- #1
Siberion
- 33
- 1
Homework Statement
Consider the following Hamiltonian
[itex]H=\frac{p^2}{2m}e^{\frac{-q}{a}}[/itex]
a: constant
m: mass of the particle
q corresponds to the coordinate, and p its momentum.
note: q' stands for the derivative of q.
a) Prove that for p(t) > 0 this system seems to describe a particle subject to a force proportional to [itex]q'^2[/itex]
b) Solve q(t) and p(t) using the initial conditions q(0)=0 and p(t)=mv , v>0
c) Find the asymptotic conditions for b), i.e. find q(t), p(t) and q'(t) for t-> infinity
d) What is the relation between this hamiltonian and the kinetic energy [itex]T=\frac{1}{2}mq'^2[/itex]
e) Determine the ratio of kinetic energy dissipation, i.e. dT/dt
I would like to ask for a little bit of help in order to understand this problem better. As far as I can tell, it corresponds to a dissipative system (hence question e), but I'm not familiarized with it. Is this a well known Hamiltonian for a common problem?
For a) I tried calculating the hamiltonian for a particle subject to a force given by [itex]F=\alpha q'^2[/itex]
I integrated the force to get a potential, thus constructing a Lagrangian with kinetic energy equal to [itex]T=\frac{1}{2}mq'^2[/itex] and potential energy [itex]V=-\frac{\alpha}{3}q'^3[/itex]
After calculating hamiltonian equations of motion, I'm kind of confused about how to get the exponential term around so I can relate it to the problem. Maybe I should start from another point?Thanks for your help and time!
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