Conditions for Integral Functionals to have same Euler's Equations

Theorem
Let $ F \left({ x, \langle y_i \rangle_{1 \le i \le n}, \langle y_i' \rangle_{1 \le i \le n} } \right)$, $\Phi \left({ x, \langle y_i \rangle_{1 \le i \le n} } \right)$, $y=y \left({ x } \right)$ be real functions.

Let $ \Phi$ be twice differentiable.

Let

Let

$ \displaystyle J_1 \left[{ \langle y_i \rangle_{1 \le i \le n} } \right] = \int_a^b F  \left({ x, \langle y_i \rangle_{1 \le i \le n},  \langle y_i' \rangle_{1 \le i \le n} } \right) \mathrm d x$,

$ \displaystyle J_2 \left[{ \langle y_i \rangle_{1 \le i \le n} } \right] = \int_a^b \left( F  \left({ x, \langle y_i \rangle_{1 \le i \le n},  \langle y_i' \rangle_{1 \le i \le n} } \right)+\Psi  \left({ x, \langle y_i \rangle_{1 \le i \le n},  \langle y_i' \rangle_{1 \le i \le n} } \right)  \right) \mathrm d x$

be functionals.

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_{y_i}- \frac{ \mathrm d }{ \mathrm d x} F_{y_i'}=0$

Equivalently, for $J_2$ we have

Furthermore:

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


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

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