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Laplace's Equation for Orbital Motion

A differential equation used to describe the motion in a central force field. While the full solution is complex, the underlying principle relates the radial and angular components of the force and motion.

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The statement of the theorem

For motion in a central force field, the radial component of the equation of motion, derived from the effective potential

V_{eff}(r) = \frac{1}{2} L^2/r^2 - \mu/r

, can be expressed using the radial coordinate

r

and the specific angular momentum

h

:\n

\frac{d^2u}{d\theta^2} + u = \frac{1}{h^2/\mu} \text{if } \mathbf{F} = -\frac{\mu}{r^2} \mathbf{\hat{r}}

\nwhere

u = 1/r

and

h

is the specific angular momentum. The solution

u(\theta)

determines the orbital geometry.

Source: Wikipedia