Question

Suppose that the potential in a problem of one degree of freedom is linearly dependent on time such that the Hamiltonian is of the form

H = p²/2m - mAtq

where m is the mass, A is a constant, q is the coordinate, p is the momentum, and t is time.
Using Hamilton's canonical equations that are given in Goldstein eq. 8.18, find the equations of motion and obtain the solution by integrating directly.

Answer 1

The equations of motion for the given Hamiltonian are q = (p/m)t - mAt²/2 and p = -mA*t + p₀.

Step-by-step explanation:

The equations of motion can be obtained by using Hamilton's canonical equations, which are given by:

q' = ∂H/∂p

p' = -∂H/∂q

Substituting the given Hamiltonian H = p²/2m - mAtq, we have:

q' = (p/m) - mA

p' = (-mAt)

To obtain the solution, we can integrate the equations of motion:

q = (p/m)t - mAt²/2

p = -mA*t + p₀

where p₀ is the initial momentum.