// Use when solving a problem by expanding around a known solution in a small parameter, including regular and singular perturbation theory.
MCTS-based autonomous physics problem solver with arXiv search, prior knowledge retrieval, and multi-agent reasoning. Use when you need to solve physics problems, search arXiv for relevant papers, or generate structured physics solutions with iterative refinement.
Use when solving problems involving Maxwell's equations, electrostatics, magnetostatics, electromagnetic waves, radiation, or relativistic electrodynamics.
Use when applying conservation of energy, momentum, angular momentum, charge, or other conserved quantities to constrain or solve a physical system.
Use when checking dimensional consistency, estimating physical scales, or deriving functional forms via the Buckingham Pi theorem.
Use when decomposing signals or fields into frequency/momentum components, applying Fourier transforms, or using spectral methods to solve differential equations.
Use when solving ordinary or partial differential equations numerically, including choosing integrators, discretization schemes, and stability analysis.
| name | perturbation_expansion |
| description | Use when solving a problem by expanding around a known solution in a small parameter, including regular and singular perturbation theory. |
Apply this skill when the problem contains a small dimensionless parameter and the solution can be constructed order-by-order as a series expansion around a known zeroth-order solution.
Systematically expand equations and their solutions in powers of a small parameter, compute corrections order by order, and assess the validity of the expansion.
unperturbed_system: The exactly solvable zeroth-order problem (Hamiltonian, equation, potential, etc.)perturbation: The small correction termsmall_parameter: The expansion parameter (epsilon, coupling constant, etc.) and its numerical value or rangeorder: The desired order of the expansioncorrections: The perturbative corrections at each requested order (energy shifts, wavefunctions, amplitudes, etc.)expanded_solution: The full solution up to the requested ordervalidity_estimate: Estimate of the regime where the expansion is reliableIdentify the small parameter and verify it is dimensionless and numerically small.
Write the full problem as: H = H_0 + epsilon * H_1 (or analogous).
Expand the solution in powers of epsilon:
Substitute into the full equation and collect terms at each order of epsilon.
For degenerate perturbation theory:
Assess convergence: