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It is also known as the Laplace vector, [13] [14] the Runge–Lenz vector [15] and the Lenz vector. [8] Ironically, none of those scientists discovered it. [ 15 ] The LRL vector has been re-discovered and re-formulated several times; [ 15 ] for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics .
The vector Laplace operator, also denoted by , is a differential operator defined over a vector field. [7] The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field , returning a vector quantity.
The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ l ə ˈ p l ɑː s /), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex-valued frequency domain, also known as s-domain, or s-plane).
The apsides are the two points of closest and furthest distance of the orbit (the periapsis and apoapsis, respectively); apsidal precession corresponds to the rotation of the line joining the apsides. It also corresponds to the rotation of the Laplace–Runge–Lenz vector, which points along the line of apsides.
In orbital mechanics, the Laplace-Runge-Lenz vector describes a quantity that is useful for studying the orbital motion of a body, for example a planet or a particle. It is a vector , defined for an orbiting body, that depends on its position and momentum.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
For any twice-differentiable real-valued function f defined on Euclidean space R n, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for R n.