Find Lagrangian and Hamaltonian equations of motion in polar coord.

In summary, the conversation discusses finding Hamilton's equations of motion for a particle attracted to a force center with a force of magnitude k/r^2. The problem specifies using plane polar coordinates and obtaining the Hamiltonian equations, but the individual is unsure about the equations for kinetic and potential energy. The individual also questions whether their obtained potential energy equation is correct and asks for input. Finally, the expert points out that the problem is 2D and thus \phi and p_{\phi} terms should be included in the Hamiltonian.
  • #1
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Homework Statement



A particle of mass m is attracted to a force center with the force center with the force of magnitude k/r2 . Use plane polar coordinates and find Hamilton’s equations of motion.

Homework Equations



(L)agrangian = T-U , U=-[tex]\int[/tex]F dr

The Attempt at a Solution



I can get the Hamiltonian equations of motion but what I am not confident about is the equations for kinetic and potential energy.
Are they correct?
Did I obtain the potential energy equation correctly?

Any input would be appreciated. Thank you.

http://b.imagehost.org/0511/tmp37F.jpg
 
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  • #2
The problem tells you to use plane polar coordinates; which means it is a 2D problem and so there are two coordinates and conjugate momentums to consider. Why haven't you included any [tex]\phi [/tex] and [tex] p_{\phi} [/tex] terms in your Hamiltonian?
 

1. What are the Lagrangian and Hamiltonian equations of motion in polar coordinates?

The Lagrangian and Hamiltonian equations of motion in polar coordinates are mathematical equations that describe the motion of a particle or system of particles in a coordinate system where the position of the particles is given in terms of polar coordinates (r, θ).

2. How do you find the Lagrangian equation of motion in polar coordinates?

To find the Lagrangian equation of motion in polar coordinates, you first need to determine the Lagrangian function, which is given by L = T - V, where T is the kinetic energy of the particle and V is its potential energy. Then, you use the Euler-Lagrange equation, which is d/dt (∂L/∂q̇) - (∂L/∂q) = 0, where q represents the generalized coordinates, to obtain the equation of motion.

3. How do you find the Hamiltonian equation of motion in polar coordinates?

To find the Hamiltonian equation of motion in polar coordinates, you need to use the Hamiltonian function, which is given by H = T + V. Then, you use the Hamilton's equations of motion, which are d/dt(q̇) = (∂H/∂p) and d/dt(p) = - (∂H/∂q), where p represents the generalized momentum, to obtain the equations of motion.

4. What are the advantages of using Lagrangian and Hamiltonian equations in polar coordinates?

The Lagrangian and Hamiltonian equations of motion in polar coordinates have several advantages, such as being able to describe the motion of a system in a non-inertial frame of reference, simplifying the equations of motion for systems with constraints, and providing a systematic approach to solving complex problems in classical mechanics.

5. Can the Lagrangian and Hamiltonian equations of motion be applied to all types of systems?

Yes, the Lagrangian and Hamiltonian equations of motion can be applied to all types of systems, including conservative and non-conservative systems, as long as the system can be described in terms of generalized coordinates and momenta.

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