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.