Hamiltonian of a Point particle on a frictionless plane

In summary, the conversation discusses a problem involving Hamiltonian expressions and circular orbits. The person is struggling with questions e and f and is unsure how to apply Hamilton's equation and use Latex. They receive advice on how to solve the problem and are reminded to use Latex in the future.
  • #1
S1000
1
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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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  • #2
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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  • #3
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.
 

1. What is the Hamiltonian of a point particle on a frictionless plane?

The Hamiltonian of a point particle on a frictionless plane is a mathematical function that describes the total energy of the particle in terms of its position and momentum. It is represented by the symbol H and is equal to the sum of the particle's kinetic and potential energies.

2. How is the Hamiltonian of a point particle on a frictionless plane calculated?

The Hamiltonian of a point particle on a frictionless plane is calculated using the Hamiltonian equation, which is H = T + V, where T represents the kinetic energy of the particle and V represents its potential energy. The values for T and V are determined by the particle's position and momentum.

3. What is the significance of the Hamiltonian in physics?

The Hamiltonian is a fundamental concept in physics and is used to describe the dynamics of a system. It is a conserved quantity and remains constant as the system evolves over time. The Hamiltonian also plays a crucial role in quantum mechanics and is used to calculate the probability of a particle's position and momentum.

4. How does friction affect the Hamiltonian of a point particle on a frictionless plane?

Friction does not affect the Hamiltonian of a point particle on a frictionless plane, as the system is assumed to be free from any external forces or resistances. In the absence of friction, the particle's kinetic energy remains constant, and the potential energy is determined solely by its position.

5. Can the Hamiltonian of a point particle on a frictionless plane change over time?

Yes, the Hamiltonian of a point particle on a frictionless plane can change over time if external forces or resistances are introduced. In this case, the Hamiltonian is no longer conserved, and the system's dynamics will be affected. However, in the absence of such external factors, the Hamiltonian remains constant, and the system's behavior is predictable.

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