variational-methods

name	variational_methods
description	Use when applying variational principles such as Lagrangian/Hamiltonian mechanics, the Rayleigh-Ritz method, or variational estimation of ground-state energies.

Variational Methods

Apply this skill when the problem can be formulated as an extremization of a functional (action, energy, entropy, etc.), or when an approximate solution is sought by optimizing a trial ansatz.

Goal

Formulate the problem variationally, derive the governing equations (Euler-Lagrange, Hamilton's equations), or obtain upper/lower bounds on physical quantities via trial functions.

Scope

• Lagrangian and Hamiltonian classical mechanics
• Euler-Lagrange equations for fields and particles
• Hamilton's principle (least action)
• Rayleigh-Ritz method for eigenvalue problems
• Variational estimation of quantum ground-state energy
• Calculus of variations with constraints (Lagrange multipliers)

Inputs

• system_description: The physical system (particles, fields, continua)
• lagrangian_or_functional: The Lagrangian, action, energy functional, or other quantity to extremize
• constraints: Any holonomic or non-holonomic constraints
• trial_function: (For Rayleigh-Ritz) A parameterized trial ansatz

Outputs

• equations_of_motion: Euler-Lagrange or Hamilton's equations derived from the variational principle
• optimized_parameters: (For Rayleigh-Ritz) Optimal values of trial parameters
• bound_estimate: Upper bound on ground-state energy or extremal value of the functional
• conserved_quantities: Quantities conserved by virtue of symmetries (via Noether's theorem)

Workflow

1. Identify the appropriate functional to extremize.

   • For mechanics: the action S = integral of L dt.
   • For quantum ground states: the energy functional E[psi] = <psi|H|psi> / <psi|psi>.

2. Choose generalized coordinates and identify constraints.

   • Incorporate constraints via Lagrange multipliers or by reducing degrees of freedom.

3. Derive the Euler-Lagrange equations.

   • delta S / delta q_i = 0 gives: d/dt (partial L / partial q_dot_i) - partial L / partial q_i = 0.
   • For field theories: partial_mu (partial L / partial (partial_mu phi)) - partial L / partial phi = 0.

4. (Rayleigh-Ritz) Substitute the trial function into the functional.

   • Minimize with respect to all variational parameters.
   • The result provides an upper bound on the true ground-state energy.

5. Identify conserved quantities.

   • For each continuous symmetry, apply Noether's theorem to derive the conserved current/charge.

6. Solve the resulting equations analytically or numerically.

Quality Checks

• The Euler-Lagrange equations must be consistent with known results (e.g., Newton's second law for simple mechanical systems).
• Rayleigh-Ritz always yields an upper bound; the variational energy must be >= the true ground-state energy.
• The number of Euler-Lagrange equations must equal the number of generalized coordinates.
• Conserved quantities should be verified by showing their time derivative vanishes on-shell.

Constraints

• The Rayleigh-Ritz method gives only an upper bound; do not claim it gives the exact energy unless the trial space contains the exact solution.
• For dissipative systems, the standard Lagrangian formulation may not apply; use Rayleigh dissipation functions or other extensions if needed.
• Non-holonomic constraints cannot always be incorporated via Lagrange multipliers in the action; use d'Alembert's principle or other appropriate methods.