Question

a.) Derive the equations of motion for an object with mass(m1) that is orbiting the second object of mass (m2>>m1) in a perfectly circular orbit with radius(R) and orbital period (T).

(Use lagrangian mechanics & show work)

b.) Find the hamiltonian of the system and describe how it is related to the system energy.

Answer 1

Taking the second object as the origin the first object is orbiting the second object with a time period T.

The radius is R.

Then the potential energy of the second object is due to the central force between the masses, which is given by

As the motion in central force will on a plane, so we can consider the polar coordinates for the problem as the displacement in polar coordinates in given by

the velocity is given by

so the square of magnitude of the velocity is

here the motion is circular with constant radius R, so, the radial element is a constant, so

so

the square of magnitude of velocity

then the kinetic energy is

so the lagrangian becomes

then the lagrangian equation of motion is given as

where the q is the generalized coordinate. in our system there is only one generalized coordinate which is , so the Lagrage's equation of motion becomes

where

and as L is not a function of

then we get from lagranges equation of motion that,

or

which is the equation of motion.

b)

the hamiltonian of the system can be found from lagrangian by changing the variables using the relations

also the using the relation

for we get

where J is the angular momentum

then

so we can write the in terms of J as

so the hamiltonian will be

which gives

where we get

which is the total energy of the system