Hamiltonian of a Point particle on a frictionless plane

AI Thread Summary
The discussion revolves around difficulties in solving parts e and f of a problem related to the Hamiltonian of a point particle on a frictionless plane. The user struggles to apply Hamilton's equations correctly and is unsure how to incorporate the momentum expressions into the Hamiltonian. A key insight shared is that for circular orbits, the radial momentum \( p_r \) equals zero, which simplifies the Hamiltonian expression significantly. The final Hamiltonian is derived as \( H = \frac{p_\theta^2}{2mr^2} + \frac{kr^2}{2} \), and the user is advised to set \( \dot{p_r} = -\frac{\partial H}{\partial r} \) to find the necessary relationships for part f. The conversation emphasizes the importance of using LaTeX for clarity in mathematical expressions.
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Homework Statement
Find the Hamiltonian of a Point particle on a frictionless plane with a given potential (kr^2)/2
Relevant Equations
L = T - V

dot rep time derivative

L = (m/2) * ( rdot^2 + r ^2 θdot^2) - V (r,θ)
Lagrange eq
d/dt (∂L/∂xdot) = ∂L/∂x
H = Σ pi qdoti - L
9efPZ.png
I am stuck on Question e and then how to proceed to f. I cannot seem to show this using the steps in the prior questions. My answers are:

a)
1673167910977.png


b)
1673167927405.png

1673167962364.png


c)

1673167979252.png


c) continued - and d) at the bottom of the page
d)
1673168014929.png
I am not sure where I have gone wrong, as I am not sure how to apply the relevant Hamilton's eq to the Hamiltonian in e)
I can sub in p_r into the Hamiltonian in e) however, I cannot use the p_theta 'dot' expression.
and then also

for f) how to show the energy is equal to kr_0.
 
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Most people here won't have the patience to wade through that ugly mess.
You'll need to learn quickly how to use Latex on this forum. :oldfrown:
(Do a search for "latex" to find instructions.)
 
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Likes PhDeezNutz and BvU
I agree with your Hamiltonian Expression, to complete the problem (the last few parts) realize that ##\dot{r} = 0## if we're talking about circular orbit ##r = r_0##. Which from the EL equation involving ##r## should tell you ##p_r = 0## (Which in turn tells you ##\dot{p_r} = 0##) (THIS IS THE KEY PART)

This cancels out a term in your Hamiltonian leaving.

## H = \frac{p_\theta^2}{2mr^2} + \frac{kr^2}{2}##

Take the expression for ##H## and find ##\dot{p_r} = - \frac{\partial H}{\partial R} ## set it equal to ##0## and solve for ##\frac{p_\theta^2}{2mr^2}## and plug it back into ##H## your answer should pop out.

Your assignment is likely turned in by now but you may find this helpful anyway.

But yes, use Latex next time, It is awesome.
 
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