CoV7

Minimize the functional from classical mechanics: $\int_{t_1}^{t_2}(\mathrm{Kinetic\,Energy} - \mathrm{Potential\,Energy})\,$

Potential Energy $U=c x^2\,$

Kinetic Energy $T=\frac{1}{2}m \dot{x}^2\,$

The functional to minimize is $\int_{t_1}^{t_2}\left(\frac{1}{2}m\dot{x}^2-cx^2\right)\,dt\,$ .

The Euler-Lagrange equation is $\frac{\partial L}{\partial x}=\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}\,$ .

Since there is no $t\,$ in the integrand, the E-L eqs reduce to

$L-\dot{x}\frac{\partial L}{\partial \dot{x}}=c_1\,$ .

$c_1=\frac{1}{2}m\dot{x}^2-cx^2-\dot{x}(m\dot{x})\,$

$c_1=\frac{-1}{2}m\dot{x}^2-cx^2\,$

$c_1\le 0\,$ and $c_1=-A^2\,$

$A^2=\frac{1}{2}m\dot{x}^2+cx^2\,$

$\dot{x}=\sqrt{\frac{2}{m}}\sqrt{A^2-cx(t)^2}\,$

$\frac{dx}{\sqrt{c}\sqrt{\frac{A^2}{c}-x(t)^2}}=\sqrt{\frac{2}{m}}dt\,$

$\frac{1}{\sqrt{c}}\sin^{-1}\left(\frac{x(t)\sqrt{c}}{A}\right)=\sqrt{\frac{2}{m}}t+c_2\,$

$x(t)=\frac{A}{\sqrt{c}}\sin\left(\sqrt{c}\left[\sqrt{\frac{2}{m}}t+c_2\right]\right)\,$