Orbit (dynamical system)

An orbit is the set of states reached from one initial state under a dynamical system's evolution. A forward orbit restricts attention to future times; a complete orbit includes backward times when the model admits them. Keeping the time ordering gives a trajectory, while forgetting the parameter emphasizes the geometry of the visited set.

For a harmonic oscillator, the phase plane orbit is an ellipse determined by its mechanical energy. The particle repeatedly visits the same positions, but its full state distinguishes motion in opposite directions. A damped oscillator instead approaches an equilibrium point along a spiralling or nonoscillatory orbit, depending on its parameters. The state-space description is therefore broader than a spatial orbital path: the variables can describe velocities, temperatures or populations as well as physical positions.

A bound orbit whose direction of closest approach changes between radial periods. For integrable central motion, closure requires a rational ratio of radial and azimuthal frequencies.

A body's position along its orbit, specified relative to a stated origin and time. Knowing only the shape and size of an orbit does not specify where the body lies on it: an initial orbital phase or timing condition is also needed. A rendezvous requires matching orbital positions at the same time as well as matching velocities.

The instantaneous distance from an attracting centre to an orbiting body. It is constant on a circular orbit but varies along an elliptic orbit. For Kepler's third law the relevant size is the semi-major axis, which equals the orbital radius only for a circular orbit.

A closed orbit returns to the same complete state and repeats its motion. For a nonconstant solution of an autonomous differential equation with unique solutions, $x(t+T)=x(t)$ for a positive period $T$ describes a periodic orbit. This usage excludes an equilibrium point, which is a stationary state rather than a moving cycle.

A harmonic oscillator has closed ellipses in its phase plane, because both position and velocity return after one oscillation. Returning to position alone is insufficient: a particle can pass the same point with a different velocity. A bound orbit also need not close; a precessing orbit can remain in a limited spatial region while changing its direction of closest approach. Specifying the complete state prevents a repeating projection from being mistaken for a repeating physical motion.

An orbit that remains within a bounded spatial region for all times for which the motion is defined. In an attractive inverse-distance central potential with zero potential energy at infinity, a noncolliding orbit is bound when its total mechanical energy is negative. That sign test depends on the potential and its energy origin; it is not a universal definition of bounded motion. A closed orbit repeats its position and velocity, whereas a general bound orbit can precess without closing.