Central Force Gallery

The Central Force Gallery is a sub-gallery of the ODE gallery. The exhibits in this gallery are examples of second order ordinary differential equations (ODEs) in two dimensions. You can look at the documentation for the ODE(2D) 2nd Order Gallery for more information on such equations. You can find more general information on ODE exhibits in the ODE documentation.

In a Central Force exhibit, the the differential equation is Newton's third law of motion, F = ma, where the force F is a "central" force, that is, one that is directed towards (or away from) the origin and whose magnitude depends only on the distance of the point from the origin. In the equation, a stands for acceleration and m is a constant that represents mass (and can be taken to be 1). The solution curves of the ODE represent the motion of a particle that moves under the influence of the given force; that is, the solutions represent "orbits" of the particle around the origin. (Central Force also models the motion of two particles in space when the force between them is directed along the line joining them and has a magnitude that depends only on their separation.)

Since acceleration is the second derivative of position and F is a function of position, the equation F = ma is a second order autonomous ODE. Initial conditions for such an equation consist of the position and velocity at a given time. Recall that is is possible to input the initial condition by clicking-and-dragging with the middle mouse button (or by holding down the ALT/Option button while dragging). This allows you to "launch" a particle in the central force field and observe its motion.

While it is possible to consider particles moving in three dimensions, the motion of the particle will always be restricted to the plane that contains the origin and the initial position and that is parallel to the initial velocity vector (since there is no force to nudge the particle out of that plane). This justifies restricting Central Force exhibits to two dimensions.