[Paper Review] Tulczyjew's triples and lagrangian submanifolds in classical field theories

Manuel de León, David Martı́n de Diego+1 more|arXiv (Cornell University)|Feb 11, 2003

Homotopy and Cohomology in Algebraic Topology16 references20 citations

TL;DR

This paper extends Tulczyjew's symplectic framework for classical mechanics to classical field theories using multisymplectic geometry. It introduces a generalized Tulczyjew triple for field theories via jet prolongations and multisymplectic forms, showing that the Euler-Lagrange and Hamilton–de Donder equations arise as local equations defining (n+1)-dimensional Lagrangian submanifolds in a multisymplectic manifold, thereby unifying the Lagrangian and Hamiltonian formulations in a geometric framework.

ABSTRACT

In this paper the notion of Tulczyjew's triples in classical mechanics is extended to classical field theories, using the so-called multisymplectic formalism, and a convenient notion of lagrangian submanifold in multisymplectic geometry. Accordingly, the dynamical equations are interpreted as the local equations defining these lagrangian submanifolds.

Motivation & Objective

• To generalize Tulczyjew's symplectic framework for classical mechanics to classical field theories.
• To develop a geometric formulation of classical field theories using multisymplectic geometry and jet prolongations.
• To interpret the Euler-Lagrange and Hamilton–de Donder equations as defining Lagrangian submanifolds in a multisymplectic setting.
• To establish a correspondence between Lagrangian and Hamiltonian formulations via a generalized Tulczyjew triple in field theory.

Proposed method

• Use the multisymplectic formalism to generalize the canonical symplectic structures of classical mechanics to field theories.
• Construct the Poincaré–Cartan (n+1)-form Ω_L from a Lagrangian density Lη on the 1-jet bundle Z.
• Define canonical diffeomorphisms α̃: J̃¹Z* → Λⁿ⁺¹₂Z and β̃: J̃¹Z* → Λⁿ⁺¹₂Z* using jet prolongations and quotienting by divergence equivalence.
• Introduce a multisymplectic structure on J̃¹Z* via the pullback of the canonical form on Λⁿ⁺¹₂Z*, leading to a multisymplectic manifold (J̃¹Z*, Ω_β).
• Represent solutions of the De Donder and field equations as (n+1)-dimensional Lagrangian submanifolds in this multisymplectic space.
• Establish equivalence between the Lagrangian and Hamiltonian formulations via the Legendre transformation and the identity Ω_α = Ω_β.

Experimental results

Research questions

• RQ1 How can Tulczyjew's triple construction in classical mechanics be extended to classical field theories?
• RQ2 What is the appropriate multisymplectic geometric structure that encodes field dynamics in a way analogous to symplectic structures in mechanics?
• RQ3 How do the Euler-Lagrange and Hamilton–de Donder equations emerge as defining equations of Lagrangian submanifolds in the multisymplectic framework?
• RQ4 What is the role of the jet prolongation and quotienting by divergence in constructing a consistent field-theoretic Tulczyjew triple?
• RQ5 How are the Lagrangian and Hamiltonian formulations related in this geometric framework?

Key findings

• The dynamical equations of classical field theories are geometrically encoded as the local defining equations of (n+1)-dimensional Lagrangian submanifolds in a multisymplectic manifold constructed via the generalized Tulczyjew triple.
• The De Donder and field equations are shown to be equivalent to the condition that the image of a horizontal section under the projection map lies in a Lagrangian submanifold of (J̃¹Z*, Ω_β).
• The canonical diffeomorphism β̃: J̃¹Z* → Λⁿ⁺¹₂Z* maps solutions of the field equations to the zero section of Λⁿ⁺¹₂Z*, confirming the Lagrangian submanifold structure.
• The multisymplectic forms Ω_α and Ω_β on J̃¹Z* are shown to be equal, establishing a geometric unification of the Lagrangian and Hamiltonian formulations.
• The Legendre transformation connects the Lagrangian and Hamiltonian formulations, and under regularity, the corresponding Lagrangian submanifolds N_L and N_h coincide.