when reading some article, i have some problem to get the force $\mathbf{F}$ related to the energy functional from Hamilton's principle.
Given the functional: $$ E=\frac{1}{2}k_{s}\int\left(\left|\frac{\partial \mathbf{X}}{\partial s}\right|-1\right)^{2}\mathrm{d}s$$ where $\mathbf{X}(s,t)$ is a position vector at different time $t$ in curvilinear coordinate $s$ for a closed curve.
The result in the article is: $$\begin{gathered} \mathbf{F}(s,t)={\frac{\partial T\hat{\boldsymbol{\tau}}}{\partial s}} \\ T=K_{s}\left(\left|{\frac{\partial\mathbf{X}}{\partial s}}\right|-1\right), \\ \hat{\tau}=\frac{\frac{\partial\mathbf{X}}{\partial s}}{\left|\frac{\partial\mathbf{X}}{\partial s}\right|}, \end{gathered}$$
Can anyone please give me some advice on vector calculus to solve this problem?
