Invariance

TL;DR

• Identifies quantities or relations that do not change when specific transformations or operations are applied.
• Lets you reason about systems without tracking irrelevant details like orientation or the form of intermediate transformations.
• Appears across geometry, algebra, mechanics, and electromagnetism to simplify analysis and prediction.

Definition

Invariance is a concept in mathematics and physics that refers to the property of certain quantities or equations remaining unchanged under certain transformations or operations.

Explanation

Invariance describes when a quantity or equation stays the same despite applying particular transformations (for example, rotations, translations, or the addition of noninteracting sources). Because invariant quantities do not depend on the specific details of a transformation, they allow statements and calculations that hold regardless of orientation, position, or how components are transformed.

Examples

Euclidean distance

Euclidean distance is a measure of the straight-line distance between two points in a Euclidean space, calculated using the Pythagorean theorem. The distance between two points remains unchanged, or invariant, regardless of how the points are rotated or translated in space. This is because the Euclidean distance formula only depends on the coordinates of the two points, which remain unchanged under rotations and translations.

Conservation of energy (mechanics)

The principle of conservation of energy states that the total amount of energy in a closed system remains constant, regardless of how the energy is transformed from one form to another. For example, if a ball is rolling down a hill and gains kinetic energy, this energy is not lost when the ball reaches the bottom of the hill and comes to a stop. Instead, the kinetic energy is converted into potential energy, which is stored in the ball’s height above the ground. The total amount of energy in the system, the sum of the kinetic and potential energies, remains unchanged.

Symmetry in geometry

Symmetry refers to the property of an object or figure that remains unchanged when it is rotated or reflected in a specific way. For example, a square has four-fold rotational symmetry because it looks the same after being rotated 90, 180, or 270 degrees. This symmetry is invariant under these rotations because the shape of the square remains unchanged.

Principle of superposition (electromagnetism)

The principle of superposition states that the total electric or magnetic field at a point in space is the sum of the fields produced by individual sources. This means that the total field is invariant under the addition of new sources, as long as the sources are not interacting with each other. For example, if two electric charges are placed at a distance from each other, the total electric field at any point between them is the sum of the fields produced by each charge. This total field remains unchanged if an additional charge is placed at the same point, as long as the additional charge does not interact with the other two charges.

Use cases

• Geometry
• Algebra
• Mechanics
• Electromagnetism

Related terms

• Euclidean distance
• Pythagorean theorem
• Conservation of energy
• Kinetic energy
• Potential energy
• Symmetry
• Superposition principle