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.
q(dot) = ∂H/∂p
-p(dot) = ∂H/∂q
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.