We can also use those tools to develop the Euler-Lagrange equation itself. In my humble opinion, this is a more straight forward derivation than is usually shown.
If L = T – V, with T a particle’s kinetic energy and V its potential energy, L is called the Lagrangian and the integral of the Lagrangian over a path in time is called the action. The principle of least action says that the path a particle actually takes is a minimum in action.
For a simple example, the Lagrangian of a particle in a uniform gravitational field is
Jumping to the answer, the Euler-Lagrange equation is
Plugging in, we eventually get
Which of course is obvious, from Newton’s second law.
Now let’s derive the equation on our own.
We have a function f that depends on x coordinate, x’s derivative in time x-dot, and t time. Time varies between t=a and t=b. We want to minimize the integral:
We can make this a discrete Riemann sum by dividing time up into even time slices, t = a, t = a + delta, t = a + 2 delta, etc.
The first term is
The second term is
The second x value is determined by x-dot1 and delta, but there is no way of determining x-dot2 from anything else (yet). So the x-dots just have to be the independent variables.
Altogether, the integral is
Note that each x-dotq appears once in the x-dot position, and appears in each x position for each i greater than q. So when we take the gradient, we get
Now we divide by delta and think about this equation. The first term is simply the partial derivative of f with respect to x-dot at some arbitrary time t. The second term is the integral (because it is a Riemann sum) of (the partial derivative of f with respect to x) from that time t to b. In other words (or symbols)
Now differentiate that with respect to t and we get
which is the equation we wanted to derive.