Hi Everyone!(adsbygoogle = window.adsbygoogle || []).push({});

I have done three years in my undergrad in physics/math and this summer I'm doing a research project in general relativity. I generally use a computer to do my GR computations, but there is a proof that I want to do by hand and I've been having some trouble.

I want to show that the Lagrangian and the Hamiltonian are conserved along the geodesic. I could be wrong about that, but I actually think they might be equal to each other in this case.

The Lagrangian is: $$L = 1/2 g_{ab}V^{a}V^{b}= V_{b}V^{b}$$

and the Hamiltonian is [itex] H = 1/2 g^{ab}V_{a}V_{b}= V_{b}V^{b}[/itex]

where V is the geodesic and equals q dot.

Where do I go from here? Can I just say [itex]A =V_{b}V^{b}[/itex] and then take the Lie derivative of A along V, or should I take a covariant derivative of A? I really need some help. Please show the steps that you would take to prove this.

Also, how long did it take you guys to get comfortable with this mathematics? I've worked through the first half of the Schutz GR book, if you have any other recommendable resources, let me know!

Thanks in advance!

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# I Two Conserved Quantities Along Geodesic

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