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The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kinetic and potential energies of all particles associated with the system. . The Hamiltonian takes different forms and can be simplified in some cases by taking into account the concrete characteristics of the system under analysis, such as single or several particles in the system, interaction ...
The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice versa. This implies that every sub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian
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 .
Hamiltonian quantum computation was the pioneering model of quantum computation, first proposed by Paul Benioff in 1980. Benioff's motivation for building a quantum mechanical model of a computer was to have a quantum mechanical description of artificial intelligence and to create a computer that would dissipate the least amount of energy allowable by the laws of physics. [1]
Action angles result from a type-2 canonical transformation where the generating function is Hamilton's characteristic function (not Hamilton's principal function ).Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian (,) is merely the old Hamiltonian (,) expressed in terms of the new canonical coordinates, which we denote as (the action angles, which are the ...
The Hamiltonian for a system of discrete particles is a function of their generalized coordinates and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field.
It does this by relating changes in the state of the system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known. All that remains is to plug the Hamiltonian into the Schrödinger equation and solve for the system state as a function of time.
Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function f ( p , q ) {\displaystyle f(p,q)} is a constant of motion. This implies that if p ( t ) , q ( t ) {\displaystyle p(t),q(t)} is a trajectory or solution to Hamilton's equations of motion , then 0 = d f d t {\displaystyle 0={\frac {df}{dt ...