PHY.7:1 - Problem frame

Use this pattern when you need to derive physical evolution from a principle that compares possible histories. The difficulty may be choosing the physical action, incorporating a constraint, or deciding which changes of a history the principle permits. A correct variation of the wrong physical formulation can give a precise but unusable equation.

The first result is an equation of motion, a boundary condition or a conserved consequence with the physical assumptions needed to use it. Sometimes the result is a particular unresolved interaction or constraint rule that must be supplied before the derivation can answer the question.

An action assigns a scalar to a history in the chosen physical description. Stationarity means that its first variation vanishes for the specified allowed changes. The physical theory supplies the reason to impose that condition. MATH.10 supplies the mathematical reasoning about variations; this method constructs and interprets its physical premises.

You need to identify the physical quantities and understand the proposed principle’s range. You also need enough mathematical preparation to interpret a variation, or another contributor who can perform it and explain its result. The examples use elementary constrained mechanics, a charged particle and a continuous field. Their branch-specific laws are stated in the cases.

Use an existing equation or PHY.6’s balances and response laws when they already answer the question economically. A variational formulation is useful when it simplifies constrained motion, carries interactions across coordinate changes, exposes a symmetry, or produces both interior and boundary equations. It need not be constructed for every physical problem.