We all know the proof, from Newtonian mechanics, that the motion of the center of mass of a system of particles can be found by treating the center of mass as a particle with all the external forces acting on it. I want to prove the same think, but within the framework of Lagrangian mechanics, and I',m having some trouble.(adsbygoogle = window.adsbygoogle || []).push({});

So to start, the Lagrangian of a system of N particles is

[tex]L=\sum_{i=1}^N \left(\frac{1}{2}m_i v_i^2 - U_{\mathrm{ext.}, i}(\vec{r}_i)\right) - U_{\mathrm{int.}}(\vec{r}_1, ..., \vec{r}_N).[/tex]

[itex]U_{\mathrm{ext.},i}[/itex] is the potential energy due to any external forces on the ith particle. With a little algebra, we can rewrite this as

[tex]L=\frac{1}{2}MV^2 - U_{\mathrm{ext.}} + \sum_{i=1}^N \frac{1}{2}m_i \tilde{v}_i^2 - U_{\mathrm{int.}}[/tex]

where [itex]M=\sum m_i[/itex] is the total mass, V is the speed of the center of mass, [itex]U_{\mathrm{ext.}} = \sum U_{\mathrm{ext.},i}[/itex], and [itex]\tilde{v}_i[/itex] is the speed of the ith particle relative to the center of mass.

I don't know where to go from here. Help?

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# Center of mass in Lagrangian mechanics

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