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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 ...
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics.It asserts that the phase-space distribution function is constant along the trajectories of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time.
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 ...
However, it is customary to rewrite the Hamiltonian in terms of the normal modes of the wavevector rather than in terms of the particle coordinates so that one can work in the more convenient Fourier space. Superposition of three oscillating dipoles- illustrate the time propagation of the common wave function for different n,l,m