Principle of least action

Laws of physics have a common form called the principle of least action. All known physical laws can be derived from this principle.

This includes the laws of gravity and electromagnetism, and even thermodynamics is based on it, although couched in the statistical methods needed to deal with large degrees of freedom.

The Lagrangian and the action

For every mechanical system, there exists a function of the generalised coordinates (and velocities) called the Lagrangian $L = L(q_i, \dot{q_i})$, which, when integrated over a period of time, defines the action, $A$, associated with the system $$A = \int_{t_1}^{t_2} L(q_i, \dot{q_i}) \textrm{d}t$$

We can apply the calculus of variations to derive the Euler-Lagrange equations of motion $$\frac{\partial L}{\partial q_i} – \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial L}{\partial \dot{q_i}} \right) = 0 \quad \forall i = 1, \dots, N$$ to find a unique trajectory defined by the coordinates $q_i$.

The Lagrangian, as defined, is very general and in this form, applies even to quantum mechanics. For all known classical systems, the Lagrangian can be written as the difference between the total kinetic energy, $T$, of the system and the potential energy, $U$.

That is, for classical systems, $$L = T - U$$

Nb. It's now apparent why we've been using the symbols $L, A$ throughout the course!

Comparison with Newton

Newton's Laws are differential equations, which are local laws - laws which tell you where you will be in the next instant, given the current position and velocity.

The principle of least action is a global law, which describes the whole trajectory, given a particular initial and final position. The result is the unique curve which minimises the action.

In fact, the two laws are completely equivalent (for classical systems) - we need only apply the principle of least action at the local level, or integrate Newton's Law over some (time) interval to convert between them.

The benefit of using the principle only really shows up when you are dealing with many degrees of freedom, or where the coordinates don't necessarily correspond to the three Euclidean components. Under these conditions, Newton's law may be practically impossible to apply.

Canonically conjugate momentum

We call the quantity $$\pi_i = \frac{\partial L}{\partial \dot{q_i}}$$ the momentum canonically conjugate to the coordinate $q_i$. This is equivalent to the usual concept of momemtum when the coordinates are Euclidean, but we shall see cases where it includes other terms.

In an analagous fashion, we define the generalised force as $$G_i = \frac{\partial L}{\partial q_i}$$

This is the Lagrangian form of $F = -\nabla U$. But it can also include other terms - for example, the centrifugal force, which isn't a force in the normal sense, but a pseudo-force due to angular motion of the system in question.

We can then rewrite the Euler-Lagrange equations in terms of these two quantities, $$G_i = \frac{\mathrm{d \pi_i}}{\mathrm{d} t}$$ which is clearly a generalisation of Newton's $$F = \frac{\mathrm{d p}}{\mathrm{d} t}$$