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The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. 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 ]
Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as "the Hamiltonian" or "the energy function." The symplectic manifold is then called the phase space. The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field.
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 ...
This turns the Hamiltonian into = +, which is in the form of the harmonic oscillator Hamiltonian. The generating function F for this transformation is of the third kind, = (,). To find F explicitly, use the equation for its derivative from the table above,
Formally, a Hamiltonian system is a dynamical system characterised by the scalar function (,,), also known as the Hamiltonian. [1] The state of the system, r {\displaystyle {\boldsymbol {r}}} , is described by the generalized coordinates p {\displaystyle {\boldsymbol {p}}} and q {\displaystyle {\boldsymbol {q}}} , corresponding to generalized ...
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.
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.
Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system; Hamiltonian path, a path in a graph that visits each vertex exactly once; Hamiltonian matrix, a matrix with certain special properties commonly used in linear algebra; Hamiltonian group, a non-abelian group the subgroups of which are all ...