Just a quick blast of calculus today.
Consider the Euler-Lagrange equation for a Lagrangian which depends on a single function :
In general, substitution of the Lagrangian into the Euler-Lagrange equation yields a second-order differential equation in . It turns out that if the Lagrangian does not explicitly depend on the independent variable , that is
the problem can be reduced to a first-order differential equation.
First we multiply the Euler-Lagrange equation through by the derivative of :
We then use a trick similar to the one used in the derivation of the Euler-Lagrange equation itself. Consider the following:
Rearranging for the second term on the right-hande side and substituting into the equation above yields
Now consider the differential of the Lagrangian:
Importantly, there is no partial derivative with respect to because the Lagrangian does not explicitly depend on it. Using this expression,
This is effectively the chain rule for a multivariable function. Using prime notation, this is the same as
But this is exactly what appears on the left-hand side of the equation in the previous paragraph. Using this result,
This is a first-order differential equation which will in general be easier to solve than the second-order Euler-Lagrange equation.