# Solving a Linear Time-Dependent Hamiltonian Problem

• tphysicsb

## Homework Statement

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.

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

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.