[Paper Review] Revisiting Legendre transformations in Finsler geometry

Ernesto Rodrigues, Iarley P. Lobo|arXiv (Cornell University)|Aug 24, 2022

Advanced Differential Geometry Research4 citations

TL;DR

This paper establishes a rigorous, coordinate-free framework for Legendre transformations between Hamiltonian mechanics and Finsler geometry, proving necessary and sufficient conditions for mapping Hamiltonian hypersurfaces in the cotangent bundle to Finsler functions on the tangent bundle such that their geodesic trajectories coincide. The key contribution is a one-to-one correspondence between compatible Hamiltonians and families of Finsler functions, with explicit characterization via Minkowski functionals and immersion conditions.

ABSTRACT

We discuss the conditions for mapping the geometric description of the kinematics of particles that probe a given Hamiltonian in phase space to a description in terms of Finsler geometry (and vice-versa).

Motivation & Objective

• To clarify the mathematical conditions under which a Hamiltonian in phase space can be mapped to a Finsler geometry description via Legendre transformation.
• To identify necessary and sufficient conditions for a Finsler function to arise from a given Hamiltonian, ensuring that their trajectories (geodesics/trajectories) coincide.
• To provide a coordinate-free, intrinsic differential geometric treatment of the Legendre transformation linking Hamiltonian and Finsler structures.
• To characterize the role of Minkowski functionals and their induced hypersurfaces (indicatrices) in establishing the duality between Hamiltonian and Finsler formulations.
• To establish the uniqueness and existence of the Finsler function induced by a Hamiltonian, under regularity and embedding conditions on the associated map.

Proposed method

• Uses the Lagrange multiplier method in infinite-dimensional calculus of variations to derive extremal curves on submanifolds of the cotangent bundle.
• Applies the theory of Minkowski functionals to define Finsler functions from hypersurfaces in the tangent bundle, ensuring homogeneity and positive definiteness.
• Introduces a map φ from the indicatrix of the Hamiltonian to the unit sphere in the tangent bundle, using the differential of the Hamiltonian to define a vector field.
• Proves that the induced Finsler function L is recovered from the image of the indicatrix under φ via the relation |L(v)| = [P(v)·v]^2, where P is the projection map.
• Establishes that the Legendre transformation is an isomorphism between the family of Hamiltonians and the family of Finsler functions when the map φ is an embedding.
• Employs differential geometry tools such as tangent spaces, normal vectors, and immersion theory to verify that the Finsler structure is well-defined and unique up to sign.

Experimental results

Research questions

• RQ1 Under what conditions can a Hamiltonian function on the cotangent bundle be Legendre-transformed into a Finsler function on the tangent bundle such that their geodesic trajectories coincide?
• RQ2 What geometric and analytic properties must a Finsler function possess to be induced by a Hamiltonian, and vice versa?
• RQ3 How can the Legendre transformation between Hamiltonian and Finsler structures be formulated in a coordinate-free, intrinsic manner?
• RQ4 What role do Minkowski functionals and their associated indicatrices play in establishing the duality between Hamiltonian and Finsler formulations?
• RQ5 What are the necessary and sufficient conditions for the existence and uniqueness of the Finsler function derived from a given Hamiltonian?

Key findings

• A Hamiltonian induces a Finsler function if and only if the associated map φ from the Hamiltonian's indicatrix to the tangent bundle is an embedding, ensuring a well-defined and unique Finsler structure.
• The Finsler function L is fully recovered from the image of the indicatrix under φ via the relation |L(v)| = [P(v)·v]^2, where P is the projection from the image to the tangent space.
• The trajectories of the Hamiltonian (defined by the Hamilton-Jacobi equations) coincide exactly with the geodesics of the induced Finsler metric, establishing physical equivalence.
• The Finsler function is uniquely determined by the Hamiltonian up to a sign, and the set of all such Finsler functions forms a one-parameter family indexed by mass, consistent with particle trajectories.
• The construction is valid for all v ∈ Λ \− L⁻¹(0), which is dense in the tangent bundle, and continuity ensures full recovery of L across the entire space.
• The condition that v is normal to the tangent space of the image surface S* at w is proven via differentiation of the constraint L(β(t)) = ±1 along a curve β in the indicatrix, confirming the geometric consistency of the transformation.