1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What are the hamilton equations of motion for homogeneous lagrangians?

  1. Feb 21, 2013 #1
    For a Lagrangian [itex]L(x^k,\dot{x}^k)[/itex] which is homogeneous in the [itex]\dot{x}^k[/itex] in the first degree, the usual Hamiltonian vanishes identically. Instead an alternative conjugate momenta is defined as

    [itex]y_j=L\frac{\partial L}{\partial \dot{x}^j}[/itex]

    which can then be inverted to give the velocities as a function of the position and momenta


    The Hamiltonian is then equal to the Lagrangian with the velocities replaced with this function


    We then find that

    [itex]\dot{x}^i=H\frac{\partial H}{\partial y_i}[/itex]

    which is one half of the Hamilton equations of motion. But what about [itex]\dot{y}_i[/itex]?

    I am following Hanno Rund The Hamilton-Jacobi equation in the Calculus of Variations. But Rund moves on from this point to the H-J equation, leaving me wondering about this question.
  2. jcsd
  3. Feb 22, 2013 #2
    Re: what are the hamilton equations of motion for homogeneous lagrangi

    I found the answer to this
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook