[Paper Review] Constrained systems and the Clairaut equation

Steven Duplij|ArXiv.org|Apr 17, 2008

Algebraic and Geometric Analysis11 references3 citations

TL;DR

This paper proposes a generalized Legendre-Clairaut transformation to handle degenerate Lagrangians in constrained Hamiltonian systems by treating the Hamiltonian construction as solving a Clairaut partial differential equation. The method yields an involutive transformation that naturally generates Dirac's primary constraints without Lagrange multipliers, unifying regular and singular cases through mixed envelope-general solutions of the Clairaut equation.

ABSTRACT

An extension of the Legendre transform to non-convex functions with vanishing Hessian as a mix of envelope and general solutions of the Clairaut equation is proposed. Applying this to systems with constraints, the procedure of finding a Hamiltonian for a degenerate Lagrangian is just that of solving a corresponding Clairaut equation with a subsequent application of the proposed Legendre-Clairaut transformation. In this way the unconstrained version of Hamiltonian equations is obtained. The Legendre-Clairaut transformation presented is involutive. We demonstrate the origin of the Dirac primary constraints, along with their explicit form, and this is done without using the Lagrange multiplier method.

Motivation & Objective

• To reformulate the Hamiltonian formalism for degenerate Lagrangians using the Clairaut equation instead of the standard Legendre transform.
• To derive Dirac's primary constraints intrinsically from the structure of the Clairaut equation, avoiding the use of Lagrange multipliers.
• To extend the Legendre transformation to non-convex, singular cases via mixed solutions of the Clairaut equation (envelope in regular variables, general solution in non-regular variables).
• To establish an involutive transformation between Lagrangian and Hamiltonian formulations in constrained systems, ensuring consistency and reversibility.
• To provide a unified framework for both regular and singular systems by treating the total Hamiltonian as a mixed solution of the Clairaut equation.

Proposed method

• Formulate the Legendre transform as a solution of the Clairaut equation, where the Hamiltonian arises from the envelope solution in regular variables.
• Introduce a mixed solution of the Clairaut equation: envelope in regular momenta and general solution in non-regular (unsolved) momenta, corresponding to primary constraints.
• Define the generalized Legendre-Clairaut transform as an involutive map between Lagrangian and Hamiltonian functions, even when the Hessian vanishes.
• Use the condition $ \mathbf{v}_1 = \frac{\partial \mathcal{H}^{Cl}}{\partial \mathbf{p}_1} $ to invert the momentum mapping and reconstruct the unconstrained Lagrangian.
• Construct the mixed Hamiltonian as $ \mathcal{H}^{Cl}_{(\mathbf{q})\text{mixed}} = \mathcal{H}^{(0)}_{(\mathbf{q})}(\mathbf{p}_1) + \mathbf{C}_{2(\mathbf{q})} \cdot \boldsymbol{\Phi}_{(\mathbf{q})}(\mathbf{p}) $, where $ \mathcal{H}^{(0)} $ is independent of non-regular variables.
• Verify involutivity by showing that applying the Legendre-Clairaut transform twice recovers the original Lagrangian, under mutual invertibility of momentum mappings.

Experimental results

Research questions

• RQ1 How can the standard Legendre transformation be generalized to handle non-convex and degenerate Lagrangians with vanishing Hessian?
• RQ2 Can Dirac's primary constraints emerge naturally from the structure of the Clairaut equation without introducing Lagrange multipliers?
• RQ3 What is the role of mixed solutions (envelope + general solution) of the Clairaut equation in describing constrained Hamiltonian systems?
• RQ4 How does the proposed Legendre-Clairaut transformation preserve the dynamics of the original Lagrangian system?
• RQ5 Is the generalized Legendre-Clairaut transformation involutive, ensuring a consistent duality between Lagrangian and Hamiltonian formulations?

Key findings

• The generalized Legendre-Clairaut transformation provides a consistent method to construct a Hamiltonian for degenerate Lagrangians by solving the Clairaut equation, even when the Hessian vanishes.
• Dirac's primary constraints emerge naturally as arbitrary constants in the general solution of the Clairaut equation, without invoking Lagrange multipliers.
• The mixed solution of the Clairaut equation—envelope in regular variables and general solution in non-regular variables—yields the total Hamiltonian of a constrained system.
• The proposed Legendre-Clairaut transformation is involutive: applying it twice recovers the original Lagrangian, provided the momentum mappings are mutually invertible.
• The resulting Hamiltonian equations of motion are equivalent to the original Lagrangian equations, even when constraints are not explicitly imposed.
• The method unifies regular and singular cases under a single framework, with the standard Legendre transform corresponding to the envelope solution and the total Hamiltonian to the mixed solution.