Hamilton's equations of motion

(redirected from Hamiltonian mechanics)
Also found in: Wikipedia.

Hamilton's equations of motion

The motion of a mechanical system may be described by a set of first-order ordinary differential equations known as Hamilton's equations. Because of their remarkably symmetrical form, they are often referred to as the canonical equations of motion of a system. They are equivalent to Lagrange's equations, but the fact that they are of first order and highly symmetrical makes them advantageous for general discussions of the motion of systems. See Lagrange's equations

Hamilton's equations can be derived from Lagrange's equations. Let the coordinates of the system be qj (j = 1, 2, . . . , f), and let the dynamical description of the system be given by the lagrangian L(q, …, t), where q denotes all the coordinates and a dot denotes total time derivative. Lagrange's equations are then given by Eq. (1). The momentum pj canonically conjugate to qj is defined by Eq. (2).


The hamiltonian H is defined by Eq. (3). Then Hamilton's canonical equations are Eqs. (4).


As they stand, Hamilton's equations are no easier to integrate directly than Lagrange's. Hamilton's equations are of great advantage in more general discussions, and they permit the making of canonical transformations which can lead to simplifications. See Canonical transformations

The hamiltonian function H of classical mechanics is used to form the quantum-mechanical hamiltonian operator.

Hamilton's equations of motion

[′ham·əl·tənz i¦kwā·zhənz əv ′mō·shən]
A set of first-order, highly symmetrical equations describing the motion of a classical dynamical system, namely j = ∂ H /∂ pj , j = -∂ H /∂ qj ; here qj (j = 1, 2,…) are generalized coordinates of the system, pj is the momentum conjugate to qj , and H is the Hamiltonian. Also known as canonical equations of motion.
References in periodicals archive ?
The opening chapters review the basics of Hamiltonian mechanics in its Hamiltonian formulation with a strong emphasis on the symplectic character of Hamiltonian flows.
Kozlov, Symmetries, Topology, and Resonances in Hamiltonian Mechanics, Springer, Berlin, Germany, 1995.
To study the thermally activated magnetization dynamics of ferromagnetic nanostructures, the authors use the principles of Hamiltonian mechanics to construct a classical microscopic description of an interacting spin system that couples with the surrounding microscopic degrees of freedom.
It is shown that classical Lagrangian mechanics is constructed completely on osculating bundles while the levels (higher order tangent bundles) provide a setting for Hamiltonian mechanics.
Sudarshan, Relation between Nambu and Hamiltonian mechanics, Phys.
Aiming to show the connection between classical and quantum mechanics, he first reviews elementary concepts in both areas, including basic math techniques and special functions, Newtonian mechanics, and Schrodinger's wave mechanics; then discusses semiclassical physics, classical periodic orbits, Lagrangian and Hamiltonian mechanics, the phenomenon of chaos, Feynman's Path Integrals, and applications of Gutzwiller's method and the trace formula to quantize chaos.
The main purpose of my article is to describe the application of Hamiltonian mechanics to measuring instrument theory.
Applications to Hamiltonian Mechanics, Kluwer, Dordrecht, FTPH 132, 2003.
The mechanics of Lagrange, the mechanics of the non-holonome systems, the Hamiltonian mechanics, and the variational principles use the analytical mechanics.
Symmetries, Topology and Resonances in Hamiltonian Mechanics," Springer-Verlag, Berlin (1995).
He covers both traditional methods and new developments; perspectives in both time and space discretization, encompassing classical Newtonian, Lagrangian, and Hamiltonian mechanics as well as new and alternate contemporary approaches and their equivalences to address various problems in engineering sciences and physics.
Remaining chapters formally describe dynamical symmetry in Hamiltonian mechanics, symmetries in classical Keplerian motion, dynamical symmetry in Schrodinger quantum mechanics, spectrum-generating Lie algebras and groups admitted by Schrodinger equations, dynamical symmetry of regularized hydrogen-like atoms, approximate dynamical symmetries in atomic and molecular physics, rovibronic systems, and dynamical symmetry of Maxwell's equations.