Consider a space ship in a stationary orbit around a planet. The potential energy of the ship is: U=-Km/r , where K=GM is the gravitational constant times the mass of the planet M, m is the mass of the ship, and r is the radial distance of the ship from the planet's center of mass. For simplicity consider the problem in two dimensions, so use a polar coordinate system with the planet at the origin.
a. Using the Lagrangian technique, derive a dynamical model for this system.
b. From the model in (a), you should have an equation for the angle ? and another equation for the radial distance r.
Consider a stationary orbit ? ?=?_0 with ?_0 constant assumed known. Add an input term u to the radial distance equation: it models the action of the ship engines in the radial direction. Using the system model derived in (a), find the equilibrium radius for the ship and then linearize the radial equation around this equilibrium radius.
c. Calculate the transfer function of the system determined in (b), by converting the state space representation of the linearized model.