In
mathematics and
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, the Poisson bracket is an important
binary operation in
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
. The Poisson bracket also distinguishes a certain class of coordinate transformations, called ''
canonical transformations'', which map
canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by
and
, respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself
as one of the new canonical momentum coordinates.
In a more general sense, the Poisson bracket is used to define a
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
, of which the algebra of functions on a
Poisson manifold
In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule
: \ = \h + g \ .
Equivalent ...
is a special case. There are other general examples, as well: it occurs in the theory of
Lie algebras, where the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the represent ...
article. Quantum deformations of the universal enveloping algebra lead to the notion of
quantum groups.
All of these objects are named in honor of
Siméon Denis Poisson
Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
.
Properties
Given two functions and that depend on
phase space and time, their Poisson bracket
is another function that depends on phase space and time. The following rules hold for any three functions
of phase space and time:
;
Anticommutativity
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
:
;
Bilinearity
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
:
;
Leibniz's rule:
;
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
:
Also, if a function
is constant over phase space (but may depend on time), then
for any
.
Definition in canonical coordinates
In
canonical coordinates (also known as
Darboux coordinates)
on the
phase space, given two functions
and
,
[ means is a function of the independent variables: momentum, ; position, ; and time, ] the Poisson bracket takes the form
The Poisson brackets of the canonical coordinates are
where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
.
Hamilton's equations of motion
Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that
is a function on the solution's trajectory-manifold. Then from the multivariable
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
,
Further, one may take
and
to be solutions to
Hamilton's equations
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
; that is,
Then
Thus, the time evolution of a function
on a
symplectic manifold can be given as a
one-parameter family of
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s (i.e.,
canonical transformations, area-preserving diffeomorphisms), with the time
being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that ''any time
'' in the solution to Hamilton's equations,
can serve as the bracket coordinates. ''Poisson brackets are
canonical invariants''.
Dropping the coordinates,
The operator in the convective part of the derivative,
, is sometimes referred to as the Liouvillian (see
Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectorie ...
).
Constants of motion
An
integrable dynamical system will have
constants of motion in addition to the energy. Such constants of motion will commute with the Hamiltonian under the Poisson bracket. Suppose some function
is a constant of motion. This implies that if
is a
trajectory
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tra ...
or solution to
Hamilton's equations of motion, then
along that trajectory. Then
where, as above, the intermediate step follows by applying the equations of motion and we assume that
does not explicitly depend on time. This equation is known as the
Liouville equation. The content of
Liouville's theorem is that the time evolution of a
measure given by a
distribution function is given by the above equation.
If the Poisson bracket of
and
vanishes (
), then
and
are said to be in involution. In order for a Hamiltonian system to be
completely integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
,
independent constants of motion must be in
mutual involution, where
is the number of degrees of freedom.
Furthermore, according to Poisson's Theorem, if two quantities
and
are explicitly time independent (
) constants of motion, so is their Poisson bracket
. This does not always supply a useful result, however, since the number of possible constants of motion is limited (
for a system with
degrees of freedom), and so the result may be trivial (a constant, or a function of
and
.)
The Poisson bracket in coordinate-free language
Let
be a
symplectic manifold, that is, a
manifold equipped with a
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
: a
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, ...
which is both closed (i.e., its
exterior derivative vanishes) and non-degenerate. For example, in the treatment above, take
to be
and take
If
is the
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 ...
or
contraction operation defined by
, then non-degeneracy is equivalent to saying that for every one-form
there is a unique vector field
such that
. Alternatively,
. Then if
is a smooth function on
, the
Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ...
can be defined to be
. It is easy to see that
The Poisson bracket
on is a
bilinear operation on
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
s, defined by
; the Poisson bracket of two functions on is itself a function on . The Poisson bracket is antisymmetric because:
Furthermore,
Here denotes the vector field applied to the function as a directional derivative, and
denotes the (entirely equivalent)
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 fi ...
of the function .
If is an arbitrary one-form on , the vector field generates (at least locally) a
flow satisfying the boundary condition
and the first-order differential equation
The
will be
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s (
canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as form invariance. It need not preserve the form of the Hamiltonian itself. Canon ...
s) for every as a function of if and only if
; when this is true, is called a
symplectic vector field In physics and mathematics, a symplectic vector field is one whose flow preserves a symplectic form. That is, if (M,\omega) is a symplectic manifold with smooth manifold M and symplectic form \omega, then a vector field X\in\mathfrak(M) in the ...
. Recalling
Cartan's identity and , it follows that
. Therefore, is a symplectic vector field if and only if α is a
closed form. Since
, it follows that every Hamiltonian vector field is a symplectic vector field, and that the Hamiltonian flow consists of canonical transformations. From above, under the Hamiltonian flow ,
This is a fundamental result in Hamiltonian mechanics, governing the time evolution of functions defined on phase space. As noted above, when , is a constant of motion of the system. In addition, in canonical coordinates (with
and
), Hamilton's equations for the time evolution of the system follow immediately from this formula.
It also follows from that the Poisson bracket is a
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
; that is, it satisfies a non-commutative version of Leibniz's
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
:
The Poisson bracket is intimately connected to the
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
of the Hamiltonian vector fields. Because the Lie derivative is a derivation,
Thus if and are symplectic, using
, Cartan's identity, and the fact that
is a closed form,
It follows that
, so that
Thus, the Poisson bracket on functions corresponds to the Lie bracket of the associated Hamiltonian vector fields. We have also shown that the Lie bracket of two symplectic vector fields is a Hamiltonian vector field and hence is also symplectic. In the language of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
, the symplectic vector fields form a
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
of the
Lie algebra of smooth vector fields on , and the Hamiltonian vector fields form an
ideal of this subalgebra. The symplectic vector fields are the Lie algebra of the (infinite-dimensional)
Lie group of
symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
s of .
It is widely asserted that the
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
for the Poisson bracket,
follows from the corresponding identity for the Lie bracket of vector fields, but this is true only up to a locally constant function. However, to prove the Jacobi identity for the Poisson bracket, it is
sufficient
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
to show that: