Conditions for Integral Functionals to have same Euler's Equations

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

Let $\map F {x, \mathbf y, \mathbf y'}$, $\map \Phi {x, \mathbf y}$ be real functions.

Let $\Phi$ be twice differentiable.

Let:

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


 * $\displaystyle J_1 \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$


 * $\displaystyle J_2 \sqbrk {\mathbf y} = \int_a^b \paren {\map F {x, \mathbf y, \mathbf y'} + \map \Psi {x, \mathbf y, \mathbf y'} } \rd x$

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

Proof
According to Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions:

Euler's Equations for functional $J_1$ are:


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

Equivalently, for $J_2$ we have

Furthermore:

Since $\Phi$ is twice differentiable, by Schwarz-Clairaut Theorem partial derivatives commute and:


 * $\Psi_{\mathbf y} - \dfrac \d {\d x} \Psi_{\mathbf y'} = 0$

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