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Mathematics
Differential Geometry
Geodesics with arbitrary parametrization
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[QUOTE="wrobel, post: 6595298, member: 593228"] Let ##x=(x^1,\ldots,x^m)## be local coordinates in a manifold ##M##; and let ##\{\Gamma^i_{jk}(x)\}## be a connection. Assume that we have a curve ##x=x(t),\quad \dot x\ne 0##. Is this curve geodesic or not? My guess is that the answer is "yes" iff for all ##k,n## the function ##x(t)## satisfies the following system $$\dot x^k(\ddot x^n+\Gamma^n_{rj}\dot x^r\dot x^j)=\dot x^n(\ddot x^k+\Gamma^k_{rj}\dot x^r\dot x^j).$$ In my taste this system looks strange. Or not? [/QUOTE]
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Forums
Mathematics
Differential Geometry
Geodesics with arbitrary parametrization
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