The trajectories of a dynamical system.

The chief object of investigation in Dynamics is the gradual change in time of the coordinates (q1, q2, …, qn) which specify the configuration of a dynamical system. When the system has three (or less than three) degrees of freedom, there is often a gain in clearness when we avail ourselves of a geometrical representation of the problem: if a point be taken whose rectangular coordinates referred to fixed axes are the coordinates (q1, q2, q3) of the given dynamical system, the path of this point in space can be regarded as illustrating the successive states of the system. In the same way when n > 3 we can still regard the motion of the system as represented by the path of a point whose coordinates are (q1, q2, …, qn) in space of n dimensions; this path is called the trajectory of the system, and its introduction makes it natural to use geometrical terms such as “intersection,” “adjacent,” etc., when speaking of the relations of different states or types of motion in the system.

Hamilton's principle, for conservative holonomic systems.

Consider any conservative holonomic dynamical system whose configuration at any instant is specified by n independent coordinates (q1, q2, …, qn), and let L be the kinetic potential which characterises its motion. Let a given are AB in space of n dimensions represent part of a trajectory of the system, and let CD be part of an adjacent are which is not necessarily a trajectory: it would however of course be possible to make CD a trajectory by subjecting the system to additional constraints.