systems by symmetries, which are generated by vector fields associated with connections in the configuration bundle of the system Q × R → R. These kind of symmetries generalize the idea of "time-invariance" (which corresponds to taking the trivial connection in the above trivial bundle).

Vector fields with distributions and invariants of ODEs

Journal of Geometric Mechanics, 2013

We study pairs (X, V) where X is a vector field on a smooth manifold M and V ⊂ T M is a vector distribution, both satisfying certain regularity conditions. We construct basic invariants of such objects and solve the equivalence problem. For a given pair (X, V) we construct a canonical connection on a certain frame bundle. The results are applied to the problem of timescale preserving equivalence of ordinary differential equations. The framework of pairs (X, V) is shown to include sprays, Hamiltonian systems, Veronese webs and other structures.

Integrable Systems In N-Dimensional Riemannian Geometry

Arxiv preprint math/0301212, 2003

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an n-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair in on+1 with the vector modified Korteweg-De Vries equation (vmKDV) ut = uxxx + 3 2 ||u|| 2 ux as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the natural or parallel frame and we show how this can be gauged by a generalized Hasimoto transformation to the (usual) Frenêt frame. If one chooses the curvature to be zero, as is usual in the context of integrable systems, then one loses information unless one works in the natural frame.

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References (6)

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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.

Integrable Systems in Three-Dimensional Riemannian Geometry

Journal of Nonlinear Science, 2002

In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable.

Evolution of curvature invariants and lifting integrability

Journal of Geometry and Physics, 2006

Given a geometry defined by the action of a Lie-group on a flat manifold, the Fels-Olver moving frame method yields a complete set of invariants, invariant differential operators, and the differential relations, or syzygies, they satisfy. We give a method that determines, from minimal data, the differential equations the frame must satisfy, in terms of the curvature and evolution invariants that are associated to curves in the given geometry. The syzygy between the curvature and evolution invariants is obtained as a zero curvature relation in the relevant Lie-algebra. An invariant motion of the curve is uniquely associated with a constraint specifying the evolution invariants as a function of the curvature invariants. The zero curvature relation and this constraint together determine the evolution of curvature invariants.

The Noether theorem for geometric actions and area preserving diffeomorphisms on the torus

Physics Letters B, 1990

We find that within the formalism of coadjoint orbits of the infinite dimensional Lie group the Noether procedure leads, for a special class of transformations, to the constant of motion given by the fundamental group one-cocycle S. Use is made of the simplified formula giving the symplectic action in terms orS and the Maurer-Cartan one-form. The area preserving diffeomorphisms on the torus T 2 = S 1 ~ Sl constitute an algebra with central extension, given by the Floratos-Iliopoulos coeycle. We apply our general treatment based on the symplectic analysis of coadjoint orbits of Lie groups to write the symplectic action for this model and study its invariance. We find an interesting abelian symmetry structure of this non-linear problem.