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Revisiting Legendre transformations in Finsler geometry

Ernesto Rodrigues, Iarley P. Lobo

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).

Revisiting Legendre transformations in Finsler geometry

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).
Paper Structure (5 sections, 4 theorems, 76 equations, 2 figures)

This paper contains 5 sections, 4 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Obtaining the Finsler Function
  3. Obtaining the Lagrangian from the Finsler Function
  4. Concluding remarks
  5. The Lagrange Multiplier

Key Result

Theorem 6

Let $S\subset T^*M$ be a subbundle of $T^*M$ such that in each of its fibres, as a hypersurface of a cotangent space, no two points have parallel tangent spaces. Assume that its function $P$ defined by Eqs.DefPT1 and DefPT2 is differentiable. Then $S$ naturally has a Lagrangian defined on it, given

Figures (2)

  • Figure 1: Square of the parabola indicatrix $F^2(u,v)=1$ for $c=0$ (parabola, blue, thick line), $c=5$ (two parallel hyperbolas, orange, dashed line) and $c=-5$ (ellipse, green, dotdashed line).
  • Figure 2: Square of the hyperbolic paraboloid indicatrix $F^2(u,v,w)=1$ for $c=5$.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4: Finsler Function
  • Definition 5: Geodesic of a Finsler Function
  • Theorem 6
  • Theorem 7
  • Definition 8
  • Theorem 9: Uniqueness and existence of $S^*$
  • Example 10: Parabola
  • ...and 4 more