conservation-laws

name	conservation_laws
description	Use when applying conservation of energy, momentum, angular momentum, charge, or other conserved quantities to constrain or solve a physical system.

Conservation Laws

Apply this skill when a problem involves identifying conserved quantities, using them to reduce degrees of freedom, or constraining kinematics and dynamics via conservation principles.

Goal

Identify applicable conservation laws, derive the conserved quantities, and use them to constrain or solve the problem.

Scope

• Energy conservation (mechanical, thermodynamic, relativistic)
• Linear and angular momentum conservation
• Charge conservation and continuity equations
• Other conserved currents from Noether's theorem (baryon number, lepton number, etc.)
• Relativistic 4-momentum conservation in scattering and decay processes

Inputs

• system_description: Description of the physical system and its interactions
• known_quantities: Masses, velocities, charges, fields, or other given data
• symmetries: Any stated or inferred symmetries (translational, rotational, gauge, etc.)

Outputs

• conserved_quantities: List of applicable conserved quantities with justification
• constraint_equations: Explicit equations relating initial and final states
• solution: Derived unknowns from the conservation constraints

Workflow

1. Identify the symmetries of the system.

   • Translational invariance -> linear momentum conservation.
   • Rotational invariance -> angular momentum conservation.
   • Time-translation invariance -> energy conservation.
   • Gauge invariance -> charge conservation.

2. Write down the conserved quantity expressions.

   • For each symmetry, write the corresponding conserved current or integral of motion.

3. Set up constraint equations.

   • Equate the conserved quantity evaluated at the initial and final states.
   • For relativistic problems, use 4-momentum conservation: sum of initial 4-momenta = sum of final 4-momenta.

4. Solve the constraint system.

   • Eliminate unknowns using the conservation equations.
   • If the system is under-determined, state which additional information is needed.

5. Verify consistency.

   • Check that the solution does not violate any conservation law.
   • For scattering problems, verify that threshold energies are satisfied.

Quality Checks

• All conservation equations must be dimensionally consistent.
• For relativistic kinematics, verify that invariant mass is preserved: (sum p_mu)^2 is Lorentz-invariant.
• Energy must be non-negative; momenta must be real-valued for physical solutions.
• If dissipative forces are present, energy conservation must account for heat or radiation losses.

Constraints

• Do not apply conservation of mechanical energy in systems with non-conservative forces without accounting for energy dissipation.
• Do not assume angular momentum conservation unless the net external torque is zero about the chosen axis.
• In relativistic problems, use 4-vectors consistently; do not mix relativistic and non-relativistic expressions.