Trajectòries

Trajectòries refers to the plural form of trajectory in certain Romance-language contexts; in English the common term is trajectories. It denotes the path traced by a moving point as it evolves in time, or the evolution of a system’s state in a given space.

In physics and classical mechanics, a trajectory is the curve described by a particle’s position r(t) as a function of time, determined by initial conditions and the forces acting on it. In celestial mechanics, trajectories describe orbits under gravity and may take the form of conic sections in appropriate models.

In mathematics and dynamical systems, a trajectory is an integral curve of a vector field, representing the evolution of a system state x(t) in a chosen state space. Trajectories provide a geometric view of how solutions to differential equations unfold over time.

In engineering, computer science, and robotics, a trajectory commonly refers to a time-parameterized curve in a configuration space that a robot or vehicle is designed to follow. Trajectory planning often separates geometric path planning from timing (the speed profile along the path), and must respect kinematic and dynamic constraints as well as environmental limits.

Measurement, estimation, and analysis of trajectories involve dealing with noise, uncertainty, and perturbations. Techniques include numerical integration, filtering, and trajectory optimization to improve feasibility, smoothness, and accuracy of the modeled path.

See also: path, orbit, integral curve, motion planning, trajectory optimization.