I'm solving a physics problem where I'm trying to find the equation of motion of this setup.
I've arrived on the equation $$\frac{u}{l}dt=\sqrt{1+2 \cos ^2 \theta} \; d\theta$$
I tried and failed to solve the resulting integral:
$$\int_{\pi}^{\alpha}\sqrt{1+2 \cos ^2 \theta} \; d\theta$$
Can anyone help me
Best Answer
Admitting that your equation is correct, you will face $$\frac u l t=\sqrt{3} E\left(\alpha \left|\frac{2}{3}\right.\right)-2 \sqrt{3} E\left(\frac{2}{3}\right)$$ where appears the elliptic integral of the second kind and the complte elliptic integral.
I suppose that you are looking for $\alpha(t)$ which is not possible.
However, we can make quite accurate approximations. For simplicity, I shall let $k=\frac u l t$.
For example, built around $\alpha=\pi$, the Taylor series of the rhs is $$\sqrt{3} (\alpha -\pi )-\frac{(\alpha -\pi )^3}{3 \sqrt{3}}+\frac{(\alpha -\pi )^5}{30 \sqrt{3}}+O\left((\alpha -\pi )^7\right)$$ which is more than decent for $\pi \leq \alpha\leq \frac 32 \pi$.
Using series reversion, $$\alpha=\pi +\frac{k}{\sqrt{3}}+\frac{k^3}{27 \sqrt{3}}+\frac{7 k^5}{2430 \sqrt{3}}+O\left(k^7\right)$$
Doing the same around $\alpha=2\pi$, the rhs would be $$2 \sqrt{3} E\left(\frac{2}{3}\right)+\sqrt{3} (\alpha -2 \pi )-\frac{(\alpha -2 \pi )^3}{3 \sqrt{3}}+\frac{(\alpha -2 \pi )^5}{30 \sqrt{3}}+O\left((\alpha -2 \pi )^7\right)$$ which is more than decent for $\frac 32 \pi \leq \alpha\leq 2 \pi$.
Using series reversion, $$\alpha=2 \pi +c+\frac{c^3}{9}+\frac{7 c^5}{270}+O\left(c^7\right)\qquad \text{where} \qquad c=\frac{k}{\sqrt{3}}-2 E\left(\frac{2}{3}\right)$$
Numerically, $ E\left(\frac{2}{3}\right) \approx 1.26119$.