- Newton’s formulation
- Minimising quantities
- Calculus of variations
- Useful theorems
- Euler-Lagrange equations
- Shortest distance between two points
- Principle of least time
We continue with Newton's formulation of the laws of motion, considering conservation of energy and momentum.
We then move on to the mathematical problem of mimimising functions in general, and minimising functionals in particular, since the action defined for a system is a functional and minimising this as-yet-undefined quantity is the key to classical mechanics.
We use the calculus of variations to derive the Euler-Lagrange equations, which are equivalent to Newton's equations, but in a more general context where Newton's equations would be way too difficult to derive for a given system.
Action is defined as an integral over time of a function called the Lagrangian, and our goal will be to find trajectories in phase space that minimise this quantity.
A note about forces. We are only considering conservative forces; gravitational forces, forces between charged particles, masses attached to each by springs, or rods etc. We do not consider forces such as friction, or resistive forces to be fundamental. Friction is in fact a statistical force - an averaging over a vast number of (conservative) forces acting at the atomic scale.