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In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton , [ 1 ] Hamiltonian mechanics replaces (generalized) velocities q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta .
The Hamiltonian is a particularly ubiquitous quantity in quantum mechanics (see Hamiltonian (quantum mechanics)). Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, which is not often used in practice but an efficient formulation for cyclic coordinates.
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, [18] Hamiltonian mechanics replaces (generalized) velocities ˙ used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.
It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. [1] Inspired by—but distinct from—the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. [2]
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
Action principles are "integral" approaches rather than the "differential" approach of Newtonian mechanics.[2]: 162 The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of a path according to the "action", a continuous sum or integral of the Lagrangian along the path.
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.
William Rowan Hamilton re-formulated Lagrangian mechanics in 1833, resulting in Hamiltonian mechanics. In addition to the solutions of important problems in classical physics, these techniques form the basis for quantum mechanics: Lagrangian methods evolved in to the path integral formulation and the Schrödinger equation builds Hamiltonian ...