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Geodesic equation

  1. Aug 12, 2013 #1

    tom.stoer

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    The geodesic equation follows from vanishing variation ##\delta S = 0## with

    ##S[C] = \int_C ds = \int_a^b dt \sqrt{g_{ab}\,\dot{x}^a\,\dot{x}^b}##

    In many cases one uses the energy functional with ##\delta E = 0## instead:

    ##E[C] = \int_a^b dt \, {g_{ab}\,\dot{x}^a\,\dot{x}^b}##

    Can this be generalized for other functions f with ##\delta F = 0## and

    ##F_f[C] = \int_a^b dt \, f\left(\sqrt{g_{ab}\,\dot{x}^a\,\dot{x}^b}\right)##
     
  2. jcsd
  3. Aug 12, 2013 #2

    martinbn

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    If the function [itex]f[/itex] is monotone.
     
  4. Aug 12, 2013 #3

    tom.stoer

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    That was my idea as well, but I don't see how to generalize the proofs used for S and E. They rely partially on the L2 norm, special case of inner product etc.

    Wikipedia writes "The minimizing curves of S ... can be obtained by techniques of calculus of variations ... One introduces the energy functional E ... It is then enough to minimize the functional E, owing to the Cauchy–Schwarz inequality ... with equality if and only if |dγ/dt| is constant"
     
    Last edited: Aug 12, 2013
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