Noether’s Theorem

  1. Write the kinetic energy for a particle in projectile motion
  2. Write the potential energy for a particle in projectile motion
  3. We can now construct the Lagrangian L=T-V. Use the Euler-Lagrange Equations (\frac{d}{dt}\frac{\partial L}{\partial \dot{x}_i}=\frac{\partial L}{\partial x_i}) to find two conserved momenta and the equation of motion for free-fall.
  4. Can you see under what general condition momentum in a given direction will be conserved by studying the Euler-Lagrange Equations?

The fact that every differentiable symmetry of a Lagrangian yields a conserved quantity is known as Noether’s Theorem and was first proved by Emmy Noether in 1915. One of the founders of Abstract Algebra and a member of the Göttingen school of Mathematics, she fled to the United States in 1933 after being expelled from her position by the Nazi party. A selection of biographies of Emmy Noether can be found here.