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Problem 4 (20 points): Prelude to our more general study of orbital motion, let's have a look at the Lagrangian for an object in a gravitational field. A satellite of mass m in the vicinity of a planet of mass M. We will study the motion in the plane containing the planet and the path of the satellite, so we can treat the problem as two-dimensional. Part A: Using polar coordinates (r, θ), write the Lagrangian for the motion of the satellite under the influence of the planet's gravity. (Make sure you use the more general 1/r form of the gravitational potential, don't assume it's constant.) Part B: Write the Euler-Lagrange equations for your Lagrangian from part A. The equations should satisfy ∂L/∂θ = 0. Identify the corresponding conserved quantity and show that this quantity is constant of the motion. Eliminate θ and obtain a single equation of motion for r. Show that there is one equilibrium value of r corresponding to a circular orbit. Part C: Geostationary orbit is a particular circular orbit in which a satellite remains fixed in position above a spot on the planet's surface, i.e. the motion of the satellite satisfies the conditions r = R and θ = θ0, where θ0 is the angular speed of the planet's rotation. Find the fixed radius R at which geostationary orbit is possible. It should only depend on the gravitational force of the planet. Now suppose the satellite is perturbed from its geostationary orbit by a small amount, i.e. take r = R + δr. Using your result from part B, Taylor expand the acceleration to first order in δr and show that the geostationary orbit is a point of stable equilibrium. What is the frequency of small oscillations in δr? (Hint: Here's a helpful calculus identity: (1 + x)^n ≈ 1 + n x for small x, where the symbol ≈ indicates terms of order x^2 and higher.) Part D: The presence of small oscillations in r will induce fluctuations in the previously constant angular speed as well. Again, working to first order in δr, solve for the modified θ(t). The orbital period T is the time required for θ to change by 2π; show that even with the added term, the orbital period is unchanged to this order.

Problem 4 (20 points): Prelude to our more general study of orbital motion, let's have a look at the Lagrangian for an object in a gravitational field. A satellite of mass m in the vicinity of a planet of mass M. We will study the motion in the plane containing the planet and the path of the satellite, so we can treat the problem as two-dimensional.

Part A:
Using polar coordinates (r, θ), write the Lagrangian for the motion of the satellite under the influence of the planet's gravity. (Make sure you use the more general 1/r form of the gravitational potential, don't assume it's constant.)

Part B:
Write the Euler-Lagrange equations for your Lagrangian from part A. The equations should satisfy ∂L/∂θ = 0. Identify the corresponding conserved quantity and show that this quantity is constant of the motion. Eliminate θ and obtain a single equation of motion for r. Show that there is one equilibrium value of r corresponding to a circular orbit.

Part C:
Geostationary orbit is a particular circular orbit in which a satellite remains fixed in position above a spot on the planet's surface, i.e. the motion of the satellite satisfies the conditions r = R and θ = θ0, where θ0 is the angular speed of the planet's rotation. Find the fixed radius R at which geostationary orbit is possible. It should only depend on the gravitational force of the planet.

Now suppose the satellite is perturbed from its geostationary orbit by a small amount, i.e. take r = R + δr. Using your result from part B, Taylor expand the acceleration to first order in δr and show that the geostationary orbit is a point of stable equilibrium. What is the frequency of small oscillations in δr?

(Hint: Here's a helpful calculus identity: (1 + x)^n ≈ 1 + n x for small x, where the symbol ≈ indicates terms of order x^2 and higher.)

Part D:
The presence of small oscillations in r will induce fluctuations in the previously constant angular speed as well. Again, working to first order in δr, solve for the modified θ(t). The orbital period T is the time required for θ to change by 2π; show that even with the added term, the orbital period is unchanged to this order.

problem 4 20 points prelude to our more general study of orbital motion let have look at the lagrangian for ohject in gravitational field satellite of mass in thc vicinity of planet of mass 51169

Added by Linda G.

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