Notes for AKT-140117/0:28:51

To demonstrate how to use the Euler-Lagrange equation in classical mechanics, we solve brachistochrone problem as an example. The problem is described in the blackboard shot, which is to find the path of a particle that minimizes the time $T$ traveled from point $P_{1}$ to point $P_{2}$ in a uniform gravitational field. In this situation, we assume there is no friction along the path; thus the energy is conserved. Let $y$ be the vertical coordinate. Then, by the conservation of energy, we have

${\frac {1}{2}}mv^{2}-{\frac {1}{2}}mv_{i}^{2}={\frac {1}{2}}mv^{2}=mgy.$

Thus, we have

$v={\sqrt {2gy}}.$

Then, the time $T$ may be described as

$T=\int _{P_{1}}^{P_{2}}{\frac {ds}{v}},$

where $ds$ is the infinitesimal arclength of the path. Then, let $x$ be the horizontal coordinate, we have $ds={\sqrt {dx^{2}+dy^{2}}}={\sqrt {1+y'^{2}}}dx.$ Thus, the above equation would be

$T=\int _{P_{1}}^{P_{2}}{\sqrt {\frac {1+y'^{2}}{2gy}}}dx.$

Now, let $L={\sqrt {\frac {1+y'^{2}}{2gy}}}$, we apply the Euler-Lagrange equation and obtain

${\frac {\partial L}{\partial y}}=-{\frac {1}{2y}}{\sqrt {\frac {1+y'^{2}}{2g}}}={\frac {d}{dx}}\left({\frac {\partial L}{\partial y'}}\right)={\frac {1}{\sqrt {2g}}}{\frac {2yy''-y'^{4}-y'^{2}}{2(y(1+y'^{2}))^{3/2}}}.$

If we rearrange the equation and integrate, we obtain the equation

$C={\frac {1}{\sqrt {2gy\left(1+y'^{2}\right)}}}$

where $C$ is some constant. Then, we rearrange the equation and obatin

$\left(1+y'^{2}\right)y={\frac {1}{2gC^{2}}}=k^{2}.$

Then, we can solve this equation with parameterization and obtain the final result

$x\left(t\right)={\frac {k^{2}}{2}}\left(t-\sin t\right),y\left(t\right)={\frac {k^{2}}{2}}\left(1-\cos t\right)$