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Lagrangian Map
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, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Manifold
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see tangent bundle#Examples, Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an element of TM can be thought of as a ordered pair, pair (x,v), where x is a point in M and v is a tangent vector to M at x . There i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential -form is thought of as measuring the flux through an infinitesimal - parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a -parallelotope at each point. Definition The exterior derivative of a differential form of degree (also differential -form, or just -form for brevity here) is a differential form of degree . If is a smooth function (a -form), then the exterior derivative of is the differential of . That is, is the unique -form such that for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew-symmetric Matrices
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ denotes the entry in the i-th row and j-th column, then the skew-symmetric condition is equivalent to Example The matrix :A = \begin 0 & 2 & -45 \\ -2 & 0 & -4 \\ 45 & 4 & 0 \end is skew-symmetric because : -A = \begin 0 & -2 & 45 \\ 2 & 0 & 4 \\ -45 & -4 & 0 \end = A^\textsf . Properties Throughout, we assume that all matrix entries belong to a field (mathematics), field \mathbb whose characteristic (algebra), characteristic is not equal to 2. That is, we assume that , where 1 denotes the multiplicative identity and 0 the additive identity of the given field. If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. * The sum of two skew-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Product
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product \iota_X \omega is sometimes written as X \mathbin \omega. Definition The interior product is defined to be the contraction of a differential form with a vector field. Thus if X is a vector field on the manifold M, then \iota_X : \Omega^p(M) \to \Omega^(M) is the map which sends a p-form \omega to the (p - 1)-form \iota_X \omega defined by the property that (\iota_X\omega)\left(X_1, \ldots, X_\right) = \omega\left(X, X_1, \ldots, X_\right) for any vector fields X_1, \ldots, X_. The interior product is the unique antiderivation of degree −1 on the exterior alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Homotopy Formula
In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product \iota_X \omega is sometimes written as X \mathbin \omega. Definition The interior product is defined to be the contraction of a differential form with a vector field. Thus if X is a vector field on the manifold M, then \iota_X : \Omega^p(M) \to \Omega^(M) is the map which sends a p-form \omega to the (p - 1)-form \iota_X \omega defined by the property that (\iota_X\omega)\left(X_1, \ldots, X_\right) = \omega\left(X, X_1, \ldots, X_\right) for any vector fields X_1, \ldots, X_. The interior product is the unique antiderivation of degree −1 on the exterior a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Derivative
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If ''T'' is a tensor field and ''X'' is a vector field, then the Lie derivative of ''T'' with respect to ''X'' is denoted \mathcal_X(T). The differential operator T \mapsto \mathcal_X(T) is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Form
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
