## 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.