Editing Calculus of variations (section)

== Beltrami's identity ==
In physics problems it may be the case that <math>\frac{\partial L}{\partial x} = 0,</math> meaning the integrand is a function of <math>f(x)</math> and <math>f'(x)</math> but <math>x</math> does not appear separately. In that case, the Euler–Lagrange equation can be simplified to the [[Beltrami identity]]<ref>{{cite web |author=Weisstein, Eric W. | url=http://mathworld.wolfram.com/Euler-LagrangeDifferentialEquation.html |title=Euler–Lagrange Differential Equation | website=mathworld.wolfram.com |publisher=Wolfram |at=Eq.&nbsp;(5)}}</ref>
<math display="block">L - f' \frac{\partial L}{\partial f'} = C \, ,</math>
where <math>C</math> is a constant. The left hand side is the [[Legendre transformation]] of <math>L</math> with respect to <math>f'(x).</math>

The intuition behind this result is that, if the variable <math>x</math> is actually time, then the statement <math>\frac{\partial L}{\partial x} = 0</math> implies that the Lagrangian is time-independent. By [[Noether's theorem]], there is an associated conserved quantity. In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system. This is (minus) the constant in Beltrami's identity.