Search results
Results From The WOW.Com Content Network
Siméon Denis Poisson. Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics.For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate the corresponding electrostatic or gravitational (force) field.
In classical mechanics, the phase space of a physical system consists of all the possible values of the position and of the momentum variables allowed by the system. It is naturally endowed with a Poisson bracket/symplectic form (see below), which allows one to formulate the Hamilton equations and describe the dynamics of the system through the phase space in time.
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
The uniqueness theorem for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. In the case of electrostatics , this means that there is a unique electric field derived from a potential function satisfying Poisson's equation under the ...
In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator . The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own ...
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson.
When λ is zero, the equation reduces to Poisson's equation. Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension n = 3 {\displaystyle n=3} , is a superposition of 1/ r functions weighted by the source function f :
In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups.