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Invariant (physics)

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Title: Invariant (physics)  
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Subject: Gauge theory, Noether's theorem, Symmetry (physics), Symmetry operation, Conservation law
Collection: Concepts in Physics, Conservation Laws
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Invariant (physics)

In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.

Note: Invariance, does not imply not varying, it pertains to a condition where there is no variation of the system under observation, and the only applicable condition, is the instantaneous condition. Invariance pertains to now(). Now(+1), to a condition where all variations are solely due the internal variables, with no external aspects imparting nor removing energy (Newton´s law of motion: a system in motion continues in motion, unless an external force imparts or removes energy). That condition is met by using the partial derivative function, ∂f(internal)xf(external) and presuming/setting f(external)=constant, leading to ∂f(external)=1 using the chain rule. Obviously, this is a model used solely for calculations, and not a reality. Reality is, that at all and every instance, energy is both removed and added to any system in observation.

In the current era, the immobility of Polaris (the North Star) under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.

Another example of a physical invariant is the speed of light under a Lorentz transformation[1] and time under a Galilean transformation. Such spacetime transformations represent shifts between the reference frames of different observers, and so by Noether's theorem invariance under a transformation represents a fundamental conservation law. For example, invariance under translation leads to conservation of momentum, and invariance in time leads to conservation of energy.

Quantities can be invariant under some common transformations but not under others. For example, the velocity of a particle is invariant when switching from rectangular coordinates to curvilinear coordinates, but is not invariant when transforming between frames of reference that are moving with respect to each other. Other quantities, like the speed of light, are always invariant.

Invariants are important in modern theoretical physics, and many theories are expressed in terms of their symmetries and invariants.

Covariance and contravariance generalize the mathematical properties of invariance in tensor mathematics, and are frequently used in electromagnetism, special relativity, and general relativity.

See also

References

  1. ^ French, A.P. (1968). Special Relativity. W. W. Norton & Company.  
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