PHY.7:4 - Solution

Choose the physical principle → represent the histories and interactions → state the allowed variations → derive the interior and boundary conditions → interpret the consequence → revise the implicated premise.

PHY.7:4.1 - Choose a principle for the physical question

State the consequence needed and the physical situation it concerns. Choose the retained participants, interactions and regime. PHY.5 helps decide which physical detail matters.

Recover an applicable variational principle from the physical theory, or propose one with a stated physical basis and conditional use. Identify what it assigns to a history and what property the realized history is required to have. A stationary action, minimum energy at equilibrium and a dissipative variational rule impose different conditions. Select the one appropriate to the question.

For a smooth classical configuration history q(t), a common form is

S[q] = integral from t0 to t1 of L(q,qdot,t) dt,

where L is the Lagrangian. A field description can instead integrate a density over space and time. The following steps show how to use such a principle; another physical principle can require a different functional or variation rule.

If the action or its physical grounds are missing, identify the interaction or assumption needed to construct them. A balance with a response law may settle the immediate question while the variational formulation remains open. The availability of a differentiation tool does not resolve that physical choice.

PHY.7:4.2 - Construct the histories and their action

Choose coordinates or fields that represent the retained physical configuration. State how they recover the physical quantities needed by the question. Include the time dependence of that recovery: for a position r=F(q,t), velocity is r_dot=F_q*qdot+F_t. The second term describes motion of the chosen mapping, such as a moving support.

For a classical mechanical description with the appropriate conservative interactions, construct kinetic energy from those velocities and potential energy from the interactions, then use L=T-V. Check the interaction rather than infer this form from the word energy. A velocity-dependent coupling, a dissipative interaction or an eliminated environment can require another term or another principle.

Keep relevant boundary contributions. Stored energy at an endpoint, an imposed load and a fixed endpoint are physically different. A field’s action can contain an interior density and separate surface or endpoint terms.

When reducing coordinates, retain how the discarded quantities or reactions can be recovered if needed. If the reduction loses a physical effect important to the question, return to PHY.5; MMP.9 handles the mathematical reduction once its physical premises are chosen.

PHY.7:4.3 - State what may vary and what must remain fixed

Specify the interval, endpoint conditions, constraints and regularity used in the comparison. For the ordinary fixed-endpoint principle, take q_epsilon=q+epsilon*eta, with eta(t0)=eta(t1)=0. A position constraint requires a family that preserves that constraint, or a justified multiplier formulation. MATH.10 distinguishes a finite admissible family from a tangent calculation valid only to first order.

The endpoint values used to derive an equation need not be known future observations. Fixing them in the variation removes its endpoint contribution. After deriving the local evolution equation, an initial-value use supplies compatible initial position and velocity; it does not add a guessed final position as another condition.

Distinguish position constraints from restrictions on velocity that cannot be integrated into position constraints. For such a restriction, varying entire constrained histories and requiring zero work of ideal reactions on selected instantaneous virtual displacements can give different equations. A virtual displacement here is a comparison of configurations at one fixed time used to state that reaction condition. Obtain the reaction or allowed-variation rule from the physical constraint model before using either construction.

For example, an ideal reaction model for A(q,t)*qdot+b(q,t)=0 may prescribe zero reaction work on displacements satisfying A*delta_q=0. Generalized forces Q are defined by their virtual work, delta W=Q dot delta_q. The Lagrange-d’Alembert equations then have the form d/dt(L_qdot)-L_q=Q+A^T*lambda, together with the velocity constraint; here Q contains the other forces and lambda determines the reactions. This result uses the ideal-reaction premise. Simply putting the velocity constraint into an action with a multiplier is a different construction and need not reproduce it. Use PHY.6 when the balance and reaction description is the sufficient route.

PHY.7:4.4 - Derive the interior and boundary conditions together

Apply MATH.10 to the allowed family. Vary the complete action, including dependent quantities and boundary terms. For the smooth finite-dimensional form above, integration by parts gives

delta S = [L_qdot dot eta] at t0,t1 + integral (L_q-d/dt(L_qdot)) dot eta dt.

With fixed endpoints and otherwise arbitrary interior variations, stationarity gives d/dt(L_qdot)-L_q=0. With restricted variations, derive the condition supported by that restricted family. Retain force terms when the selected principle includes their virtual work.

For a field, perform the corresponding integration in space as well as time. First identify the boundary terms, then decide which vanish because the boundary value is prescribed and which produce a condition because its variation is free. An endpoint force or boundary energy can change the latter condition without changing the interior equation.

Differentiate symbolically or computationally when helpful, but retain the variables held fixed and the substitution rules. A result from varying only part of the action answers that smaller calculation. It does not justify omitting a physical term.

Stationarity alone does not establish a minimum. Use the stronger comparison only when required and supported. MATH.10 gives both a minimizing free-particle case and a stationary oscillator history with changes of either sign in the action.

PHY.7:4.5 - Recover the physical consequence and preparation

Interpret the equations in the original physical quantities. Supply the independent initial, boundary and driving conditions needed for the intended use. Check their compatibility with constraints. An equation of motion can be the sufficient first result; obtaining a trajectory or a peak can require the subsequent computation in C.29.2.

Derive a conservation claim from the applicable symmetry and its conditions. For the ordinary unconstrained Lagrangian with no additional generalized force, absence of a coordinate from L gives a constant corresponding L_qdot. Absence of explicit time dependence gives a constant qdot dot L_qdot-L. Interpret these expressions physically; a velocity-dependent interaction can make a canonical momentum differ from mass times velocity. MATH.13 supplies the broader symmetry-to-consequence reasoning and PHY.4 supplies the physical grounds for the symmetry.

Use a discriminating comparison when it can change the result’s use. A force balance can reveal a missing coupling; boundary work can reveal an omitted load; a change of coordinates can expose a missing velocity term. Agreement establishes that comparison under its premises. It does not independently validate the physical principle.

PHY.7:4.6 - Return through the premise that changed

When an interaction changes, revise its action term or force contribution. When a support or boundary changes, revise the histories, velocity mapping and allowed variations before reusing the equations. When a coordinate description changes, transform the whole expression while keeping the physical history recoverable.

Equivalent expressions can describe the same motion. In the ordinary fixed-endpoint principle, adding dF(q,t)/dt changes the action only by endpoint values, so it preserves the interior equations. A use with different endpoint freedoms must also carry the changed boundary term. Distinguish this redescription from introducing another physical interaction.

If a computed consequence fails, locate whether the fault is the physical principle, admissible comparison, mathematical derivation or obtaining procedure. B.5.MPC.R provides the combined return. Seek additional physical evidence only where its possible result could alter the decision or action; a useful conditional derivation can already be used with its stated limits.