Least Action, Principle of

one of the variational principles of mechanics; according to this principle, the actual motion for a given class of comparable motions of a mechanical system will be that motion for which the physical quantity called the action is a minimum or, more exactly, is an extremum. The principle of least action is usually used in one of two forms.

(1) The principle of least action in the Hamilton-Ostrogradskii form asserts that, of all the kinematically possible displacements of a system from one configuration to a configuration close to the first one and carried out during the same time interval, the actual displacement will be that for which the action S in the sense of Hamilton is least (a minimum). The mathematical expression for the principle of least action in this case has the form δS = 0, where δ is a symbol for the variation of S between adjacent displacements carried out in the same time interval.

(2) The principle of least action in the Maupertuis-Lagrange form asserts that, of all the kinematically possible displacements of a system from one configuration to a configuration adjacent to the first one and carried out with the total energy of the system conserved, the actual displacement will be that for which the action W in the sense of Lagrange is a minimum. In this case, the mathematical expression for the principle of least action has the form ΔW = 0, where Δ is the symbol for total variation. (Unlike the case of the Hamilton-Ostrogradskii principle, here there is a variation not only in the coordinates and velocities but also in the time during which the system is displaced from the first configuration to the second.) In this case, the principle of least action holds only for conservative systems that are also holonomic. In form (1), on the other hand, the principle of least action is more general and, in particular, can be extended to nonconservative systems.

Principles of least action are used to formulate the equations of motion of mechanical systems and to study the general properties of these motions. Applications to continuum mechanics, electrodynamics, quantum mechanics, and other fields have been found by appropriately generalizing the concepts of the principle of least action. (SeeVARIATIONAL PRINCIPLES OF MECHANICS.)