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.

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