Movement of planet in central force - hamilton mechanics

[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!

 

[gabbagabbahey]

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)

 

[player1_1_1]

Thanks for answer;]
one of assumption was that the planet is not moving in third dimension - 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