Part A - Construct and constrain a physical account

PHY.4 - Constrain an Unknown Physical Law by Transforming the Situation

Type: Method Status: Usable, evolving Normativity: Normative

PHY.4:1 - Problem frame

Use this pattern when an observed or proposed physical effect needs an account, but the interaction law is not supplied. A detailed mechanism may be unresolved and a familiar analogue may be inadequate. You can still ask how the situation could be changed, what would remain physically equivalent, and which responses those comparisons allow.

Start with one physical response or relation needed by the work. Describe its participants, preparation and relevant surroundings. Construct one comparison that could rule out a proposed dependence or restrict its form. A conditional restriction, together with the dependence it leaves unresolved, is a useful first result.

The method uses transformations justified for the physical situation: changes of position, orientation, scale, preparation, participant assignment or other relevant conditions. It combines their consequences with applicable dimensions, balances and dissipation. The question selects which of these contributions is useful.

You need enough physical understanding to propose the comparison and explain its conditions. Obtain an unfamiliar physical premise from a suitable account or collaborator. A qualitative comparison can already change the next move. The vector example additionally uses rotations and dot products; the chamber example uses elementary algebra and particle balance. Their branch-specific preparation belongs to those examples.

If an established interaction law already answers the question under the intended conditions, apply it. PHY.1 develops physical similarity across changed conditions, PHY.2 constructs a physical analogue, and PHY.3 derives a performance limit from permitted transformations. Return here when the unresolved contribution is the form of the interaction law itself.

PHY.4:2 - Problem

A formula can be easy to calculate with while admitting a physically impossible direction, an unjustified dependence or an omitted influence. Choosing the formula first can hide those differences.

Conversely, an appeal to symmetry or conservation can appear to determine a law while leaving important functions or parameters free. The task is to obtain the restriction that follows from the physical comparison and carry its remaining freedom into the next construction or inquiry.

PHY.4:3 - Forces

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.

PHY.4:5 - Archetypal Grounding

These are constructed physical accounts. They illustrate how physical premises constrain an unknown law and how a changed premise changes the result.

PHY.4:5.1 - Restrict an unknown resistive force

Seek the instantaneous mean resistive force on a body moving through a homogeneous isotropic medium at a fixed material state, in a classical three-dimensional regime. Assume the body and preparation introduce no preferred direction, relevant memory is negligible, and relative velocity w is the only varying input. These assumptions define the proposed comparison; their adequacy in a particular experiment remains a physical question.

Rotating the entire relevant situation rotates w and the force together. Because there is no further directional input, F(Qw)=QF(w) for every spatial rotation Q. For nonzero w, rotations about w rule out a perpendicular force component. Equal-length velocities are related by rotation, so write:

F(w)=-a(|w|^2)*w for w different from zero.

At w=0, rotational symmetry gives F(0)=0. The coefficient a is an unknown scalar function at the fixed material conditions. Its dimension is mass divided by time. Memoryless passive resistance gives:

F(w) dot w=-a(|w|^2)*|w|^2 <= 0,

so a(s)>=0 for s>0.

The result determines a direction and sign while leaving the speed dependence unresolved. Both a(s)=a0 and a(s)=b0*sqrt(s) with suitable nonnegative dimensional constants satisfy these restrictions. They predict different force magnitudes when speed changes. The comparison alone therefore provides neither a linear nor a quadratic drag law.

For a direction-only question, use the result directly. To predict stopping time, supply further information about a over the speeds involved, or obtain a sufficient bound over the admissible family. MMP.11 keeps that unresolved relation visible instead of inserting a familiar drag coefficient.

PHY.4:5.2 - Add a physical direction

Now the body is oriented or the environment has an aligned surface. Include its unit direction n in the input. Under a rotation of the complete situation, both w and n change, so the condition becomes:

F(Qw,Qn)=QF(w,n).

A rotation fixing w can now change n. The earlier argument that fixed all inputs while rotating a perpendicular output component is no longer available.

For example, one admissible linear resistive family is:

F(w,n)=-alpha*w-beta*(n dot w)*n.

For unit n, its power is -alpha*|w|^2-beta*(n dot w)^2. Resolving w into parts parallel and perpendicular to n shows passivity for every w when alpha>=0 and alpha+beta>=0. This is a possible family, not an exhaustive determination of the changed law.

With consistent units, take alpha=1, beta=3, w=(1,1,0) and n=(1,0,0). The force is (-4,-1,0) and power is -5. The force resists motion while pointing in a direction different from -w. This possibility is excluded by the earlier isotropic account and allowed by the additional physical input.

If n is fixed in the laboratory while only the body motion changes, retain that fixed n in the experiment’s account. Rotating the coordinate system changes the components of both quantities; it supplies no premise that removes the material or environmental direction.

PHY.4:5.3 - Exchange two participants

Two identical chambers at the same temperature exchange one kind of particle through a symmetric passage. Let a and b be their current concentrations and let J(a,b) be the instantaneous mean transfer rate from the first chamber to the second. Assume the proposed regime needs no additional passage state, and the surroundings introduce no directional bias.

Keep the chamber labels and the positive counting direction fixed, and exchange the concentrations in the two preparations. Because the chambers and passage are symmetric and the surroundings supply no bias, this physical exchange predicts a reversed measured transfer rate:

J(b,a)=-J(a,b).

At equal concentrations, J(a,a)=0. A proposed law J(a,b)=k*(a-b)^2 with k>0 fails the interchange condition: it gives a positive rate in the same counted direction after the preparations are exchanged.

Both J(a,b)=k1*(a-b) and J(a,b)=k3*(a-b)^3 satisfy the interchange condition when their constants have the corresponding units. Symmetry has not chosen between them or determined their coefficients. Directional thermodynamic claims would additionally use the physical driving potentials and the applicable dissipation law.

If the passage stores no particles, the chamber particle numbers satisfy dN1/dt=-J and dN2/dt=J, preserving their sum. If appreciable particles accumulate in the passage, include its particle number and separate inlet and outlet rates. The earlier two-chamber balance then omits a relevant participant.

This comparison uses exchange and balance rather than rotation. It opens the same kind of result: an admissible law family and the physical condition that would require revising it.

PHY.4:6 - Bias-Annotation

A familiar formula can conceal a missing physical premise. An elegant symmetry argument can conceal the same omission. Recover the participants, preparation and surroundings before deciding which transformations the situation admits.

A conditional family is useful when its remaining freedom is stated. Supplying an unexplained familiar coefficient would replace an unresolved physical question with apparent precision. Obtain further information only for the consequence that needs it.

PHY.4:7 - Conformance Checklist

For the restriction being used:

• The physical response or relation, participants, preparation and relevant surroundings are recoverable.
• The transformation states what changes and what remains physically comparable.
• The mathematical input and output transformations follow that physical account.
• Each dimensional, balance or dissipation constraint has applicable physical premises.
• The derived consequence distinguishes what is constrained from what remains unknown.
• The result changes a construction, interpretation, prediction or next inquiry.
• A changed physical condition returns to the affected premise and constraint.

PHY.4:8 - Common Anti-Patterns and How to Avoid Them

Dropping the surroundings from the transformation. Rotating a body relative to a fixed material direction can change its response. Transform the complete relevant account and distinguish that experiment from a coordinate change.

Selecting a magnitude law from direction symmetry. The linear and quadratic resistance possibilities in :5.1 satisfy the same directional restriction. Carry the unknown scalar function until a further physical contribution constrains it.

Applying an instantaneous dissipation inequality to an energy-storing interaction. Recover stored energy and the exchange balance. Temporary return of stored energy changes the instantaneous power without establishing an active energy source.

Using a balance with an omitted participant. Accumulation in the passage changes the two-chamber balance in :5.3. Include the storage and its exchanges before computing the chamber changes.

PHY.4:9 - Consequences

Useful physical restrictions become available before a complete mechanism or fitted law is known. They can guide formulation, reject an incompatible model or direct a discriminating inquiry. The remaining freedom also becomes a specific task rather than an implicit assumption.

The result is conditional on the physical comparison. An added direction, stored state or changed regime can reopen the law family. This makes the dependence of the conclusion inspectable and supports retaining the portions that still apply.

PHY.4:10 - Architectural Rationale

The physical construction precedes the mathematical symmetry calculation. MATH.13 can derive what a supplied transformation preserves; the present method supplies and criticizes the physical grounds for choosing that transformation and its inputs.

The unknown relation remains explicit through the work. This supports relational formulations as well as response functions, and allows later symbolic, numerical or learned models to use the same physical restrictions.

Similarity, analogy and performance bounds remain separately usable methods. Their results may settle the physical question directly. When a law is still missing, this method constrains its form and identifies the remaining contribution without requiring a detailed mechanism first.

PHY.4:11 - SoTA-Echoing

Feynman’s discussion of symmetry in physical laws, especially §52-2, is a historical methodological anchor for transforming an experiment together with its relevant surroundings. The adopted contribution is the physical construction of the comparison. The particular transformation and its regime are selected from the physical account being used.

Villar and colleagues’ Scalars are universal, Proposition 4 and Appendix H, provides a contemporary constructive connection between specified symmetry actions, scalar invariants and equivariant response families. Its distinctions between rotation and reflection groups and between different input quantities matter when selecting a model. The paper’s mathematical representation results are used under their hypotheses; the present physical method establishes which input and transformation account is appropriate.

When formulating an unfamiliar interaction law, a common alternative is to select a familiar constitutive formula and fit its coefficients. With only the physical premises in :5.1, that choice would insert a speed dependence the premises do not determine. The transformation method in :4.2-:4.4 first establishes the allowed direction and sign, leaving the scalar function open. For a direction-only question, this supplies the needed answer without obtaining a detailed law or fitting its coefficients. For a stopping-time prediction, :4.5 requires further information about that function or a sufficient bound; the restriction alone leaves the prediction unresolved.

An established constitutive account or reliable analogue is preferable when it applies to the intended regime and supplies the needed consequence with less work. The present method then helps inspect its physical assumptions or a proposed change of conditions. Reopen this choice when a new physical input or regime invalidates the comparison, an improved account changes its grounds or offers an easier answer, or the receiving question requires information that the constrained family leaves open.

PHY.4:12 - Relations

• PHY.1 constructs physical similarity and scale-dependent comparisons.
• PHY.2 constructs a physical analogue or proposed mechanism from interactions.
• PHY.3 derives a physical performance limit from the permitted transformations.
• MATH.13 derives mathematical consequences of a supplied symmetry.
• MMP.10 and MMP.11 formulate compatible conditions and relations with unresolved dependence.
• A.3.3.TR supplies common state and joint-change construction.
• C.29.1, C.29.2 and C.29.3 connect correspondence, computational formulation and physical realization.
• C.16.IR and MMP.7 support interpreted indications and the probability law used for recorded observations.
• C.11.DUA compares the value and effort of resolving a remaining physical question.

PHY.4:End

PHY.5 - Choose an Effective Physical Description by Scales and Couplings

Type: Method Status: Usable, evolving Normativity: Normative

PHY.5:1 - Problem frame

Use this pattern when a physical question leaves you uncertain which motions, interactions or material details its description must retain. Resolving every microscopic change may be impractical. Omitting a fast motion or a small interaction may change the answer that matters.

Begin with one consequence: a response over a stated interval, a propagation speed, an accumulated exchange, or a distribution of outcomes. Choose one proposed omission and follow how it could affect that consequence. A simpler account with a justified range of use, or a reason to restore the omitted contribution, is a useful first result.

An effective physical description here describes the physical behavior needed at chosen scales without resolving every underlying process. Unresolved processes can still contribute through response coefficients, constraints, dependence on earlier states or fluctuations. The method helps choose and examine that description across physical branches.

You need to identify the physical participants, explain the interactions relevant to the question and compare their characteristic sizes or rates. A qualitative separation can direct the next construction. The examples add their own mathematical preparation: elementary differential equations for the motor, waves for the discrete chain, and covariance for thermal motion.

When a known physical account already answers the question under the intended conditions, use it. PHY.1 supplies similarity between changed arrangements; PHY.2 constructs an analogue from physical interactions. MMP.9 supplies mathematical reduction once the governing relations and retained quantities have been chosen. Use this pattern for the physical choice of those quantities, relations and conditions.

PHY.5:2 - Problem

A detailed account can obscure the mechanism that determines a response. A convenient coarse account can erase the same mechanism. The relevant distinction depends on the question: a description may preserve a mean displacement while losing its spread, or preserve slow motion while losing an initial torque.

A comparison of isolated time scales also misses how parts interact. A rapidly adjusting part can exert a sustained force. Unresolved molecular motion can produce both drag and continuing fluctuations. A small length can become consequential when the imposed variation approaches that length.

The task is to retain the physical influence required by the intended result, while choosing how much of its underlying process to describe.

PHY.5:3 - Forces

PHY.5:4 - Solution

Choose the consequence → locate its physical influences → compare their scales in the coupled situation → retain the needed effects of unresolved processes → test the resulting answer → use it or restore the consequential difference.

PHY.5:4.1 - Choose the physical consequence and its resolution

State what the answer will let someone interpret, choose or do. Identify the relevant quantity and how it is used: an instantaneous value, a time integral, a spatial average, a peak, a correlation or a probability can require different descriptions of the same situation.

Include the preparation, forcing, spatial range and observation interval when they affect that answer. For example, the first moment after switching can include a motion that has already decayed when a later reading is taken. A question about a wave’s arrival concerns a propagating disturbance and the wavelengths present in it.

Use the working question to choose adequate resolution. If a bound already determines the choice, recovering a detailed trajectory may add nothing to that choice. C.11.DUA helps when the value of further inquiry is uncertain. B.5.MPC connects the mathematical result and its conditions to the physical question.

PHY.5:4.2 - Locate the interactions and compare their scales

Follow the influence from preparation or input to the consequence. Identify what stores, transports, exchanges or dissipates the quantities involved. Include a boundary, contact, surrounding medium or measuring interaction when it carries an influence on that consequence.

Obtain the governing physical relations at the detail needed for this comparison. A studied theory, measured response or working hypothesis can supply them; retain which of these is being used. PHY.4 helps constrain an unresolved law. MMP.11 constructs its remaining mathematical family.

Derive characteristic sizes or rates from these relations and the proposed regime. Compare contributions to the same physical change. For time scales, compare relaxation with the fastest relevant forcing and with the interval of use. For spatial scales, compare a variation length with the scale of material structure or transport. An energy comparison may determine which states can be appreciably excited.

Inspect the coupled arrangement. A coupling can change a relaxation rate or create a collective mode. For a proposed rapidly adjusting state, ask whether small departures from its proposed response actually decay while the retained variables change. A response near loss of stability can become slow even though one component, considered alone, relaxes rapidly.

Scaling the equations as in PHY.1 makes these comparisons explicit. Keep the physical reason for each scale: a narrow gap, driving period or relaxation distance can matter more than the size of the whole apparatus.

PHY.5:4.3 - Choose how unresolved processes enter the retained account

Select the variables and interactions needed by the consequence. They may describe individual participants or collective quantities such as a displacement field, concentration or slowly changing amplitude.

For a part that adjusts rapidly, first determine what it adjusts to. Substitute that response into its coupling with the retained part. The resulting force, constraint or transport remains in the effective account. In the motor example, eliminating rapidly changing current retains its effect on torque and damping.

For a collection of unresolved processes, determine which of their effects remain:

• a mean response changes the retained forces or transport coefficients;
• a delayed response makes current change depend on earlier states;
• fluctuations produce a spread of possible changes around a mean response;
• a persistent unresolved mode may require another retained state.

Choose among these from the physical preparation and interactions. Eliminating variables mathematically can expose a memory term even in deterministic dynamics. A statistical description additionally needs grounds for the distribution of unresolved states. MMP.7 supplies probability composition for a specified information and recording situation; it does not choose the physical preparation.

An effective coefficient can be obtained from a more detailed description, existing measurements or a constrained response law. Compare the predicted quantity in a regime where the two descriptions apply and choose the coefficient to preserve that contribution. This matching can be useful even when no complete microscopic theory is available. Carry the range and remaining uncertainty that change its later use.

PHY.5:4.4 - Construct the approximation and its first omitted contribution

Use MMP.9 for the mathematical elimination or retained-state evolution. Keep the physical assumptions that permit the construction alongside its result.

For a rapidly relaxing state, examine its initial departure and response to changing inputs. Replacing it by a steady response is useful when the remaining departure has sufficiently little effect on the requested consequence. The motor calculation below derives that departure instead of assigning it zero at the initial instant.

For a spatial or energy expansion, choose the ratio in which the description is expanded and compare the first omitted contribution with the retained ones. The ratio and its powers come from the physical relations. In the chain example, expansion in the small ratio of lattice spacing a to wavelength lambda, or equivalently in q*a=2*pi*a/lambda, yields a continuum wave description and its leading correction.

When unresolved correlations persist over the interval of interest, retain their delayed influence or add variables that reproduce it. Replacing a delayed response by an instantaneous one needs a comparison of that response time with the retained motion and driving. A mathematically equivalent auxiliary state is another way to calculate the memory; its interpretation as a material component requires a separate physical account.

Maintain the relevant exchanges and constraints through the approximation. Recover the condition for a balance or an allowed motion when changing the variables or boundaries. C.29.BB supplies the common balance construction.

PHY.5:4.5 - Judge the influence on the requested answer

Follow the approximation through to the actual output. A small state error can grow under differentiation, accumulation, feedback or a sensitive later choice. MATH.20 supplies bounds. If the consequence is obtained in another mathematical account, C.29.1 establishes what transfers between the accounts.

Examine conditions suggested by the construction itself. An omitted initial transient matters when the reading moves into it. A spatial approximation becomes suspect when the wavelength approaches the unresolved structure. A weak loss can determine a resonant response. A nearly unstable mode can defeat the earlier time-scale separation. Use the conditions that affect the proposed account.

For a statistical description, compare the statistic the work uses. Preserving an average response need not preserve a variance or a transition probability. In thermal motion, eliminating velocity while retaining random forcing can preserve long-time displacement statistics. Deleting that forcing gives a different result.

A derivation, an existing limiting result or a suitable comparison can settle the choice. If a physical premise remains unresolved, identify a feasible change of preparation or readout for which the plausible accounts give different useful answers. Obtain that comparison when its possible outcomes warrant the effort. A conditional physical conclusion can remain useful while a stronger claim is unresolved.

PHY.5:4.6 - Use the account and reopen the affected physical choice

Return the answer with the conditions needed to use it. These may be expressed in a short derivation, an annotated model or an ordinary explanation.

Choose what the work now permits: use the simpler computation, interpret a measurement, change a drive, retain a fluctuation model, or restore an interaction. If several accounts provide complementary consequences, retain their respective uses. When a choice among available descriptions matters, C.11 supplies the comparison, including whether more inquiry is worth its cost. Use G.5 when the retained set itself must be stated: distinguish alternatives for later choice from descriptions used together for a named result. E.23 supports improvement when the chosen characteristics can judge a change.

For a new question or failed prediction, return to the influence whose omission is implicated. Change the physical variables, coupling, preparation or range, then redo the affected reduction and interpretation. B.5.MPC.R helps separate a physical-account failure from a mathematical or computational one.

The result can also change the method of work. A team may calculate a slow response with one model and delegate a short transient to another. ME.7 describes those contributions and their joins; ME.12 examines the claims needed for the combination. Each result retains the physical conditions that make it usable by the next participant.

PHY.5:5 - Archetypal Grounding

PHY.5:5.1 - Keep a motor’s torque while eliminating fast current dynamics

Consider an ideal linear motor over a range in which resistance R, inductance L, inertia J, damping b and conversion constant k are positive and constant. In consistent SI units, use the same k for torque per current and back voltage per angular speed:

L*i' = V - R*i - k*omega

J*omega' = k*i - b*omega

The electrical and mechanical balances describe the assumed device, including its load in J and b. Saturation, variable load or a different drive would require the corresponding physical relations.

Suppose the work needs the slow speed response. The electrical relaxation time is tau=L/R. The current toward which the electrical part relaxes is q(t)=(V(t)-k*omega(t))/R. Replacing i by q gives

J*omega' = (k/R)*V - (b+k^2/R)*omega.

The eliminated current still supplies driving torque and additional damping. The slow response time of this candidate is J/(b+k^2/R). Compare tau with that time and the drive’s variation time.

To examine the neglected response, set e=i-q. The full electrical equation gives tau*e'=-e-tau*q'. If |q'|<=K on the interval, integration yields

|e(t)| <= |e(0)|*exp(-t/tau) + tau*K*(1-exp(-t/tau)).

The bound on q’ can come from |q'|<=(|V'|+k*|omega'|)/R and the allowed drive and acceleration, using the full physical account where needed. A bound inferred only by assuming the proposed approximation would leave that assumption unresolved.

For example, let R=2 ohms, L=0.02 henry, k=0.1 in the stated SI convention, J=0.02 kg m² and b=0.01 N m s per radian. Then tau=0.01 s and the candidate slow time is about 1.33 s. With |e(0)|<=0.5 ampere and K=1 ampere per second, the current departure at 0.05 s is at most 0.01331 ampere, giving a torque departure at that instant of at most 0.001331 N m relative to kq.

The slow speed error is a different output. With the same initial speed and drive, let delta be full speed minus reduced speed. It satisfies

J*delta' + (b+k^2/R)*delta = k*e.

Let B(s) denote the current-departure bound above. Since the initial speed difference is zero and the response kernel is positive, integration gives

|delta(t)| <= (k/J)*integral_0^t exp(-(b+k^2/R)*(t-s)/J)*B(s) ds.

For the stated values, this speed-departure bound at 0.05 s is about 0.026064 rad/s, hence less than 0.02607 rad/s. An allowed error of 0.03 rad/s therefore permits the reduced calculation for that speed reading. The integral carries the earlier transient into the answer; the small current departure at the final instant alone would not give this bound.

Changed work. If the next question concerns torque immediately after switching, the bound includes the initial current departure. Use the electrical transient. If the drive varies on the electrical relaxation time, recompute its departure instead of extending the slow-drive approximation. These returns change which physical response is retained.

PHY.5:5.2 - Decide when a chain can be treated as a continuous medium

Consider an infinite ideal one-dimensional chain with identical masses m, spacing a and linear springs of stiffness kappa. Each mass moves a small distance u_j from its reference position. The balance is

m*u_j'' = kappa*(u_(j+1)-2*u_j+u_(j-1)).

For waves with wavenumber q in 0<q*a<pi, substitution of a sinusoidal wave gives

omega^2 = (4*kappa/m)*sin^2(q*a/2).

For wavelengths long compared with a, expanding the neighboring displacements gives the continuum equation u_tt=c^2*u_xx with c=a*sqrt(kappa/m). It predicts both phase and group speed c. The chain’s phase speed divided by c is sin(q*a/2)/(q*a/2); its group speed divided by c is cos(q*a/2). Expanding those ratios gives v_phase/c = 1-(q*a)^2/24+O((q*a)^4) and v_group/c = 1-(q*a)^2/8+O((q*a)^4), exposing their different first corrections.

At q*a=0.2 these ratios are about 0.99833 and 0.99500. A half-percent allowance for these speeds accommodates this ideal comparison, subject to the question’s waveform and other physical premises. The group-speed difference is already larger than the phase-speed difference.

Changed work. A disturbance containing wavelengths near the shortest traveling waves of the chain probes q*a near pi. The chain’s group speed tends to zero, while the continuum account keeps c. A question about that disturbance’s propagation requires retaining the discrete dispersion or an adequate extension. Making the computation of the uncorrected continuum equation more accurate cannot recover the omitted physical dependence.

This case concerns an ideal linear chain. It demonstrates choosing spatial resolution from the wave that matters. It supplies no claim that every material, boundary or large deformation obeys the same chain law.

PHY.5:5.3 - Retain fluctuations after fast velocity has relaxed

For a one-dimensional Brownian particle in a uniform equilibrium bath, take mass m, drag coefficient gamma and temperature T, with no applied force. The underdamped account has position x, velocity v and thermal forcing. Let tau=m/gamma and D=k_B*T/gamma. With an initially equilibrated velocity, its velocity covariance is (k_B*T/m)*exp(-|t-s|/tau).

Integrating that covariance over the two times gives

E[(x(t)-x(0))^2] = 2*D*(t-tau*(1-exp(-t/tau))).

At times large compared with tau, the overdamped diffusion description gives 2*D*t. Its omitted contribution to this mean-square displacement is bounded by 2*D*tau. This calculation states which long-time consequence the reduction preserves.

Setting the mean velocity to zero and deleting the forcing instead gives no displacement spread. The unresolved bath continues to transfer random impulses after the velocity’s preparation has relaxed. For the displacement distribution, retain their diffusion effect.

Changed work. Change the bath to a spatially varying temperature and ask about entropy production. The uniform-bath calculation no longer answers the question. Celani and coauthors show a further distinction: under their smooth-temperature and small-inertia conditions, the overdamped position process has the appropriate limit, while the mean rate of entropy production retains an additional positive contribution absent from the naive overdamped expression. Return to the thermodynamic observable and its limiting calculation. Position accuracy alone cannot decide that use. Their 2012 paper states the preparation and the contribution.

PHY.5:6 - Bias-Annotation

The worked calculations emphasize classical continuous-time descriptions with explicit laws. The method also applies to choosing physical states and interactions when only a constrained response or statistical account is available. In such cases the first result can be a qualified regime choice or a discriminating physical question.

Time-scale separation is especially convenient and can dominate the choice too early. The spatial chain and changed thermodynamic observable show other reasons to retain a contribution. Quantum coherence, rare transitions and collective behavior require their own physical grounds; the classical examples do not decide them.

PHY.5:7 - Conformance Checklist

• The intended consequence specifies the physical quantity, relevant preparation, range and interval.
• The comparison uses the coupled physical arrangement and explains the characteristic scales.
• Each omitted process retains any mean response, constraint, delayed influence or fluctuation needed by that consequence.
• The approximation follows from stated physical relations or remains an identified hypothesis.
• A bound or comparison reaches the actual requested output, including a relevant initial or boundary contribution.
• A changed condition that can defeat the selected description leads back to the implicated physical choice.
• Existing results are used when sufficient; further inquiry is selected for what it could change.

PHY.5:8 - Common Anti-Patterns and How to Avoid Them

PHY.5:9 - Consequences

A useful effective account makes a physical mechanism easier to reason about and calculate with. Its stated regime supports a targeted return when the forcing, observation or intended action changes.

The choice costs physical analysis and sometimes a comparison with a richer account. It saves work when those results permit a cheaper calculation or a clearer explanation. A range without adequate scale separation can require a coupled, memory-bearing or more resolved description.

PHY.5:10 - Architectural Rationale

This method organizes the choice by physical consequence and coupling. Comparing parameter sizes alone would miss how an omitted process enters the result. Starting from the most detailed available theory can impose work that the question does not need.

Effective-theory construction supplies a useful way to retain selected influences and improve an approximation systematically. Projection and coarse-graining show why unresolved motion can remain as memory and fluctuations. The observable-specific limit shows why their adequacy is judged at the receiving physical result.

MMP.9 owns the mathematical elimination; PHY.1 uses an effective account to compare changed physical arrangements; PHY.2 uses it when constructing an analogue. The contribution here is choosing and revising the physical resolution and remaining influence of unresolved processes. The motor, chain and Brownian particle demonstrate that contribution in different comparisons.

PHY.5:11 - SoTA-Echoing

Stewart’s introduction to effective field theory (MIT, 2013, lecture transcript pp. 2-4) is a historical teaching source for selecting relevant degrees of freedom, scales and an improvable leading description. This pattern adapts that construction beyond its quantum-field setting. It retains the need to identify an expansion and its range; improvement by further terms remains conditional on that expansion being useful.

Dalton and coauthors, Memory and Friction: From the Nanoscale to the Macroscale (2025; accessible manuscript, section 3) supplies the contemporary account of retained observables, memory-dependent friction and simulation through auxiliary variables. Adopt the distinction between eliminating explicit variables and discarding their influence. An instantaneous response saves computation when its conditions hold; resolved memory preserves effects that such a response misses. The review’s stated projection, equilibrium and preparation conditions govern its particular equations.

Celani, Bo, Eichhorn and Aurell (2012) provide the observable-specific counterexample used in :5.3. Adopt the comparison of the physical result before and after the limit. Their thermal-gradient result qualifies an otherwise successful position reduction; it does not rule out overdamped modeling for the position questions it answers.

The synthesis combines controlled omission, retained coupling and use-specific comparison. It favors the least costly description that supplies the needed consequence under the established physical conditions. Memory kernels, white-noise diffusion and a fully resolved model are alternative constructions with different requirements, selected by the question.

PHY.5:12 - Relations

• B.5.TU and PHY.4: supply application of a physical theory and constraints on an unresolved law.
• A.3.3.TR and C.29.BB: supply common state, interaction and balance constructions used by the physical account.
• MMP.9: supplies elimination and reduced evolution; MMP.7/.11 supply probability composition and constrained response families.
• MATH.20 bounds the unresolved contribution; C.29.1 establishes transfer of a bound or consequence between mathematical accounts.
• PHY.1/.2/.3: use the chosen description in similarity, analogue construction and physical limits.
• C.11.DUA and C.11: examine the value of further inquiry and choose among available descriptions.
• G.5 and E.23: state retained alternatives or jointly used descriptions when needed, and guide improvement by the chosen characteristics.
• B.5.MPC.R and ME.7/.12: return to a failed contribution and revise the method that uses the physical result.

PHY.5:End

PHY.6 - Construct Physical Evolution from Balances and Response Laws

Type: Method Status: Usable, evolving Normativity: Normative

PHY.6:1 - Problem frame

Use this pattern when you can identify physical participants and their interactions, but cannot yet say how their coupled situation will evolve. You may know what is conserved while lacking the law that determines an exchange rate. You may have equations for each part while their connection or initial preparation remains inconsistent.

Start with the consequence you need: an initial response, a trajectory, a transported amount, a settled state or a distribution of outcomes. Choose the participants, exchanges and response laws needed for that consequence. A usable first result is a coupled physical description from which the consequence follows, or an identified missing response or incompatible condition that tells you what to resolve next.

Here a response law relates a physical interaction or change to the conditions on which it depends. A constitutive law is one such relation for a material or component. The method constructs an account from physical balances, response laws and preparation; it works across physical branches.

You need to identify the physical quantities and interpret the proposed relations and their conditions. The worked cases add their own preparation: elementary motion and energy calculations, a spatial continuity equation, and charge storage. These are different demonstrations of the same construction.

If a known physical account already supplies the required result, use it. If a balance alone settles the question, stop there. C.29.BB supplies that general balance method. Use PHY.5 when choosing which physical detail to retain; use this pattern when assembling the retained physical relations. Once that account is settled, MMP.10 and C.29.2 help formulate and obtain its mathematical consequences.

PHY.6:2 - Problem

A balance says how accumulation relates to transfer and production. It can leave the transfer itself unknown. Adding a convenient rate formula may complete an equation system while describing the wrong physical interaction.

Combining individually useful descriptions adds another difficulty. A contact may store a quantity, a connection may impose a constraint, and two parts may use different signs or reference frames. Initial values acceptable to each isolated part can be impossible after they are connected. A solver can then fail for a physical reason, or return a result for a preparation different from the intended one.

The task is to construct mutually compatible physical relations and preparation, then derive the consequence that the work needs.

PHY.6:3 - Forces

PHY.6:4 - Solution

Choose the consequence → identify participants and exchanges → supply response laws → connect the relations → make the preparation consistent → derive and use the needed consequence.

PHY.6:4.1 - Choose the consequence and physical participants

State what the answer will let someone interpret, choose or do. Identify the quantity, interval and preparation that matter. A peak response, a total transferred amount and a settled value can require different accounts of the same arrangement.

Choose the physical participants and their boundaries at a useful resolution. Locate the interactions that can affect the answer, including supports, surrounding media and measurement when relevant. Distinguish a quantity stored in a participant from a quantity passing through its boundary. Include storage in a contact or field when omitting it would change the consequence.

Use PHY.5 to resolve a consequential choice of scale or omitted interaction. A spatial field, a few aggregate variables and individual particles are possible descriptions; their physical adequacy depends on the question and regime.

PHY.6:4.2 - Construct the balances with physical meanings

Use C.29.BB to choose the additive quantity, common interval, signs, transfers and internal production. Write each term with its physical meaning and compatible units. For vector quantities, use a common frame or an explicit transformation between frames.

For a fixed spatial region, a useful form is

rate of stored quantity = inward transfer - outward transfer + internal production.

A local continuity equation expresses the same relation for a density and its flux. The choice of quantity determines the production term: a chemical species can be consumed while the atoms it contains remain in reaction products. A closed boundary alone does not make every selected quantity constant.

Identify which terms remain undetermined. A momentum balance may still need forces; a species balance may need transport and reaction rates. If the requested total already follows without those details, retain that consequence and avoid completing an unnecessary model.

PHY.6:4.3 - Supply the response and configuration relations

For each unresolved interaction that affects the answer, state how its response depends on the physical conditions. Use an applicable theory, an established material response, an available measurement or a stated working hypothesis. PHY.4 helps constrain a law whose form is unknown; MMP.11 helps represent its remaining freedom.

Recover the conditions under which the relation is used. Does it describe the current state, dependence on earlier states, a spatial gradient, an average response or fluctuations? Does it assume a settled contact, a constant material parameter or a particular preparation? Supply the configuration relations needed to connect that response to the retained variables.

Keep physical restrictions that the chosen description relies on. A passive damping law transfers mechanical energy into other forms; it does not destroy total energy. If that heating changes the response during the intended use, include the resulting dependence. A fitted or learned response can also supply a relation, but its physical restrictions and usable range must come from its construction or grounds, not from the fact that it produces numbers.

When several physically plausible laws remain, derive what each changes in the requested consequence. A bound or a conditional answer may be sufficient. Seek another observation or a more detailed account when its possible result can change the work; C.11.DUA helps decide that inquiry.

PHY.6:4.4 - Connect the physical relations before choosing a solving order

State what a connection makes common and what it transfers. Match quantities, units, frames and orientations at that connection. Equal values require a physical reason: two locations in contact can still have a finite resistance or an intervening store. If the connection has its own consequential dynamics, describe those dynamics.

Combine the part balances through the common transfer. Transfers internal to the combined boundary cancel when they describe the same exchange over the same interval. Retain conversions between quantities and energy forms. For an energetic connection, derive its power expression from the physical variables; variable names such as potential and flow alone do not establish their product as power.

Keep the equations as simultaneous relations while constructing the physical account. They can include derivatives, algebraic constraints, spatial dependence or statistical response. In a differential-algebraic description, some relations constrain values while others describe change. Which variable is solved for is a later mathematical or computational choice; changing that order need not change the physical interaction.

Eliminate a variable only while retaining the relation or reconstruction needed by later use. MMP.9 supplies mathematical reduction and MMP.10 supplies a formulation for the chosen analysis. If two parts disagree at their connection, return to their quantity meanings and physical assumptions before changing a solver.

PHY.6:4.5 - Make the preparation compatible with the connected account

Supply the independent initial, boundary and driving conditions needed by the requested evolution. Substitute the proposed preparation into the connected relations. Solve for dependent initial values and reactions. Preserve the difference between a physical initial condition and a numerical starting guess.

A motion constraint also constrains admissible initial velocity. An electrical connection can constrain initial potentials. A spatial description needs boundary conditions appropriate to its transport and response. Determine which conditions can be chosen freely and which follow from the others.

An equation count can expose an omitted relation, but equal counts do not establish a consistent or uniquely determined problem. Examine the dependence of the actual relations and the existence conditions needed for the intended result. C.29.2 supplies the subsequent computational formulation; a solver diagnostic can help locate a problem without deciding whether the physical preparation should change.

If the preparation is incompatible, identify the conflict and its physical alternatives. Correct a mistaken initial value, relax an unjustified ideal constraint, or describe the interaction that establishes the new state. Choose among these from the actual preparation. Do not silently substitute an easier initial state.

PHY.6:4.6 - Derive the consequence and return through the implicated premise

Obtain the first result at the resolution the question needs. This may be an initial derivative, an integrated balance, a limiting state or a computed evolution. B.5.MPC connects the mathematical result and its conditions to the physical question.

Use checks that can distinguish a wrong construction for this use. Combining part balances can expose a duplicated exchange; an energy calculation can expose a sign error or an omitted conversion; substituting the preparation can expose an impossible constraint. A limiting case or available observation can test a disputed physical premise. Passing one such check establishes only what it examines.

Distinguish a consequence of the stated equations from evidence that those equations describe the intended situation. Numerical accuracy concerns how the chosen consequence was obtained. Physical adequacy concerns the premises, preparation and interactions represented. Keep those qualifications with the result, without requiring new evidence when the conditional result already serves the work.