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Use Hamiltons canonical equations and integrate them to find expressions for q and p

  1. Nov 13, 2011 #1
    1. The problem statement, all variables and given/known data
    Suppose the potential in a problem of one degree of freedom is linearly dependent on time such that the Hamiltonian has the form:

    H= p^2/2m - mAtq

    where m is the mass of the object and A is contant

    Using Hamilton's canonical equations that are give below. Find the equations of motion and obtain the solution by integrating directly.


    2. Relevant equations

    q(dot) = ∂H/∂p
    -p(dot) = ∂H/∂q


    3. The attempt at a solution

    Finding q(dot) = ∂H/∂p = p/m → integrating q=q(knot) + pt/m

    Finding p(dot) = -∂H/∂q = -(-mAt) → integrating p= p(knot) +mAt^2/2

    the initial conditions were p(knot)= p and q(knot) = q at t=0


    Subbing the expression for p into the formula for q

    q= q(knot) + [p(knot) + mAt^2/2]*t/m = q(knot) + p(knot)t/m + 1/2*(At^3)




    The solution I obtained for my expression for q does not match the desired that was given by my professor of q(knot) + p(knot)t/m + 1/6*(At^3).



    I am trying to determine if I made a mistake somewhere. It appears to me that my solution is correct. Any guidance would be greatly appreciated.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 13, 2011 #2
    Re: Use Hamiltons canonical equations and integrate them to find expressions for q a

    I failed to realize p is also a function of time and i need to integrate it with respect to t as well. i.e. ∫ p(knot) + 1/2 mAt^2 dt
     
  4. Nov 13, 2011 #3

    dextercioby

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    Science Advisor
    Homework Helper

    Re: Use Hamiltons canonical equations and integrate them to find expressions for q a

    Yes, usually one of the 2 Hamilton's equations will help you eliminate P in favor of Q and turn the system of ODE's into a single ODE.
     
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