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List Of Things Named After Joseph-Louis Lagrange
Several concepts from mathematics and physics are named after the mathematician and astronomer Joseph-Louis Lagrange, as are a crater on the moon and a street in Paris. Lagrangian * Lagrangian analysis *Lagrangian coordinates * Lagrangian derivative * Lagrangian drifter * Lagrangian foliation * Lagrangian Grassmannian * Lagrangian intersection Floer homology *Lagrangian mechanics **Relativistic Lagrangian mechanics *Lagrangian (field theory) *Lagrangian system * Lagrangian mixing * Lagrangian point *Lagrangian relaxation *Lagrangian submanifold * Lagrangian subspace * Nonlocal Lagrangian * Proca lagrangian * Special Lagrangian submanifold Lagrange *Euler–Lagrange equation * Green–Lagrange strain *Lagrange bracket *Lagrange–Bürmann formula * Lagrange–d'Alembert principle * Lagrange error bound * Lagrange form * Lagrange form of the remainder *Lagrange interpolation *Lagrange invariant *Lagrange inversion theorem *Lagrange multiplier **Augmented Lagrangian method *Lagran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CLaMS
Clam is a common name for several kinds of bivalve molluscs. The word is often applied only to those that are edible and live as infauna, spending most of their lives halfway buried in the sand of the seafloor or riverbeds. Clams have two shells of equal size connected by two adductor muscles and have a powerful burrowing foot. They live in both freshwater and marine environments; in salt water they prefer to burrow down into the mud and the turbidity of the water required varies with species and location; the greatest diversity of these is in North America. Clams in the culinary sense do not live attached to a substrate (whereas oysters and mussels do) and do not live near the bottom (whereas scallops do). In culinary usage, clams are commonly eaten marine bivalves, as in clam digging and the resulting soup, clam chowder. Many edible clams such as palourde clams are ovoid or triangular; however, razor clams have an elongated parallel-sided shell, suggesting an old-fashioned s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Error Bound
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ''k'' of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange–d'Alembert Principle
D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alembert's principle generalizes the principle of virtual work from static to dynamical systems by introducing ''forces of inertia'' which, when added to the applied forces in a system, result in ''dynamic equilibrium''. The principle does not apply for irreversible displacements, such as sliding friction, and more general specification of the irreversibility is required. D'Alembert's principle is more general than Hamilton's principle as it is not restricted to holonomic constraints that depend only on coordinates and time but not on velocities. Statement of the principle The principle states that the sum of the differences between the forces acting on a system of massive particles and the time derivatives of the momenta of the system itself pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Inversion Theorem
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. Statement Suppose is defined as a function of by an equation of the form :z = f(w) where is analytic at a point and f'(a)\neq 0. Then it is possible to ''invert'' or ''solve'' the equation for , expressing it in the form w=g(z) given by a power series : g(z) = a + \sum_^ g_n \frac, where : g_n = \lim_ \frac \left left( \frac \right)^n \right The theorem further states that this series has a non-zero radius of convergence, i.e., g(z) represents an analytic function of in a neighbourhood of z= f(a). This is also called reversion of series. If the assertions about analyticity are omitted, the formula is also valid for formal power series and can be generalized in various ways: It can be formulated for functions of several variables; it can be extended to provide a ready formula for for an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Bracket
Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph Louis Lagrange in 1808–1810 for the purposes of mathematical formulation of classical mechanics, but unlike the Poisson brackets, have fallen out of use. Definition Suppose that (''q''1, …, ''q''''n'', ''p''1, …, ''p''''n'') is a system of canonical coordinates on a phase space. If each of them is expressed as a function of two variables, ''u'' and ''v'', then the Lagrange bracket of ''u'' and ''v'' is defined by the formula : u, v = \sum_^n \left(\frac \frac - \frac \frac \right). Properties * Lagrange brackets do not depend on the system of canonical coordinates (''q'', ''p''). If (''Q'',''P'') = (''Q''1, …, ''Q''''n'', ''P''1, …, ''P''''n'') is another system of canonical coordinates, so that :: Q=Q(q,p), P=P(q,p) :is a canonical transformation, then the Lagrange bracket is an invariant of the transformation, in the sen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green–Lagrange Strain
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. Displacement The displacement of a body has two components: a rigid-body displacement and a deformation. * A rigid-body displacement consists of a simultaneous translation (physics) and rotation of the body without changing its shape or size. * Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration \kappa_0(\mathcal B) to a current or deformed configuration \kappa_t(\mathcal B) (Figure 1). A change in the confi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler–Lagrange Equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Lagrangian Submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proca Lagrangian
In physics, specifically field theory and particle physics, the Proca action describes a massive spin-1 field of mass ''m'' in Minkowski spacetime. The corresponding equation is a relativistic wave equation called the Proca equation. The Proca action and equation are named after Romanian physicist Alexandru Proca. The Proca equation is involved in the Standard Model and describes there the three massive vector bosons, i.e. the Z and W bosons. This article uses the (+−−−) metric signature and tensor index notation in the language of 4-vectors. Lagrangian density The field involved is a complex 4-potential B^\mu = \left (\frac, \mathbf \right), where \phi is a kind of generalized electric potential and \mathbf is a generalized magnetic potential. The field B^\mu transforms like a complex four-vector. The Lagrangian density is given by: :\mathcal=-\frac(\partial_\mu B_\nu^*-\partial_\nu B_\mu^*)(\partial^\mu B^\nu-\partial^\nu B^\mu)+\fracB_\nu^* B^\nu. where c is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlocal Lagrangian
In field theory, a nonlocal Lagrangian is a Lagrangian, a type of functional \mathcalphi(x) containing terms that are ''nonlocal'' in the fields \phi(x), i.e. not polynomials or functions of the fields or their derivatives evaluated at a single point in the space of dynamical parameters (e.g. space-time). Examples of such nonlocal Lagrangians might be: * \mathcal = \frac\big(\partial_x \phi(x)\big)^2 - \fracm^2 \phi(x)^2 + \phi(x) \int \frac \,d^ny. * \mathcal = -\frac\mathcal_\left(1 + \frac\right)\mathcal^. * S = \int dt \,d^dx \left psi^*\left(i\hbar \frac + \mu\right)\psi - \frac\nabla \psi^*\cdot \nabla \psi\right- \frac\int dt \,d^dx \,d^dy \, V(\mathbf - \mathbf) \psi^*(\mathbf) \psi(\mathbf) \psi^*(\mathbf) \psi(\mathbf). * The Wess–Zumino–Witten action. Actions obtained from nonlocal Lagrangians are called ''nonlocal actions''. The actions appearing in the fundamental theories of physics, such as the Standard Model, are local actions; nonlocal actions play a par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Subspace
In mathematics, a symplectic vector space is a vector space ''V'' over a Field (mathematics), field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a map (mathematics), mapping that is ; bilinear form, Bilinear: linear map, Linear in each argument separately; ; alternating form, Alternating: holds for all ; and ; Nondegenerate form, Non-degenerate: for all implies that . If the underlying field (mathematics), field has characteristic (algebra), characteristic not 2, alternation is equivalent to skew symmetry, skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric bilinear form, symmetric form, but not vice versa. Working in a fixed basis (linear algebra), basis, ''ω'' can be represented by a matrix (mathematics), matrix. The conditions above are equivalent to this matrix being skew-symmetric matrix, skew-symmetri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
