The Lie group approach to solving differential equations

Abstract

Certain ideas recur in many areas of mathematics. One example is groups of symmetries, which appear in the Galois theory of equations and in Lie groups. Lie groups are of great value in physics, where Noether’s theorem enables us to derive a conservation law for every case in which a function known as the Lagrangian is invariant under a one-parameter Lie group. The importance of this approach can be seen from the fact that the laws of the conservation of energy, linear momentum and angular momentum are all outcomes of Noether’s theorem, though they can of course be derived by simpler methods. The full power of Noether’s approach is shown in its applications to quantum field theory, where it can be used to find conserved currents and charges.