Patricia Crane

2022-01-18

Treacherous Euler-Lagrange equation
${\left({y}^{\prime }\right)}^{2}=2\left(1-\mathrm{cos}\left(y\right)\right)$ where y is a function of x subjected to boundary conditions , how might I find all its solutions?

Karen Robbins

Expert

$2y{}^{″}\left(x\right){y}^{\prime }\left(x\right)=2{y}^{\prime }\left(x\right)\mathrm{sin}y\left(x\right)$
which implies $y{}^{″}\left(x\right)=\mathrm{sin}y\left(x\right)$. After substitution $y\left(x\right)=\pi -\theta \left(x\right)$ this translates into $\theta {}^{″}\left(x\right)=-\mathrm{sin}\theta \left(x\right)$ which is the pendulum equation. Your boundary condition require that . Hence the solution is not periodic.

Becky Harrison

Expert

Use $1-\mathrm{cos}y=1-\left({\mathrm{cos}}^{2}\frac{y}{2}-{\mathrm{sin}}^{2}\frac{y}{2}\right)=2{\mathrm{sin}}^{2}\frac{y}{2}$.

alenahelenash

Expert

$1-\mathrm{cos}\left(y\right)=2{\mathrm{sin}}^{2}\left(\frac{y}{2}\right)$. Hence, ${y}^{\prime 2}=4{\mathrm{sin}}^{2}\left(\frac{y}{2}\right)⇒{y}^{\prime }=±2\mathrm{sin}\left(\frac{y}{2}\right)$