Conditions for Integral Functionals to have same Euler's Equations

Theorem
Let $ \mathbf y $ be a real $ n $-dimensional vector mapping.

Let $ F \left( { x, \mathbf y, \mathbf y' } \right ) $, $ \Phi \left( { x, \mathbf y } \right) $ be real functions.

Let $ \Phi $ be twice differentiable.

Let

Let $ J_1 $, $ J_2 $ be functionals such that:

$ \displaystyle J_1 \left[ { \mathbf y } \right ] = \int_a^b F  \left( { x, \mathbf y,  \mathbf y' } \right ) \mathrm d x $,

$ \displaystyle J_2 \left[ { \mathbf y } \right] = \int_a^b \left ( F \left( { x, \mathbf y,  \mathbf y' } \right ) + \Psi \left ( { x, \mathbf y,  \mathbf y' } \right ) \right ) \mathrm d x $

Then $ J_1 $ and $ J_2 $ have same Euler's Equations.

Proof
According to the necessary conditions for an integral functional dependent on N functions to have an extremum for given function,

Euler's Equations for functional $ J_1 $ are:


 * $ \displaystyle F_{ \mathbf y } - \frac{ \mathrm d }{ \mathrm d x } F_{ \mathbf y' } = 0 $

Equivalently, for $ J_2 $ we have

Furthermore:

Since $ \Phi $ is twice differntiable, by Schwarz-Clairaut theorem partial derivatives commute and


 * $ \displaystyle \Psi_{ \mathbf y } - \frac{ \mathrm d }{ \mathrm d x } \Psi_{ \mathbf y' } = 0 $

Therefore, $ J_1 $ and $ J_2 $ have same Euler's Equations.