Lagrangian (disambiguation)
Encyclopedia
The term Lagrangian refers to any of several mathematical concepts developed by Joseph Louis Lagrange
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Mathematical sciences
Physics
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
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Mathematical sciences
- In optimization theory, the Lagrangian function is used to solve constrained minimization problems; see Lagrange multipliersLagrange multipliersIn mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...
.- The method of approximating a difficult constrained problem with an easier problem having an enlarged feasible set (allowing but penalizing some violation of constraints), is called Lagrangian relaxationLagrangian relaxationIn the field of mathematical optimization, Lagrangian relaxation is a relaxation method which approximates a difficult problem of constrained optimization by a simpler problem...
. - The problem of maximizing the value of the Lagrangian function, in terms of the Lagrange-multiplier variable, is the Lagrangian dual problemDual problemIn constrained optimization, it is often possible to convert the primal problem to a dual form, which is termed a dual problem. Usually dual problem refers to the Lagrangian dual problem but other dual problems are used, for example, the Wolfe dual problem and the Fenchel dual problem...
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- The method of approximating a difficult constrained problem with an easier problem having an enlarged feasible set (allowing but penalizing some violation of constraints), is called Lagrangian relaxation
- In the calculus of variations, the Lagrangian is a functional whose extrema are to be determined; see Calculus of variationsCalculus of variationsCalculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
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- In symplectic geometry, a Lagrangian submanifold is a special class of submanifolds, with dimension half the dimension of the ambient space and where the symplectic form vanishes identically.
Physics
- In physics, the LagrangianLagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
function characterizes the dynamics of a system. - In orbital mechanics, the Lagrangian points are stable and meta-stable points of a two body system.
- In continuum mechanics, Lagrangian coordinates are a way of describing the motions of particles of a solid or fluid.