Donghao.ouyang: Created page with "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,..."

2018-05-23T17:17:20Z

Created page with "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,..."

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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 <math>T</math> traveled from point <math>P_1</math> to point <math>P_2</math> in a uniform gravitational field. In this situation, we assume there is no friction along the path; thus the energy is conserved. Let <math>y</math> be the vertical coordinate. Then, by the conservation of energy, we have

<math>\frac{1}{2}mv^2-\frac{1}{2}mv_i^2=\frac{1}{2}mv^2=mgy.</math>

Thus, we have

<math>v=\sqrt{2gy}.</math>

Then, the time <math>T</math> may be described as

<math>T=\int_{P_1}^{P_2}\frac{ds}{v},</math>

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

<math>T=\int_{P_1}^{P_2}\sqrt{\frac{1+y'^2}{2gy}}dx.</math>

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

<math>\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}}.</math>

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

<math>C=\frac{1}{\sqrt{2gy\left(1+y'^2\right)}}</math>

where <math>C</math> is some constant. Then, we rearrange the equation and obatin

<math>\left(1+y'^2\right)y=\frac{1}{2gC^2}=k^2.</math>

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

<math>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)</math>

Notes for AKT-140117/0:28:51 - Revision history

Donghao.ouyang: Created page with "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,..."