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Old Nov 24th 2016, 02:52 PM   #1
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Newton's 2nd law in curved space

Hi,

I am struggling to find information on applying Newton's second law to a particle moving freely (ma=0) on a 3D curved surface like, say, the surface of a torus. What would the trajectory of such a particle undergoing free motion be if it was restricted to travel on a curved surface? So far I found the so called geodesic equations (http://cs.stanford.edu/people/jbaek/18.821.paper2.pdf, pg.8). Do these equations predict the free motion of a point particle restricted to travel on a curved surface? If so, why do we need the term containing the Christofel Symbols added to the term $\displaystyle ddot x = 0$?

Thanks
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Old Nov 24th 2016, 03:50 PM   #2
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Originally Posted by crazyCoconut View Post
Hi,

I am struggling to find information on applying Newton's second law to a particle moving freely (ma=0) on a 3D curved surface like, say, the surface of a torus. What would the trajectory of such a particle undergoing free motion be if it was restricted to travel on a curved surface? So far I found the so called geodesic equations (http://cs.stanford.edu/people/jbaek/18.821.paper2.pdf, pg.8). Do these equations predict the free motion of a point particle restricted to travel on a curved surface? If so, why do we need the term containing the Christofel Symbols added to the term $\displaystyle ddot x = 0$?

Thanks
The general solution can be found using Differential Geometry (upon which GR is based.) If you are doing a simple problem, like that of motion on the surface of a sphere, you simply need to add a constraint of the form f(x, y, z) = constant to the set of equations. If you have to use a full blown metric in some kind of space-time then you would have to do the full GR treatment.

The Christoffel symbols are easy enough to see once you know what they do. One of the main principles of GR is that we can erect a locally Minkowski coordinate system at any point on the manifold we are using. The Christoffel symbol "sews" the motion in the coordinate system at each point into a patchwork that covers the whole manifold. (This is actually too simple of an explanation but I don't know what your level is.)

-Dan
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