# Movement of planet in central force - hamilton mechanics

• player1_1_1
In summary, the conversation discusses the speaker's attempt to find motion equations for a mass moving around a central force, where the central force is represented by a mass in the middle of a Cartesian system. The speaker is using Hamilton mechanics and has generalized coordinates in a polar system. They have found the potential of centrifugal and central forces but are unsure of what to do with the Coriolis effect. They are also using plane polar coordinates to simplify the problem. The conversation concludes with the speaker realizing they do not need to include the Coriolis effect in their Lagrangian.
player1_1_1
Hello, sorry for my English:D

## Homework Statement

I am trying to find motion equations for a mass moving around a big mass (ex. planet around sun), assumption is that the mass in middle is static (so this can be reduced to moving of mass around central force in middle of cartesian system), and everything would be good but I don't know what to do with Coriolis effect (I am using polar system with angle and radius as generalised coordinates), using Hamilton mechanics

## Homework Equations

Hamilton equations, differential equations, motion equations

## The Attempt at a Solution

I have generalized coordinates in polar system: $$r,\phi,p_r,p_\phi$$ and I going to find lagrangial and hamiltonian, depending on general definition of both functions
$$\mathcal{L}\left(r,\phi,\dot r,\dot\phi\right)=\frac{m}{2}\left(\dot r^2+r^2\dot\phi^2\right)-U(r)$$
now I find potential of centrifugal and central force
$$\mathcal{L}\left(r,\phi,\dot r,\dot\phi\right)=\frac{m}{2}\left(\dot r^2+r^2\dot\phi^2\right)+\frac{GMm}{r}-\frac{\ell^2}{2mr^2}-U\left(F_c\right)$$
and now my problem is, what to do with general potential of Coriolis force? I know I can write is similar to electromagnetic force potential, but I don't know if Coriolis effect exist in this kind of motion (this system is inertial and Coriolis effect is connected with not inertial systems), when I know this, it will not be a big problem to find hamilton equations and finish divagations, so I only need answer for this, with explanation please;] thank you!

player1_1_1 said:
I have generalized coordinates in polar system: $$r,\phi,p_r,p_\phi$$ and I going to find lagrangial and hamiltonian, depending on general definition of both functions
$$\mathcal{L}\left(r,\phi,\dot r,\dot\phi\right)=\frac{m}{2}\left(\dot r^2+r^2\dot\phi^2\right)-U(r)$$
now I find potential of centrifugal and central force
$$\mathcal{L}\left(r,\phi,\dot r,\dot\phi\right)=\frac{m}{2}\left(\dot r^2+r^2\dot\phi^2\right)+\frac{GMm}{r}-\frac{\ell^2}{2mr^2}-U\left(F_c\right)$$
and now my problem is, what to do with general potential of Coriolis force?

Why are you even bothering to add in these pseudo-potentials to your Lagrangian? Just work with the gravitational potential alone and derive the general equations of motion...if you want to then assume a circular or elliptical orbit and see what happens to your equations, it is a straight forward matter of adding this constraint into your Lagrangian by writing it in terms of a single independent variable ($\rho$ and $\phi$ will have a specific relationship to each other for any given orbit you choose to look at)

Also, why are you using plane polar coordinates (as opposed to spherical polar coordinates)...isn't this really a 3D problem? (The gravitational potential you are using is the 3D- potential)

one of assumption was that the planet is not moving in third dimention - only in one plane; for simplify divagations. Of course, I know that I can work only with gravity force potential, but I added centrifugal force potential to lagrangian because I also need to express this function only by $$r,\dot r$$ coordinates. I didnt write all the calculations because I only needed to know what to do with Coriolis effect. Summing up, it means that since coriolis force is pseudo force, I don't need it in my lagrangian (only gravity force potential)? thanks for help!
edit: well, I already found out an answer for this;] I must say - I asked stupid question :D

Last edited:

## 1. What is central force in Hamiltonian mechanics?

Central force in Hamiltonian mechanics refers to a conservative force that acts on an object towards a fixed point or center. This force depends only on the distance between the object and the center, and not on its direction.

## 2. How does central force affect the movement of a planet?

Central force affects the movement of a planet by causing it to move in a curved path around the center of force. This path is known as an orbit and is determined by the strength of the central force and the planet's initial velocity.

## 3. What is Hamiltonian mechanics and how does it relate to the movement of planets?

Hamiltonian mechanics is a mathematical framework used to describe the motion of objects in a conservative system, such as the movement of planets. It uses the Hamiltonian function to calculate the position and velocity of an object at any given time.

## 4. Can Hamiltonian mechanics be used to accurately predict the movement of planets?

Yes, Hamiltonian mechanics can be used to accurately predict the movement of planets as it takes into account the conservation of energy and momentum, which are key factors in planetary motion.

## 5. How does the Hamiltonian function change when considering the movement of planets in a central force field?

The Hamiltonian function for the movement of planets in a central force field includes an additional term for the potential energy of the central force. This term is dependent on the distance between the planet and the center of force, and can be used to calculate the planet's orbit and velocity at any given time.

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