Nöther's Theorem

This theorem derives conservation laws from the variational principle

If you have a theory based on a variational principle, Nöther's Theorem furnishes you with conservation laws that come from the invariance of the principle under continuous groups of transformations. For each parameter of the transformation, a conserved quantity is found. If you need a review of variational principles, see The Beautiful Theory, which takes you from generalized coordinates to Hamiltonian mechanics. Emmy Nöther was a mathematician at Göttingen, and revealed her theorem in 1918. It has been of great aid in quantum field theory.

As a concrete example, consider the dynamics of a system of particles, described by the Langrangian L(q,q'), where q stands for the coordinates, and q' for their time derivatives, the velocities. L is unchanged if a new time variable t' is introduced through t = t' + α, since t appears nowhere in L. This means that any time origin can be used for discussing the problem. Now suppose that α is a function of t', vanishing at the limits of the variational integral. This messing around with the time origin will change the velocities, so if we express the variational principle in terms of t' instead of t, we will find an added term proportional to the time derivative of α, α'. Now, α'dt' = δα is a variation at our disposal, since the function α(t') was arbitrary. The variational principle is still valid, so that the Euler-Lagrange equation corresponding to α is simply (d/dt)(q'dL/dq' - L) = 0, or pq' - L = constant. pq' - L we recognize as the Hamiltonian, or total energy if L is independent of t. This is the conservation law given by Nöther's Theorem for invariance under time displacement.

Alternatively, we could keep α constant, which would change the limits of the variational integral. If α is infinitesimal, the boundary terms that arise at the limits would then give the same result, that a certain quantity (the variation in the integrand) would take the same value at the limits. This quantity would be the same one found in the preceding paragraph.

Let us suppose that L is invariant under a common displacement δx of the x-coordinates. Now δL = [Σ(dL/dx')]d(δx)/dt, the sum over all the particles. Integrated between the limits t2 and t1, this gives [(Σmx')2 - (Σmx')1]δx = 0, or Σmx' = constant. Σmx' = Px, the x-component of the momentum. Considering all three axes, we find that P = constant. A similar result is obtained when infinitesimal rotations are considered, and the conserved quantity is the component of the angular momentum along the axis of the rotation.

These are all familiar results from mechanics, but the Theorem is useful wherever a variational principle exists. In electromagnetism, gauge invariance of the potential, which amounts to adding the gradient of a function to the vector potential, and a time derivative of the function to the scalar potential, which does not change the fields derived from the potentials, gives the conservation of charge. In quantum mechanics, invariance under a phase change of the wave function (multiplication by e), gives the conservation of particle number.

References

C. V. Lanczos, The Variational Principles of Mechanics (Toronto: U. of Toronto Press, 1970), pp. 401-405, 384-385.