[Paper Review] Geodesic equivalence and integrability

Peter Topalov, Vladimir S. Matveev|arXiv (Cornell University)|Nov 10, 1999

Homotopy and Cohomology in Algebraic Topology4 references4 citations

TL;DR

This paper introduces a construction that generates integrals of motion for Hamiltonian systems from trajectorial diffeomorphisms, with a primary application to geodesically equivalent metrics. It proves that the existence of a non-trivial geodesically equivalent metric implies Liouville integrability and provides explicit formulae for the associated integrals, establishing a deep link between metric equivalence and dynamical systems integrability.

ABSTRACT

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.

Motivation & Objective

• To establish a general construction that produces integrals of motion from trajectorial diffeomorphisms between Hamiltonian systems.
• To investigate the implications of geodesic equivalence between Riemannian metrics for the integrability of the corresponding geodesic flows.
• To provide explicit formulae for integrals of motion arising from geodesic equivalence.
• To demonstrate that the existence of a non-trivial geodesically equivalent metric leads to Liouville integrability of the geodesic flow.
• To bridge differential geometry and integrable systems by connecting metric equivalence to the existence of conserved quantities.

Proposed method

• Utilizes trajectorial diffeomorphisms—diffeomorphisms that preserve the orbits of Hamiltonian vector fields up to reparametrization.
• Applies the construction to the case of geodesic flows on Riemannian manifolds to derive conserved quantities from geodesic equivalence.
• Derives explicit formulae for integrals of motion using the Jacobian and Hessian of the generating diffeomorphism.
• Relies on symplectic geometry and the structure of Hamiltonian systems to ensure the integrals are in involution.
• Employs the notion of Liouville integrability, requiring the existence of sufficiently many Poisson-commuting integrals.
• Uses the framework of differential geometry and the theory of integrable systems to analyze the spectral and dynamical consequences of metric equivalence.

Experimental results

Research questions

• RQ1 How can a trajectorial diffeomorphism between two Hamiltonian systems be used to generate conserved quantities?
• RQ2 What dynamical consequences arise when two Riemannian metrics are geodesically equivalent?
• RQ3 Under what conditions does geodesic equivalence imply Liouville integrability of the geodesic flow?
• RQ4 Can explicit formulae for integrals of motion be derived from the geodesic equivalence of metrics?
• RQ5 What is the relationship between the geometric structure of geodesically equivalent metrics and the algebraic structure of integrable systems?

Key findings

• A non-trivial geodesically equivalent metric to a given Riemannian metric implies the Liouville integrability of the geodesic flow on the manifold.
• The construction yields explicit formulae for the integrals of motion in terms of the Jacobian and Hessian of the diffeomorphism mapping geodesics of one metric to those of another.
• The integrals obtained are in involution, satisfying the conditions for Liouville integrability.
• The method applies generally to any pair of trajectorially diffeomorphic Hamiltonian systems, not only geodesic flows.
• The results establish a direct geometric mechanism for generating integrals from metric equivalence, enriching the theory of integrable systems.
• The framework provides a systematic way to construct integrals from geometric data, offering new tools for studying the dynamics of Riemannian manifolds.