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Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations, ˙ = ˙ = ˙ = ˙ = Momentum , which corresponds to the vertical component of angular momentum = ˙ , is a constant of motion. That is a consequence of the rotational symmetry of the ...
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.
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.
Created Date: 8/30/2012 4:52:52 PM
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
Numerous other concepts and objects in mechanics, such as Hamilton's principle, Hamilton's principal function, the Hamilton–Jacobi equation, Cayley-Hamilton theorem are named after Hamilton. The Hamiltonian is the name of both a function (classical) and an operator (quantum) in physics, and, in a different sense, a term from graph theory .
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A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can be studied in both Hamiltonian mechanics and dynamical systems theory.