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Symmetry and invariance

Specify a transformation, identify what stays unchanged, and distinguish symmetry of an object from invariance of a physical model.

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After reading, you can

Describe a translation, rotation or reflection precisely enough to compare two cases.
Identify the particular feature or quantity that a transformation leaves unchanged.
Distinguish a change of coordinate description from an active change of a physical arrangement.
Check spatial-translation invariance in a stipulated constant-velocity model.
Keep the relevant environment, conditions and evidence attached to a physical symmetry claim.

Start with a specified transformation

A transformation is a rule for changing an arrangement or its description. A translation shifts it by a stated distance and direction. A rotation turns it through a stated angle about a specified centre or axis. A reflection makes its mirror arrangement across a specified line or plane. A quarter turn is 90°, and a half turn is 180°. Naming the operation includes these choices.

Invariance means that something specified remains unchanged under the transformation. For a quantity, compare its values; for a pattern, compare the relevant features. A symmetry of an object maps those features back onto themselves. The operation can move individual points even when the whole outline occupies exactly the same region afterward.

Say which feature has the symmetry

An ideal plain square outline has quarter-turn symmetry about its centre: the turned outline coincides with the original. A particular corner moves to another corner's former position. If a unique coloured mark is attached to one corner, the decorated pattern has changed after that turn. The outline and the decoration are different features to check.

Retaining one quantity does not establish symmetry of every feature. An ideal rectangle retains its area during a rigid rotation, but a 4 cm by 2 cm rectangle does not coincide with its original region after a quarter turn about its centre. State both the transformation and the target of the comparison. The table uses ideal geometric arrangements and the motion model introduced below.

The same word unchanged can refer to different comparisons
Specified comparison	What remains unchanged	What can change
Quarter-turn an ideal plain square about its centre	The occupied outline	The position of an individual corner
Quarter-turn a non-square rectangle about its centre	Its area and side lengths	The region occupied by its outline
Shift the coordinate origin while keeping its direction and unit	The physical arrangement and separation of fixed markers	Their reported position values
Translate every position in the stipulated zero-force motion by one constant	Its velocity and the constant-velocity relationship	Its initial and later positions in the fixed coordinates

Separate coordinate changes from physical changes

A coordinate assigns a position relative to an origin and a chosen positive direction. On our straight line, positive means right, and position x is expressed in metres. Let a second origin be fixed h metres to the right of the first, with the same direction and unit. The same point then has coordinate x' = x - h. The prime mark names the new description. Moving these coordinate labels leaves the markers where they are. This is a passive transformation.

For an active translation, keep the original coordinate system and compare an arrangement with a second arrangement shifted h metres right. Each corresponding point has x_new = x + h. The markers' physical positions differ between these arrangements. Both comparisons can retain the separation between markers, but they describe different changes. State which comparison you intend before interpreting the numbers.

Test the relationship, not equality of every number

Symmetry of a physical law or model concerns the governing relationship under a specified transformation. Transform the quantities and conditions consistently and check whether permitted motions remain permitted. Position values or directional components may change while the relationship keeps its form and validity. This differs from asking whether one object has a symmetric outline.

Use an ideal Newtonian particle of fixed positive mass moving on an unlimited straight line in an inertial frame, a frame in which zero net external force allows constant velocity. Stipulate zero net external force at every position throughout the times considered. The rule is then constant velocity, including the possibility of rest. In one dimension v denotes signed velocity: positive is right and negative is left.

Let x0 be the initial position at time zero and t the elapsed time. For a constant v, the position is x = x0 + v t: velocity multiplied by elapsed time gives the signed change of position. Add the same constant h to every position in a possible motion, keeping v and t unchanged. Then x_new = (x0 + h) + v t. It has the same constant-velocity form with initial position x0_new = x0 + h. This works for any constant shift in the stipulated unlimited zero-force model, so that model has spatial-translation symmetry.

Keep the environment and conditions in the comparison

A physical symmetry claim must specify how the relevant surroundings are treated. Moving an apparatus while leaving a source of force behind can change its behaviour. The symmetry comparison includes the relevant influences, initial conditions and boundaries. Changing a setup's orientation relative to gravity, for example, changes an influence that matters to many experiments.

Our translated particle motions satisfy the same zero-force condition everywhere because it is an explicit model assumption. A real cart moved into a region with resistance or a wall would need those interactions included. A new coordinate origin changes neither contact with that wall nor the physical forces. Check the setup before treating two different outcomes as evidence against a symmetry of the underlying law.

Keep invariance tied to its transformation

A quantity can be unchanged under one transformation and change under another. The rectangle's area is unchanged by its rigid rotation, but doubling both side lengths multiplies that area by four. Geometric similarity retains shape while size can change. An invariant quantity must therefore be named together with the transformation being considered.

Invariance under a transformation also differs from constancy as time passes. The translated particle model retains its constant-velocity relationship, while a moving particle's position changes with time. A check of one numerical case illustrates a relationship; the algebra for an arbitrary constant h establishes this model's translation property. Neither step establishes an exact symmetry for every real apparatus or every possible law.

Make a symmetry claim complete

A useful claim identifies the system or description, the transformation, the feature or relationship to retain, and the applicable conditions. For an ideal outline, geometry can settle the comparison. For a physical model, check its transformed equations and assumptions. For an experiment, compare corresponding observations with the relevant measurement limitations.

Use a precise conclusion: the plain outline has this half-turn symmetry; this separation is invariant under this origin shift; or this zero-force model admits the translated motion. These statements explain what has been checked and leave further claims to their own evidence.

Two shifts with the same separation

1. Original arrangement
A is at 1.20 m and B at 1.90 m. Their separation is 1.90 m - 1.20 m = 0.70 m.
2. Passive: origin 0.50 m right
Keep the markers fixed and subtract the origin shift. Their new labels are 0.70 m and 1.40 m. Separation remains 0.70 m.
3. Active: pair 0.50 m right
Keep the original origin and compare the shifted pair. Their positions are 1.70 m and 2.40 m. Separation remains 0.70 m.
4. Name what remains unchanged
The separation is invariant under both specified comparisons. Position labels alone do not tell you whether the arrangement or its description changed.

Fixed markers have supplied exact original positions A = 1.20 m and B = 1.90 m, with right positive. The coordinate-origin shift and the active pair shift are separate alternatives. Both preserve separation, while only the active comparison changes the markers' physical positions.

Worked examples

An outline and its decoration can differ

An ideal rectangular outline is 4 cm wide and 2 cm high, centred at a fixed point. A unique coloured dot is attached to its upper-right corner. Compare a half turn and a quarter turn about the centre. What remains unchanged?

After a half turn, the plain outline occupies the same region. Its former upper-right corner is now at the lower-left corner's former position.
The unique dot moves to that lower-left position, so the complete decorated pattern does not coincide with the original after the half turn.
After a quarter turn, the outline is 2 cm wide and 4 cm high relative to the fixed directions. It occupies a different region, although the side lengths are retained.
The area in all cases is 4 cm × 2 cm = 8 cm². Unchanged area alone does not establish coincidence of the outline or decoration.

The plain outline has half-turn symmetry about its centre, while the uniquely decorated pattern does not. A quarter turn preserves area but does not return this non-square rectangle's outline to its original region. The comparison uses exact ideal geometry.

Use different formulas for different shifts

Fixed markers A and B have supplied exact coordinates 1.20 m and 1.90 m on a line with right positive. First describe them from an origin 0.50 m to the right. Separately, compare an actively translated pair shifted 0.50 m right using the original origin. What changes in each case?

For the new origin, subtract the origin shift: x'_A = 1.20 m - 0.50 m = 0.70 m and x'_B = 1.90 m - 0.50 m = 1.40 m. The physical markers stay fixed.
For the active comparison in the original coordinates, add the shift: x_new,A = 1.20 m + 0.50 m = 1.70 m and x_new,B = 1.90 m + 0.50 m = 2.40 m.
The positive separations are 1.90 - 1.20 = 0.70, 1.40 - 0.70 = 0.70 and 2.40 - 1.70 = 0.70, all in metres.

Both comparisons retain the 0.70 m separation. The passive shift changes only position labels. The active shift compares pairs at different physical positions. The calculations are separate alternatives with stipulated exact inputs.

Check a translated motion in the stated model

In the unlimited zero-force Newtonian model, a particle starts at x0 = 0.40 m with constant velocity +0.30 m/s. Compare a second possible motion whose position is always 1.00 m to the right at the same elapsed time. Use supplied exact values over 0 to 2.00 s. Do both motions fit the model?

The first endpoint is x = 0.40 m + 0.30 m/s × 2.00 s = 1.00 m.
The translated initial position is x0_new = 0.40 m + 1.00 m = 1.40 m. With the same velocity, its endpoint is 1.40 m + 0.30 m/s × 2.00 s = 2.00 m.
Their signed displacements are 1.00 m - 0.40 m = +0.60 m and 2.00 m - 1.40 m = +0.60 m.
At every time in the interval, adding 1.00 m changes position without changing the constant velocity. Both retain the stipulated zero-force condition.

Both possible motions fit the same constant-velocity model with different initial positions. The model relationship is invariant under this spatial translation. These are hypothetical model histories; the process of relocating a real particle or apparatus would require its own physical description.

Try it yourself

An ideal 6 cm by 3 cm rectangle is quarter-turned about its centre with the surrounding coordinate directions fixed. Its area is unchanged. Does that establish that its outline occupies its original region?
Show the answer and reasoning
No. The area stays 18 cm², while the horizontal and vertical extents exchange from 6 cm by 3 cm to 3 cm by 6 cm.
Area is the invariant quantity in this comparison. Coincidence of the occupied outline is a different feature, which fails for this non-square rectangle and this turn.
A fixed marker is at x = 2.30 m. A new origin is 0.80 m to the right, with the same unit and direction. What is its new coordinate, and has the marker moved?
Show the answer and reasoning
x' = 1.50 m; the physical marker has not moved.
Subtract the new origin's original coordinate: 2.30 m - 0.80 m = 1.50 m. This is a passive relabelling of the same point. An active rightward shift would be a different comparison.
In the same unlimited zero-force model, x0 = 0.80 m and v = +0.20 m/s. Translate the possible motion 2.00 m right in fixed coordinates. Find both positions after 3.00 s and both displacements, using exact supplied values.
Show the answer and reasoning
Original position 1.40 m; translated position 3.40 m. Both displacements are +0.60 m.
The original is 0.80 m + 0.20 m/s × 3.00 s = 1.40 m. The new initial position is 2.80 m, giving 3.40 m. Subtract each motion's own initial position: 1.40 - 0.80 = 0.60 and 3.40 - 2.80 = 0.60, in metres. Both retain the same velocity and model conditions.
A particle's reported positions are 1.00 m and 1.60 m at two times. Reverse the coordinate direction while keeping the origin fixed: x' = -x. What happens to its signed displacement and the distance between these two positions?
Show the answer and reasoning
The signed displacement changes from +0.60 m to -0.60 m. The distance between the positions remains 0.60 m.
The new position labels are -1.00 m and -1.60 m, so the new signed change is -1.60 - (-1.00) = -0.60 m. Distance is the nonnegative separation. Changing these labels does not reverse the physical motion. This is a coordinate comparison, without a claim about mirror symmetry of every physical interaction.
Two apparatuses happen to give equal readings after a spatial shift. Has this one comparison established that every physical law is invariant under every transformation?
Show the answer and reasoning
No. It supplies a result for a particular setup, transformation and measured quantity.
Check the relevant conditions and measurement limitations, and state the narrow comparison. A universal claim covers additional systems and transformations that this result has not examined.
A cart's first model motion assumes zero net force and no obstacles. A physically shifted version reaches a wall left fixed in the room. Can the original free-motion symmetry comparison predict what happens during contact?
Show the answer and reasoning
No. Contact introduces an interaction outside that model's zero-force conditions.
Include the wall and its force in the new physical description. A changed setup can change an outcome without establishing that the underlying law lacks a spatial symmetry. Moving the coordinate origin alone would leave the physical contact unchanged.

What to carry forward

A symmetry claim specifies a transformation and the feature or relationship it leaves unchanged.
An object's outline, its decoration and a quantity such as area can have different invariance properties.
A passive coordinate change relabels the same arrangement; an active transformation compares changed arrangements.
A law or model can retain its relationship while individual position values change.
The unlimited zero-force constant-velocity model admits spatially translated motions with consistently changed initial positions.
Physical comparisons require the relevant environment and conditions; one matching result does not establish every symmetry.

References

Sources for the concepts and teaching approaches in this article.

The Feynman Lectures on Physics Volume I — Symmetry and vectorsRichard P. Feynman, Robert B. Leighton and Matthew Sands — California Institute of Technology, official online editionTextbook chapterThe Feynman Lectures on Physics, Volume I, Chapter 52: Symmetry in Physical Laws The Feynman Lectures on Physics, Volume I, Chapter 11: Vectorsfeynmanlectures.caltech.edu
Ganita Prakash, Grade 6, Chapter 9: SymmetryNational Council of Educational Research and Training (NCERT), IndiaTextbook chapter · PDFncert.nic.inRead source
University Physics Volume 1, 5.2 Newton's First LawWilliam Moebs, Samuel J. Ling and Jeff Sanny — OpenStax / Rice UniversityTextbook chapteropenstax.orgRead source
Computational Chemistry from Laptop to HPC: A notebook exploration of quantum chemistry — Symmetry, Active or passive point of viewT. Fransson, M. Delcey, I. E. Brumboiu, M. Hodecker, X. Li, Z. Rinkevicius, A. Dreuw, Y. M. Rhee and P. Norman (book-level author team) — KTH Royal Institute of TechnologyTextbook chapterkthpanor.github.ioRead source