Trajectory

In ordinary language, a trajectory is the path followed by a body moving through space. For instance, the path taken by a falling body or the orbit of a planet are examples of trajectories. For bodies moving in space according to the laws of classical physics, the trajectory is a curve in space with a specific parametrization.

In the case of a projectile launched under the influence of a constant force field (modeling the effect of gravity over very short distances), the trajectory can be described by a (possibly degenerate) parabola. More generally, the movement of projectiles requires taking into account gravitional forces which are not constant and possibly other forces such as drag and wind. This is the focus of the discipline of ballistics.

In a wider sense, trajectory refers to the ordered set of intermediate states assumed by a dynamical system as a a result of time evolution. It is also used metaphorically, for instance, to describe an individual's career.

Physics of trajectories

Consider a particle of mass m moving in potential field; the motion of the particle is governed by the second-order differential equation

<math> \mathbf{m} \frac{d^2 \vec{x}(t)}{dt^2} = -\operatorname{grad} V(\vec{x}(t)) <math>

One of the remarkable achievements of Newtonian mechanics was the derivation of the fact that in the case of the gravitational field of a point mass, the trajectories must be conic sections on some plane of motion.