Homogeneity (physics)

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For other uses, see homogeneous.

In physics, homogeneity can mean several things. Two completely distinct major meanings are

• the quality of having all properties independent of the position, i.e. translational invariance.
• the quality of an equation of having quantities of the same units on both sides, also referred to as dimensional homogeneity.

Contents 1 Translation invariance 1.1 Consequences 2 Dimensional homogeneity 3 See also 4 References

[edit] Translation invariance

Main article: translational invariance.

By translation invariance one means, the independence of the (absolute) position, especially when referring to a law of physics, or to the evolution of a physical system.

Indeed, fundamental laws of physics should not (explicitely) depend on the position in space, not only in view of the fact that this would make them quite useless. In some sense, this is also linked to the requirement that experiments should be reproducible. This principle is indeed true for all basic laws of mechanics (Newton's law, ...), electrodynamics, quantum mechanics, etc.

In practice, this principle is usually violated, since one studies only a small subsystem of the universe, which of course "feels" the influence of rest of the universe. This situation gives rise to "external fields" (electric, magnetic, gravitational,...) which make the description of the evolution of the system depending on the position (potential wells, ...). However, this only stems from the fact that the objects creating these extenal fields are not considered as ("dynamical") part of the system.

Translational invariance as described above is equivalent to shift invariance in system analysis, although here it is most commonly used in linear systems, whereas in physics the distinction is not usually made.

The notion of isotropy, for properties independent of direction, is not a consequence of homogeneity. For example, a uniform electric field (i.e. which has the same strength and the same direction in each point) would be compatible with homogeneity (in each point physics will be the same), but not with isotropy, since the field singles out one "preferred" direction.

[edit] Consequences

In the Lagrangian formalism, homogeneity (in space) implies conservation of momentum, and homogeneity in time implies conservation of energy. This is shown, using variational calculus, in standard textbooks like the classical reference [Landau & Lifshitz] cited below. This is a particular application of Noether's theorem.

[edit] Dimensional homogeneity

As said in the introduction, dimensional homogeneity is the quality of an equation of having quantities of same units both sides. A valid equation in physics must be homogeneous, since equality cannot hold between quantities of different nature. This can be used to spot errors in formulae or calculations. For example, if one is calculating a speed, units must always combine to [length]/[time]; if one is calculating an energy, units must always combine to [mass]·[length]²/[time]², etc. For example, the following formulae could be valid expressions for some energy:

if m is a mass, v and c are velocities, p is a momentum, h is Planck's constant, λ a length. On the other hand, the units of the right hand side do not combine to [mass]·[length]²/[time]², it cannot be a valid expression for some energy.

However, if the equation is homogeneous, it doesn't necessarily mean the equation will be true, since it does not take into account numerical factors. For example, E = m v² could be or could not be the correct formula for the energy of a particle of mass m travelling at speed v, and one cannot know if h c/λ should be divided or multiplied by 2π.

Nevertheless, this is a very powerful tool in finding characteristic units of a given problem, see dimensional analysis.

Theoretical physicists tend to express everything in natural units given by constants of nature, for example by taking c = ħ = k = 1; once this is done, one loses the possibility of the above checking.

[edit] See also

• Translational invariance
• Noether's theorem
• dimensional analysis

[edit] References

• Landau - Lifschitz: "Theoretical physics - I. Mechanics", chapter one.