PHY.4:4 - Solution

Local mantra: recover the physical situation; construct its transformations; justify the comparison; restrict the relation; retain its unknown part; use the consequence or revise the premise.

PHY.4:4.1 - Recover the participants and the relation being sought

Name what interacts, what is prepared, what can be exchanged and what response matters. Include surrounding bodies, material state, fields or boundaries when they can change that response. An omitted orientation or stored state may matter more than an additional numerical parameter.

Choose candidate quantities from that physical account. Explain how each quantity describes a participant, preparation or interaction. A.3.3.TR supports constructing a state that distinguishes different possible continuations; here the physical account supplies what retains and changes that state.

Decide what kind of relation is being proposed. An instantaneous mean response y=F(x) assumes that the chosen current inputs determine that mean. Memory, unresolved states or several possible responses can require a history, additional state, a probability law or a relation admitting several outputs. MMP.10 and MMP.11 express those mathematical choices after the physical conditions are identified.

Retain the status of the premises. A studied physical law, an idealization and a proposed hypothesis can all support reasoning, with different conditions on its use. Use the resulting conditional consequence when it already answers the present question.

PHY.4:4.2 - Construct the transformation of the situation

Describe what happens to every relevant participant and condition under the proposed change. If an apparatus is rotated, what happens to gravity, nearby surfaces, an applied field and its preparation? If two participants are exchanged, which material properties and connections move with them?

Separate two comparisons:

• Changed coordinates: the same physical situation is expressed with different components or labels. Transform every quantity representing that situation consistently.
• Changed physical situation: participants or conditions are changed. Explain which physical premise predicts corresponding behavior in the new situation.

For an isotropic material, all spatial directions are physically equivalent under the stated conditions. An oriented material can also be described in rotated coordinates; its material direction must then rotate in the description. These are different premises for restricting a response.

Select the actual transformation class. A justified rotation condition supplies rotation consequences. Reflection, time reversal, scaling or exchange requires its own physical grounds when used. PHY.1 supplies the detailed work for a scaling comparison.

A thought comparison can be enough. When a physical premise is unsettled, identify a rival case that could make the responses differ. C.11.DUA helps decide whether resolving that difference is worth the work needed now.

PHY.4:4.3 - Turn the comparison into a constraint on the unknown relation

Specify how inputs and responses transform. If x changes by T and y by U, a proposed response function has the comparison condition:

F(T(x))=U(F(x)).

For a reversible symmetry of a relation R, the corresponding condition is R(x,y) iff R(T(x),U(y)). This permits constraining an account before choosing a direction in which to solve it.

Derive the restriction. One useful move is to hold the input fixed under some allowed transformations. The output must then remain fixed under its corresponding transformations. Another is to compare inputs related by a transformation: a value chosen at one constrains the value at the other. MATH.13 develops these mathematical consequences once the physical action has been supplied.

For example, in three-dimensional space, rotations about a nonzero vector w leave w fixed. A vector response determined only by w and unchanged physical conditions must therefore lie along w: any perpendicular component would turn. Rotations between equal-length velocities then make the scalar coefficient depend only on their length. The physical work is establishing that no additional direction or state must also be supplied; :5.1 makes those assumptions explicit.

If a proposed input was held fixed even though the physical transformation changes it, restore that input and repeat the derivation. The anisotropic case in :5.2 shows how the changed argument opens additional response directions.

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.

PHY.4:4.5 - Obtain the needed consequence and choose the next use

Use the constraint for the original question. It may reject an impossible response direction, locate a missing input, restrict a learned or symbolic model, bound a consequence, or identify a condition under which competing laws disagree.

MMP.11 supplies a mathematical family respecting the physical constraints. When a quantitative value is needed, an interaction theory, interpreted observations or another appropriate method can constrain its remaining freedom. C.16.IR handles inference from interpreted indications; MMP.7 formulates a probability law for recorded data when the inference uses that form. C.29.2 supplies a computational formulation.

Choose further work from the unresolved consequence. A direction or family-wide bound may already suffice. If a missing magnitude changes the decision, obtain the needed contribution at a useful range and precision. A proposed experiment or simulation should discriminate something that matters to that next move.

When a changed situation gives a response outside the family, inspect the physical comparison before adding arbitrary terms. A preferred direction, external drive, stored state or different regime can change the family itself. Retain earlier consequences where their premises still hold.