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Physics
Special and General Relativity
Solving Geodesic Eq.: Mysterious Conservation Eq. (Sec. 5.4 Carroll)
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[QUOTE="vanhees71, post: 6311441, member: 260864"] This also follows from the fact that one possible Lagrangian (afaik the most convenient one) for the geodesic equation is $$L=\frac{1}{2} g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}.$$ Since this is not explicitly dependent on the world-line parameter ##\lambda## (derivatives wrt. ##\lambda## are denoted with a dot), the corresponding "Hamliltonian" is again ##L## itself, i.e., ##L=\text{const}## in this form of the action principle for geodesics you get automatically a parametrization with an affine parameter. For timelike (spacelike) geodesics you can simply set ##L=\pm 1## and for lightlike ones ##L=0##. The Euler-Lagrange equations are the usual geodesic equations for an affine parametrization, $$\mathrm{D}_{\lambda} x^{\mu}=\ddot{x}^{\mu} + {\Gamma^{\mu}}_{\nu \rho} \dot{x}^{\nu} \dot{x}^{\rho}$$ with $$\Gamma_{\mu \nu \rho}=\frac{1}{2} (\partial_{\nu} g_{\mu \rho} + \partial_{\rho} g_{\mu \nu} -\partial_{\mu} g_{\nu \rho}), \quad {\Gamma^{\sigma}}_{\nu \rho} =g^{\mu \sigma} \Gamma_{\mu \nu \rho}.$$ [/QUOTE]
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Forums
Physics
Special and General Relativity
Solving Geodesic Eq.: Mysterious Conservation Eq. (Sec. 5.4 Carroll)
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