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1-D Lagrange and Hamilton equation gives different results.

  1. May 3, 2016 #1
    1. The problem statement, all variables and given/known data
    This was supposed to be an easy question. I have a question here that wants you to describe a yoyo's acceleration (in one dimension) using Lagrangian mechanics. I did and got the right answer. Now I want to use Hamilton's equations of motion but I get a wrong number. Here is the excerpt

    "A yoyo attached to a mass less string is suspended vertically from a fixed point and the other end is wrapped several times around a uniform cylinder of mass, m and radius R. When the cylinder is released it moves vertically down, rotating as the string unwinds. Write down the Lagrangian equation using the distance x as your generalized coordinate. Find the Lagrange equation of motion and show that the cylinder accelerates downward with #\ddot{x}# = 2g/3"
    2. Relevant equations
    I used Hamilton equation xdot = partial H/partial P
    To help,
    L = (3/4) m xdot^2 + mgx
    and
    H = (3p^2)/(4m) - mgx

    3. The attempt at a solution

    I used Hamilton's equation and took the derivative
    d/dt (xdot) = xddot = d/dt(3p/4m)
    xddot = (3 xddot)/2
     
  2. jcsd
  3. May 3, 2016 #2

    TSny

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    The momentum, ##p##, that enters into the Hamiltonian should be the "canonical momentum" which is determined from the Lagrangian. You should find that ## p \neq m \dot{x}##.
     
  4. May 3, 2016 #3
    I get ##p = \partial L / \partial \dot{q} = (6/4) m \dot{x}## but I dont know how/where to replace this ##p## into. Back into L and then re equate the Hamiltonian then use the Hamiltonian equation?
     
  5. May 3, 2016 #4

    TSny

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    OK for your result for ##p##.

    I don't understand what you mean when you say "re equate the Hamiltonian". There is a specific prescription for constructing the Hamiltonian from the Lagrangian and the canonical momentum.
     
  6. May 3, 2016 #5
    We im stuck at ##p = (6/4)m \dot{x}## do I take this result and somehow put ##p## back into the lagrange?
     
  7. May 3, 2016 #6

    TSny

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  8. May 3, 2016 #7
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