(adsbygoogle = window.adsbygoogle || []).push({}); Question:

A particle of massmstarting from rest at x=1 moves along the x axis toward the origin. Its potential energy is [itex] V=\frac{1}{2}mlnx[/itex]. Write the Lagrange equation and integrate it to find the time required for the particle to reach the origin.

Lagrange Equation in 1-D:

[tex]\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}-\frac{\partial L}{\partial x}=0[/tex]

[tex]L = T - V = \frac{1}{2}mv^{2}-\frac{1}{2}mlnx =\frac{1}{2}m\dot{x}^{2}-\frac{1}{2}mlnx [/tex]

Substitute L in Lagrange Equation:

[tex]\frac{d}{dt}\frac{\partial}{\partial\dot{x}}\left(\frac{1}{2}m\dot{x}^{2}\right)-\frac{\partial}{\partial x}\left(-\frac{1}{2}mlnx\right)=0[/tex]

[tex]\frac{d}{dt}\frac{\partial}{\partial\dot{x}}\left(\frac{1}{2}m\dot{x}^{2}\right)=\frac{\partial}{\partial x}\left(-\frac{1}{2}mlnx\right)[/tex]

[tex]\frac{d}{dt}m\dot{x}=-\frac{m}{2x}[/tex]

... And I don't really know what to do from here. The answer is given and it is supposed to be [itex]\Gamma\left(\frac{1}{2}\right)[/itex]. Can someone tell me where to go from where I left off? Thank you!

-Rick

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# Homework Help: Gamma function application

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