Suppose a particle with mass $m$ moves freely in $\mathbb{R}^{n}$ under the influence of a conservative force field with potential $\Phi\in C^{2}([t_{1},t_{2}]\times\mathbb{R}^{n})$; that is, its trajectory $u:[t_{1},t_{2}]\to\mathbb{R}^{2}$ satisfies $m\ddot{u}=-\Phi_{u}(t,u)$ for $t\in [t_{1},t_{2}]$.
I'm trying to show that for fixed boundary values $u$ is critical for the action functional $J$ given by $$J(u)=\int_{t_{1}}^{t_{2}}F(t,u(t),\dot{u}(t))dt,\qquad F(t,u,v)=\frac{m}{2}|v|^{2}-\Phi(t,u).$$
Thus, we have $$F(t,u(t),\dot{u}(t))=\frac{m}{2}|\dot{u}(t)|^{2}-\Phi(t,u(t)).$$
Let $u(t_{1})$ and $u(t_{2})$ be fixed. Then $$\begin{aligned} J(u)&=\int_{t_{1}}^{t_{2}}\left(\frac{m}{2}|\dot{u}(t)|^{2}-\Phi(t,u(t)\right)dt \\ &=\frac{m}{2}\int_{t_{1}}^{t_{2}}|\dot{u}(t)|^{2}dt-\int_{t_{1}}^{t_{2}}\Phi(t,u(t))dt.\end{aligned}$$
Note that $\Phi_{u}(t,u(t))=\frac{\partial}{\partial u}\Phi(t,u(t))=-m\ddot{u}(t)$. Therefore $$\begin{aligned}\int\frac{\partial}{\partial u}\Phi(t,u(t))du&=-m\ddot{u}(t)\int du \\ & =-m\ddot{u}(t)\cdot u(t)+C =\Phi(t,u(t))\end{aligned}.$$
Hence $$J(u)=\frac{m}{2}\int_{t_{1}}^{t_{2}}|\dot{u}(t)|^{2}dt+\int_{t_{1}}^{t_{2}}(m\ddot{u}(t)\cdot u(t)+C)dt.$$
However, I feel as though I am going in the wrong direction.
Edit: Using the correction: $$\int\frac{\partial}{\partial u}\Phi(t,u(t))du = -m\int\ddot{u}(t)\frac{du}{dt}dt = -m\int\ddot{u}(t)\dot u(t)dt = -\frac m2 \dot u^2 + C$$
I still end up with
$$\begin{aligned} J(u)&=\frac{m}{2}\int_{t_{1}}^{t_{2}}|\dot{u}(t)|^{2}dt+\int_{t_{1}}^{t_{2}}\left(\frac{m}{2}\dot{u}(t)^{2}+C\right)dt \\ &=\frac{m}{6}[|u(t)|^{3}]^{t_{2}}_{t_{1}}+[\frac{m}{6}u(t)^{3}+Ct]^{t_{2}}_{t_{1}} \\ &=\frac{m}{6}(|u(t_{2})|^{3}-|u(t_{1})|^{3})+\frac{m}{6}(u(t_{2})^{3}-u(t_{1})^{3})+C(t_{2}-t_{1})\end{aligned}$$
which is not equal to zero.
$\ddot u$ is not independent of $u$. I.e., $$\int\frac{\partial}{\partial u}\Phi(t,u(t))du \ne-m\ddot{u}(t)\int du$$
In fact $$\int\frac{\partial}{\partial u}\Phi(t,u(t))du = -m\int\ddot{u}(t)\frac{du}{dt}dt = -m\int\ddot{u}(t)\dot u(t)dt = -\frac m2 \dot u^2 + C$$