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To demonstrate how to use the Euler-Lagrange equation in classical mechanics, we solve brachistochrone problem as an example. The problem is described in the blackboard shot, which is to find the path of a particle that minimizes the time
traveled from point
to point
in a uniform gravitational field. In this situation, we assume there is no friction along the path; thus the energy is conserved. Let
be the vertical coordinate. Then, by the conservation of energy, we have
Thus, we have
Then, the time
may be described as
where
is the infinitesimal arclength of the path. Then, let
be the horizontal coordinate, we have
Thus, the above equation would be
Now, let
, we apply the Euler-Lagrange equation and obtain
If we rearrange the equation and integrate, we obtain the equation
where
is some constant. Then, we rearrange the equation and obatin
Then, we can solve this equation with parameterization and obtain the final result