Central Forces

General formalism for orbits under central forces. Covers: Effective Potential and Radial Equation and Binet Equation and Orbits.

General idea of the topic

A central force always points toward a fixed point and its magnitude depends only on the distance to it. Gravity is the main example. Since the moment of the force about that point is zero, angular momentum is conserved and the motion takes place in a plane.

That conservation reduces a three-dimensional problem to one with a single variable, the distance r. The topic develops that reduction through an effective potential and then describes the shape of the orbits with Binet's equation.

What you will truly understand

By the end you will be able to write the radial equation with the effective potential, which adds the real potential and the centrifugal barrier, and use it to interpret whether an orbit is bound, circular or unbound.

You will also be able to obtain the shape of the orbit, not its time dependence, with Binet's equation, and recognize that for 1/r² conic sections result.

Recommended learning route

1. Effective Potential and Radial Equation: reduction of the problem to one variable, effective energy, turning points and circular orbits.
2. Binet's Equation and Orbits: differential equation for u = 1/r as a function of the angle and its solution for the gravitational force.

Physical criterion you must acquire

The essential criterion is to start from the constants of motion, angular momentum and energy, and use them to eliminate variables. A bound orbit exists where the total energy lies above the effective potential between two turning points, and the circular one corresponds to its minimum. For a non-gravitational force, the orbit may not close.

Connection with real problems

• orbits of planets and satellites.
• scattering of particles by a central field.
• motion of an electron in a classical Coulomb field.
• transfer maneuvers between orbits.

Final goal of the topic

The final goal is to analyze motion under a central force with the effective potential and deduce the shape of orbits from the constants of motion.