Hamilton's equations of motion

Article By:

Safko, John L. Formerly, Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina.

Stehle, Philip Department of Physics, University of Pittsburgh, Pittsburgh, Pennsylvania.

Last reviewed:August 2020

DOI:https://doi.org/10.1036/1097-8542.307300

A set of first-order ordinary differential equations that may be used to describe the motion of a mechanical system. Because of their remarkably symmetrical form [which appears in Eqs. (4), below], they are often referred to as the canonical equation of motion (where “canonical” is used in the sense of designating a simple general set of standard equations). The Lagrangian formulation of a system of f degrees of freedom generates f differential equations of second order in the time derivatives of the variables. Hamilton's equations, which are equivalent to Lagrange's equations, consist of 2f first-order and highly symmetrical equations. These properties make Hamilton's equations very useful for general discussions of the motion of systems. See also: Degree of freedom (mechanics); Differential equation; Lagrange's equations

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