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Nonlocal Mechanics

Carlos Heredia, Josep Llosa

TL;DR

This work delivers a comprehensive Hamiltonian formulation for nonlocal Lagrangian systems that does not rely on infinite-derivative expansions. It develops a trajectory-based variational principle, a generalized Noether theorem, and a presymplectic structure on an extended kinematic space, then extracts a proper phase space by enforcing EL equations as constraints. The approach yields a functional, nonlocal analog of the Legendre transformation and provides explicit Hamiltonians and canonical coordinates for several models, including a finite nonlocal oscillator, the fully nonlocal Pais-Uhlenbeck system, and a delayed oscillator, all without solving the EL equations explicitly. By clarifying the role of constraints and offering a practical toolkit for nonlocal dynamics, the framework paves a direct route to quantization and extends standard local techniques to nonlocal and field-theoretic contexts.

Abstract

We introduce a Hamiltonian framework for nonlocal Lagrangian systems without relying on infinite-derivative expansions. Starting from a (trajectory-based) variational principle and a generalized Noether theorem, we define the canonical momenta and energy. Moreover, we construct a (pre)symplectic form on the kinematic space, and show that its restriction to the phase space (by implementing the constraints) yields a true (pre)symplectic structure encoding the dynamics. Three examples -- a finite nonlocal oscillator, the fully nonlocal Pais-Uhlenbeck model, and a delayed harmonic oscillator -- demonstrate how phase space and the Hamiltonian emerge without explicitly solving the Euler-Lagrange equations.

Nonlocal Mechanics

TL;DR

This work delivers a comprehensive Hamiltonian formulation for nonlocal Lagrangian systems that does not rely on infinite-derivative expansions. It develops a trajectory-based variational principle, a generalized Noether theorem, and a presymplectic structure on an extended kinematic space, then extracts a proper phase space by enforcing EL equations as constraints. The approach yields a functional, nonlocal analog of the Legendre transformation and provides explicit Hamiltonians and canonical coordinates for several models, including a finite nonlocal oscillator, the fully nonlocal Pais-Uhlenbeck system, and a delayed oscillator, all without solving the EL equations explicitly. By clarifying the role of constraints and offering a practical toolkit for nonlocal dynamics, the framework paves a direct route to quantization and extends standard local techniques to nonlocal and field-theoretic contexts.

Abstract

We introduce a Hamiltonian framework for nonlocal Lagrangian systems without relying on infinite-derivative expansions. Starting from a (trajectory-based) variational principle and a generalized Noether theorem, we define the canonical momenta and energy. Moreover, we construct a (pre)symplectic form on the kinematic space, and show that its restriction to the phase space (by implementing the constraints) yields a true (pre)symplectic structure encoding the dynamics. Three examples -- a finite nonlocal oscillator, the fully nonlocal Pais-Uhlenbeck model, and a delayed harmonic oscillator -- demonstrate how phase space and the Hamiltonian emerge without explicitly solving the Euler-Lagrange equations.

Paper Structure

This paper contains 25 sections, 1 theorem, 235 equations.

Table of Contents

  1. Introduction
  2. Standard Lagrangian systems. An outline
  3. Noether Theorem
  4. Hamiltonian formalism
  5. Higher-order Lagrangian systems. An outline
  6. Noether theorem for higher-order Lagrangian systems
  7. Hamiltonian formalism
  8. Infinite-order Lagrangians
  9. The phase space
  10. Some examples
  11. Noether theorem for infinite-order Lagrangians
  12. Hamiltonian formalism
  13. An inconsistency
  14. Nonlocal Lagrangians
  15. The variational principle
  16. ...and 10 more sections

Key Result

Theorem 1

Let the infinitesimal transformation nl6 be a symmetry of the nonlocal action, in the sense that it preserves the Lagrangian. Then, the associated conserved quantity for dynamical trajectories is where $p(q,\xi)$ is defined in nl9c and $E(q)$ is the energy functional given in nl14.

Theorems & Definitions (1)

  • Theorem 1