Pierre Louis Maupertuis proposed that the path taken by a particle between two points in space is such that certain quantities are minimized.
He named this quantity the "action" of the path. His definition of action was the following:
where is the mass of the particle, is the velocity of the particle, and is the distance traveled by the particle.
Essentially, if the object is heavier, it will have a greater action, and if it moves faster, it will also have a greater action.
The principle of least action states that the path taken by a particle between two points in space is such that this action is minimized.
The definition of is not the most general form of the principle of least action, but it is a good starting point to understand the concept.
Imagine a particle moving from point to point in space, where the path is expressed as .
In order to allow for continuous paths, we integrate over the velocity and displacement instead of just multiplying them:
Next, since , we can rewrite the action as:
Notice that is just double the kinetic energy of the particle:
Next, is equal to the total energy minus the potential energy :
The second term, , is just the total energy of the system multiplied by the time taken to travel the path.
If we assume that the total energy of the system is constant, then this term is just a constant and does not affect the path taken by the particle.
Hence, we can ignore this term and focus on the first term, which is the difference between the kinetic and potential energies of the system.
We call this quantity the Lagrangian of the system:
The inputs to the Lagrangian are the position , the velocity , and the time .
The action can then be stated as follows:
The principle of least action states that the path taken by a particle between two points in space is such that the action is minimized (more accurately, it is stationary).
How do we write this mathematically?
In normal calculus, a stationary point of is one where the derivative of the function is zero.
This means that at the stationary point, an infinitesimal change in the input does not change the output .
We can apply a similar concept to the action . This time, instead of a function of a single variable, it has a function as an input.
is hence called a functional.
Imagine adding a small perturbation to the path , denoted as .
It is zero at the ends of the path, so that the path still starts at and ends at .