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kcirick
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Question:
A particle of mass m starting 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
A particle of mass m starting 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