Poisson bracket - Citizendia

In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In a more general setting, the Poisson bracket is used to define a Poisson algebra, of which the Poisson manifolds are a special case. In Mathematics, a Poisson algebra is an Associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is the bracket In Mathematics, a Poisson manifold is a Differential manifold M such that the algebra C &infin( M) of Smooth functions These are all named in honour of Siméon-Denis Poisson. Siméon-Denis Poisson (21 June 1781 &ndash 25 April 1840 was a French Mathematician, Geometer, and Physicist.

1 Canonical coordinates
2 Equations of motion
3 Constants of motion
4 Definition
5 Lie algebra
6 See also
7 References

Canonical coordinates

In canonical coordinates $(q i, p j)$ on the phase space, given two functions $f(p_i,q_i,t)\,$ and $g(p_i,q_i,t)\,$ , the Poisson bracket takes the form

$\{f,g\} = \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right].$

Equations of motion

The Hamilton-Jacobi equations of motion have an equivalent expression in terms of the Poisson bracket. In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System In Physics, the Hamilton–Jacobi equation (HJE is a reformulation of Classical mechanics and thus equivalent to other formulations such as Newton's laws of This may be most directly demonstrated in an explicit coordinate frame. Suppose that $f (p, q, t)$ is a function on the manifold. Then one has

$\frac {\mathrm{d}}{\mathrm{d}t} f(p,q,t) = \frac{\partial f}{\partial t} + \frac {\partial f}{\partial p} \frac {\mathrm{d}p}{\mathrm{d}t} + \frac {\partial f}{\partial q} \frac {\mathrm{d}q}{\mathrm{d}t}.$

Then, by taking $p = p (t)$ and $q = q (t)$ to be solutions to the Hamilton-Jacobi equations $\dot{q}={\partial H}/{\partial p}$ and $\dot{p}=-{\partial H}/{\partial q}$ , one may write

$\frac {\mathrm{d}}{\mathrm{d}t} f(p,q,t) = \frac{\partial f}{\partial t} + \frac {\partial f}{\partial q} \frac {\partial H}{\partial p} - \frac {\partial f}{\partial p} \frac {\partial H}{\partial q} = \frac{\partial f}{\partial t} +\{f,H\}.$

Thus, the time evolution of a function f on a symplectic manifold can be given as a one-parameter family of symplectomorphisms, with the time t being the parameter. In Mathematics, a flow formalizes in mathematical terms the general idea of "a variable that depends on time" that occurs very frequently in Engineering In Mathematics, a symplectomorphism is an Isomorphism in the category of Symplectic manifolds Formal definition Specifically Dropping the coordinates, one has

$\frac{\mathrm{d}}{\mathrm{d}t} f= \left(\frac{\partial }{\partial t} - \{\,H, \cdot\,\}\right)f.$

The operator $- \{\,H, \cdot\,\}$ is known as the Liouvillian. In Physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian

Constants of motion

An integrable dynamical system will have constants of motion in addition to the energy. In Mathematics and Physics, there are various distinct notions that are referred to under the name of integrable systems. In Mechanics, a constant of motion is a quantity that is conserved throughout the motion imposing in effect a constraint on the motion Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function $f (p, q)$ is a constant of motion. This implies that if $p (t), q (t)$ is a trajectory or solution to the Hamilton-Jacobi equations of motion, then one has that $0=\frac{\mathrm{d}f}{\mathrm{d}t}$ along that trajectory. Trajectory is the path a moving object follows through space The object might be a Projectile or a Satellite, for example In Physics, the Hamilton–Jacobi equation (HJE is a reformulation of Classical mechanics and thus equivalent to other formulations such as Newton's laws of Then one has

$0 = \frac {\mathrm{d}}{\mathrm{d}t} f(p,q) = \frac {\partial f}{\partial p} \frac {\mathrm{d}p}{\mathrm{d}t} + \frac {\partial f}{\partial q} \frac {\mathrm{d}q}{\mathrm{d}t} = \frac {\partial f}{\partial q} \frac {\partial H}{\partial p} - \frac {\partial f}{\partial p} \frac {\partial H}{\partial q} = \{f,H\}$

where, as above, the intermediate step follows by applying the equations of motion. This equation is known as the Liouville equation. In Physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian The content of Liouville's theorem is that the time evolution of a measure (or "distribution function" on the phase space) is given by the above. Liouville's theorem has various meanings all mathematical results named after Joseph Liouville: In Complex analysis, see Liouville's theorem In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with This article describes the distribution function as used in physics

In order for a Hamiltonian system to be completely integrable, all of the constants of motion must be in mutual involution. In Mathematics and Physics, there are various distinct notions that are referred to under the name of integrable systems.

Definition

Let M be symplectic manifold, that is, a manifold on which there exists a symplectic form: a 2-form $ω$ which is both closed ( $d ω = 0$ ) and non-degenerate, in the following sense: when viewed as a map $\omega: \xi \in \mathrm{vect}[M] \rightarrow i_\xi \omega \in \Lambda^1[M]$ , $ω$ is invertible to obtain $\tilde{\omega}: \Lambda^1[M] \rightarrow \mathrm{vect}[M]$ . In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is Here $d$ is the exterior derivative operation intrinsic to the manifold structure of M, and $i ξ θ$ is the interior product or contraction operation, which is equivalent to $θ(ξ)$ on 1-forms $θ$ . In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Mathematics, the interior product is a degree &minus1 derivation on the Exterior algebra of Differential forms on a Smooth manifold In Multilinear algebra, a tensor contraction is an operation on one or more Tensors that arises from the natural pairing of a finite- Dimensional

Using the axioms of the exterior calculus, one can derive:

i [v, w] ω = d (i v i w ω) + i v d (i w ω) - i w d (i v ω) - i w i v d ω

Here $[v, w]$ denotes the Lie bracket on smooth vector fields, whose properties essentially define the manifold structure of M. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the

If v is such that $d (i v ω) = 0$ , we may call it $ω$ -coclosed (or just coclosed). Similarly, if $i v ω = d f$ for some function f, we may call v $ω$ -coexact (or just coexact). Given that $d ω = 0$ , the expression above implies that the Lie bracket of two coclosed vector fields is always a coexact vector field, because when v and w are both coclosed, the only nonzero term in the expression is $d (i v i w ω)$ . And because the exterior derivative obeys $d \circ d = 0$ , all coexact vector fields are coclosed; so the Lie bracket is closed both on the space of coclosed vector fields and on the subspace within it consisting of the coexact vector fields. In the language of abstract algebra, the coclosed vector fields form a subalgebra of the Lie algebra of smooth vector fields on M, and the coexact vector fields form an algebraic ideal of this subalgebra. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Algebra (mathematics, the word "algebra" usually means a Vector space or module equipped with an additional bilinear operation In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

Given the existence of the inverse map $\tilde{\omega}$ , every smooth real-valued function f on M may be associated with a coexact vector field $\tilde{\omega}(df)$ . (Two functions are associated with the same vector field if and only if their difference is in the kernel of d, i. e. , constant on each connected component of M. ) We therefore define the Poisson bracket on $(M,ω)$ , a bilinear operation on differentiable functions, under which the $C^\infty$ (smooth) functions form an algebra. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. It is given by:

$\{f,g\} = i_{\tilde{\omega}(df)} dg = - i_{\tilde{\omega}(dg)} df = -\{g,f\}$

The skew-symmetry of the Poisson bracket is ensured by the axioms of the exterior calculus and the condition $d ω = 0$ . In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms Because the map $\tilde{\omega}$ is pointwise linear and skew-symmetric in this sense, some authors associate it with a bivector, which is not an object often encountered in the exterior calculus. In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p In this form it is called the Poisson bivector or the Poisson structure on the symplectic manifold, and the Poisson bracket written simply $\{f,g\} = \tilde{\omega}(df, dg)$ . In Mathematics, a Poisson manifold is a Differential manifold M such that the algebra C &infin( M) of Smooth functions In Mathematics, a Poisson manifold is a Differential manifold M such that the algebra C &infin( M) of Smooth functions

The Poisson bracket on smooth functions corresponds to the Lie bracket on coexact vector fields and inherits its properties. It therefore satisfies the Jacobi identity:

{f,{g, h}} + {g,{h, f}} + {h,{f, g}} = 0

The Poisson bracket ${f,_}$ with respect to a particular scalar field f corresponds to the Lie derivative with respect to $\tilde{\omega}(df)$ . In Mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the Consequently, it is a derivation; that is, it satisfies Leibniz' law:

{f, g h} = {f, g} h + g {f, h}

It is a fundamental property of manifolds that the commutator of the Lie derivative operations with respect to two vector fields is equivalent to the Lie derivative with respect to some vector field, namely, their Lie bracket. In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. The parallel role of the Poisson bracket is apparent from a rearrangement of the Jacobi identity:

{f,{g, h}} - {g,{f, h}} = {{f, g}, h}

If the Poisson bracket of f and g vanishes ( ${f, g} = 0$ ), then f and g are said to be in mutual involution, and the operations of taking the Poisson bracket with respect to f and with respect to g commute.

Lie algebra

The Poisson brackets are anticommutative. In mathematics anticommutativity refers to the property of an operation being anticommutative, i Note also that they satisfy the Jacobi identity. In Mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation This makes the space of smooth functions on a symplectic manifold an infinite-dimensional Lie algebra with the Poisson bracket acting as the Lie bracket. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie The corresponding Lie group is the group of symplectomorphisms of the symplectic manifold (also known as canonical transformations). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a symplectomorphism is an Isomorphism in the category of Symplectic manifolds Formal definition Specifically In Hamiltonian mechanics, a canonical transformation is a change of Canonical coordinates (\mathbf{q} \mathbf{p} t \rightarrow (\mathbf{Q} \mathbf{P} t

Given a differentiable vector field X on the tangent bundle, let $P X$ be its conjugate momentum. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent The conjugate momentum mapping is a Lie algebra anti-homomorphism from the Poisson bracket to the Lie bracket:

$\{P_X,P_Y\}=-P_{[X,Y]}.\,$

This important result is worth a short proof. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Lie bracket can refer to Lie algebra Lie bracket of vector fields Write a vector field X at point q in the configuration space as

$X_q=\sum_i X^i(q) \frac{\partial}{\partial q^i}$

where the $\partial /\partial q^i$ is the local coordinate frame. "Configuration space" may also refer to PCI Configuration Space. The conjugate momentum to X has the expression

$P_X(q,p)=\sum_i X^i(q) \;p_i$

where the $p i$ are the momentum functions conjugate to the coordinates. One then has, for a point $(q, p)$ in the phase space,

$\{P_X,P_Y\}(q,p)= \sum_i \sum_j \{X^i(q) \;p_i, Y^j(q)\;p_j \}$

$=\sum_{ij} p_i Y^j(q) \frac {\partial X^i}{\partial q^j} - p_j X^i(q) \frac {\partial Y^j}{\partial q^i}$

$= - \sum_i p_i \; [X,Y]^i(q)$

$= - P_{[X,Y]}(q,p). \,$

The above holds for all $(q, p)$ , giving the desired result. In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System

References

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 In Physics, the Moyal bracket is the suitably normalized antisymmetrization of the star product. In Theoretical physics, the Peierls bracket is an equivalent description of the Poisson bracket. In Mathematics, a Poisson superalgebra is a Z 2- graded generalization of a Poisson algebra. In Mathematics, a Poisson superalgebra is a Z 2- graded generalization of a Poisson algebra. The Dirac bracket is a generalization of the Poisson bracket developed by Paul Dirac to correctly treat systems with second class constraints in Hamiltonian

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents