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Eliminating velocities from Lagrange E.o.M

  1. Jun 15, 2012 #1
    Hello, this doubt is related to Generalized Hamiltonian Dynamics paper by Dirac.

    Consider the set of n equations : pi = [itex]\partial[/itex]L/[itex]\partial[/itex]vi,

    (where vi is qi(dot) = dqi/dt, or time derivative of qi)(L is the lagrangian, q represent degrees of freedom in configuration space)

    Now Dirac says : "If the n quantities [itex]\partial[/itex]L/[itex]\partial[/itex]vn on the right-hand side of the given equations are NOT independent functions of the velocities, we can eliminate the v's from (the above-given) set of equations and obtain one or more equations:
    j(q, p) = 0, (j = 1, 2, ... ,m if there are m such constraints)"

    Could anyone please explain how this comes about? I can't understand how the v's can be necessarily eliminated, and if the p's are not all independent, then we can simply obtain relations like Ʃaipi = 0 (where a's are non-zero coefficients), not involving q's at all.

    Thanks.
     
  2. jcsd
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