PX285 - B1 - variational principles

• allows quantities such as path lengths, and functions to be extremized

• considering a person taking different paths through strips of tarmac, beach and sea
• the time taken is:

variational principles in physics:

• optics: fermat's law, $l(x)$ is extremal
• waves: destructive interference is small for paths that interfere constructively (ie: are at the same phase), such that their phase difference is minimized
• relativity: trajectories in free space, in a gravitational field for general relativity, are those that extremize (maximize here) elapsed proper time:

- classical paths in flat space minimize velocity
- straight line paths give lower velocities, and hence higher values of $\tau$

• newtonian mechanics: trajectories extremize the action
• electromagnetism:

• path integrals:
  • quantum mechanics: (non-examinable)
  $$\psi (x,t)=\sum _{paths}\exp \left(\frac{1}{\hslash }{\int }_{t_0}^{t_1}Ldt\right)$$
  • equivalent to finding the solutions of schrödinger's equation
  • $\exp (i\varphi )$ is a quantum mechanical phase, with phase angle
  $$\varphi =\frac{1}{\hslash }A=\frac{1}{\hslash }{\int }_{t_0}^{t_1}Ldt$$
  • paths, $x(t)$, that minimize the action are paths for which the quantum mechanical phase is nearly the same for nearby paths $⟹$ constructive interference
  • in the limit of macroscopic particles, $N\sim {10}^{20}T$, interference is very strong $implies$ only this 'classical' trajectory is realized