References: Tom Lancaster and Stephen J. Blundell, Quantum Field Theory for the Gifted Amateur, (Oxford University Press, 2014) – Section 1.4.
A fundamental application of functional derivatives is in the derivation of the principle of least action and the Euler-Lagrange equation. A particle with mass following some trajectory will have a potential energy and a kinetic energy at each point on the trajectory. If the particle follows the trajectory between times and , then these energies have averages given as functionals
From example 1 in the earlier post, we have
From example 2 in the same post, we have
In classical physics, all forces are ultimately represented by conservative forces (gravity and electromagnetism), so a force can be represented by the gradient of a potential: . Newton’s law then says
so in terms of functional derivatives
That is, the variation in the average kinetic energy over the path is equal to the variation in the average potential energy. This suggests that the quantity
known as the Lagrangian, is in some sense fundamental, since the variation in its integral over a path is zero. To give a name to this integral, we define the action as
giving rise to the condition
Technically, this means that the action is stationary when the trajectory is the actual trajectory followed by a particle. In pracitice, the action nearly always turns out to be a minimum, so this is known as the principle of least action. Varying the trajectory slightly causes the action to increase and gives a path that is not followed by the particle.
If we know the Lagrangian for a particle (or more generally, for a system of particles), we can work out the equation(s) of motion by applying the principle of least action. To do this, we consider the Lagrangian as a function of a generalized coordinate and its first derivative :
We can then apply the principle of least action:
This has the same form as example 2 from the earlier post that we used above, so we get
Thus we get the Euler-Lagrange equation for a single particle moving in one dimension
Example As a simple example, suppose we have a particle whose potential and kinetic energies are given by
the equation of motion is given by
This is just the equation of motion for a mass on a spring with spring constant .