Beltrami Identity - Derivation

Derivation

The following derivation of the Beltrami identity starts with the Euler-Lagrange equation,

Multiplying both sides by u′,

According to the chain rule,

where u′′ = du′/dx = d2u / dx2. Rearranging,

Substituting this expression for u′ ∂L/∂u into the second equation of this derivation,

using chain rule again,

and rearranging,

For the case of ∂L / ∂x = 0,

and taking the antiderivative results in the Beltrami identity,

where C is a constant.