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:

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

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

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

Proof
According to 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:


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

Equivalently, for $J_2$ we have

Furthermore:

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


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

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