[Paper Review] Legendre transformation for regularizable Lagrangians in field theory

Olga Krupková, Dana Smetanová|ArXiv.org|Nov 1, 2001

Geophysics and Gravity Measurements10 references3 citations

TL;DR

This paper introduces a generalized Legendre transformation for regularizable Lagrangians in field theory using Lepagean equivalents, enabling Hamiltonian formulations for singular Lagrangians—such as those of the Dirac and electromagnetic fields—that are typically non-regular under standard theory. The key contribution is a consistent, constraint-free Hamiltonian framework for affine and quadratic Lagrangians in first-order field theories.

ABSTRACT

Hamilton equations based not only upon the Poincare--Cartan equivalent of a first-order Lagrangian, but rather upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton--De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed, and Hamilton equations, equivalent with the Euler--Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.

Motivation & Objective

• To develop a Legendre transformation for Lagrangians that are singular under standard regularity conditions but regularizable via Lepagean equivalents.
• To establish Hamilton equations (specifically Hamilton $p_2$-equations) equivalent to the Euler–Lagrange equations for such regularizable Lagrangians.
• To demonstrate that all Lagrangians affine or quadratic in first derivatives are regularizable, avoiding constraints in the Hamiltonian formulation.
• To provide explicit momenta and Hamiltonians for the Dirac and electromagnetic fields using this generalized approach.
• To show that under certain conditions, the Hamilton $p_2$-equations coincide with standard Hamilton equations of an equivalent Lagrangian.

Proposed method

• Uses Lepagean equivalents of Lagrangians that are at most 2-contact forms, extending beyond the Poincaré–Cartan form.
• Applies the Hamilton $p_2$-equations, a generalized form of Hamiltonian dynamics, derived from the Lepagean equivalent instead of the standard Poincaré–Cartan form.
• Introduces a regularity condition based on the non-degeneracy of the matrix $\left(\frac{\partial^2 L}{\partial y^\sigma_i \partial y^\nu_k} + \frac{\partial^2 L}{\partial y^\sigma_k \partial y^\nu_i}\right)$, which generalizes the standard Hessian condition.
• Derives the Legendre transformation by inverting the momentum map defined via the generalized Hessian, ensuring invertibility for affine and quadratic Lagrangians.
• Constructs the Hamiltonian in Legendre coordinates using the inverse transformation of momenta, ensuring equivalence with the Euler–Lagrange equations.
• Applies the formalism to the Dirac field and electromagnetic field, computing explicit momenta and Hamiltonians, and verifies equivalence with known field equations.

Experimental results

Research questions

• RQ1 Can a Legendre transformation be consistently defined for Lagrangians that are singular under the standard regularity condition but regularizable via Lepagean equivalents?
• RQ2 Do Hamilton $p_2$-equations derived from Lepagean forms yield equations equivalent to the Euler–Lagrange equations for singular but regularizable Lagrangians?
• RQ3 Are all Lagrangians affine or quadratic in first derivatives regularizable under this generalized framework, and do they admit a constraint-free Hamiltonian formulation?
• RQ4 Can the generalized Legendre transformation recover standard Hamiltonian structures for specific physical fields like the Dirac and electromagnetic fields?
• RQ5 Under what conditions do the Hamilton $p_2$-equations coincide with the standard Hamilton equations of an equivalent Lagrangian?

Key findings

• All Lagrangians that are affine or quadratic in the first derivatives of field variables are regularizable under the proposed framework, as the generalized Hessian matrix is non-degenerate.
• For the electromagnetic field Lagrangian $L = \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, the matrix $\left(\frac{\partial^2 L}{\partial y^\sigma_i \partial y^\nu_k} + \frac{\partial^2 L}{\partial y^\sigma_k \partial y^\nu_i}\right)$ is non-degenerate, confirming regularizability.
• The momenta for the electromagnetic field are given by $p^\mu_\nu = 4\partial_\nu A^\mu - 3\partial^\mu A^\nu$ for $\mu \ne \nu$, and $p^\mu_\mu = 4\sum_{\alpha \ne \mu} \partial_\alpha A^\mu$, with the inverse transformation explicitly computed.
• The resulting Hamiltonian for the electromagnetic field in Legendre coordinates is $H = -\frac{1}{12}\sum (p^\mu_\mu)^2 - \frac{1}{16}\sum (p^\mu_\nu)^2 - \frac{3}{8}\sum p^\mu_\nu p^\nu_\mu$, with additional terms from the dedonderization.
• For the Dirac field in $X = \mathbb{R}^2$, the momenta are $p^1_1 = 4y^2_2$, $p^1_2 = -3y^1_2 + y^2_1$, $p^2_2 = 4y^1_1$, $p^2_1 = y^1_2 - 3y^2_1$, and the Hamiltonian is $H = \frac{1}{4}p^1_1 p^2_2 - \frac{3}{8}p^1_2 p^2_1 - \frac{1}{16}(p^1_2)^2 - \frac{1}{16}(p^2_1)^2$.
• The Hamilton $p_2$-equations for the electromagnetic field are equivalent to the Maxwell equations, confirming consistency with standard field theory.