Alternative Forms to the Basic Equations of Motion for a Particle in a Central Force Field. Recall the basic equations of motion as they will be our starting point: we derived the following constant of the motion: r2θ˙=h= constant This constant of the motion will allow you to determine the θ component of motion, provided you know the r component of motion. However, (8) and (9) are coupled (nonlinear) equations for the r and θ components of the motion. How could you solve them without solving for both the r and θ components? This is where alternative forms of the equations of motion are useful. Let us rewrite (8) in the following form (by dividing through by the mass m ):r¨−r θ˙2=mf(r) Now, using (10), (11) can be written entirely in terms of r:r¨− r 3 h 2​=mf(r) We can use (12) to solve for r(t), and the use (10) to solve for θ(t) Equation (12) is a nonlinear differential equation. There is a useful change of variables, which for certain important central forces, turns the equation into a linear differential equation with constant coefficients, and these can always be solved analytically. Here we describe this coordinate transformation.