PHY.4:4.4 - Combine applicable dimensions, balances and dissipation

Use the physical restrictions relevant to the requested consequence:

• Dimensions: terms combined as one physical quantity need compatible units. Determine the dimensions of remaining coefficients and arguments. PHY.1 supplies dimensionless similarity groups when scale is part of the question.
• Balance: account for the relevant quantity retained, transferred, supplied or lost across the chosen boundary. A flux entering and leaving a region can differ when the region stores the quantity.
• Dissipation or passivity: identify the exchange whose sign is constrained and the regime in which the constraint holds. For an instantaneous passive resistive force at relative velocity w, the mechanical power condition is F(w) dot w <= 0.

Derive each condition at its stated scope. A storage element can temporarily return energy that it received earlier; its instantaneous power need not satisfy the memoryless resistance condition. Including storage changes the applicable balance and inequality.

Combine the constraints and examine whether any candidate remains. If they conflict, return to the assumptions or allowed class that caused the conflict. If they leave several laws, express the unresolved function, parameter or state dependence. Symmetry and dimensions often narrow a family without selecting one member.