In
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 ...
, the tautological one-form is a special
1-form defined on the
cotangent bundle of a
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 n ...
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
with
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 ...
(on the manifold
).
The
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 res ...
of this form defines 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 ...
giving
the structure of a
symplectic manifold. The tautological one-form plays an important role in relating the formalism of
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 ...
and
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lou ...
. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the
canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical example ...
one-form, or the symplectic potential. A similar object is the
canonical vector field on the
tangent bundle
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 ...
.
To define the tautological one-form, select a coordinate chart
on
and a
canonical coordinate
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of ...
system on
Pick an arbitrary point
By definition of cotangent bundle,
where
and
The tautological one-form
is given by
with
and
being the coordinate representation of
Any coordinates on
that preserve this definition, up to a total differential (
exact form In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diff ...
), may be called canonical coordinates; transformations between different canonical coordinate systems are known as
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.
The canonical symplectic form, also known as the Poincaré two-form, is given by
The extension of this concept to general
fibre bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
s is known as the
solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
. By convention, one uses the phrase "canonical form" whenever the form has a unique, canonical definition, and one uses the term "solder form", whenever an arbitrary choice has to be made. In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
the term "canonical" is discouraged, due to confusion with the
canonical class In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
, and the term "tautological" is preferred, as in
tautological bundle In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector ...
.
Physical interpretation
The variables
are meant to be understood as
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
, so that a point
is a point in
configuration space. The tangent space
corresponds to velocities, so that if
is moving along a path
the instantaneous velocity at
corresponds a point
on the tangent manifold
for the given location of the system at point
Velocities are appropriate for the
Lagrangian formulation of classical mechanics, but in the
Hamiltonian formulation, one works with momenta, and not velocities; the tautological one-form is a device that converts velocities into momenta.
That is, the tautological one-form assigns a numerical value to the momentum
for each velocity
and more: it does so such that they point "in the same direction", and linearly, such that the magnitudes grow in proportion. It is called "tautological" precisely because, "of course", velocity and momenta are necessarily proportional to one-another. It is a kind of
solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
, because it "glues" or "solders" each velocity to a corresponding momentum. The choice of gluing is unique; each momentum vector corresponds to only one velocity vector, by definition. The tautological one-form can be thought of as a device to convert from Lagrangian mechanics to Hamiltonian mechanics.
Coordinate-free definition
The tautological 1-form can also be defined rather abstractly as a form on
phase space. Let
be a manifold and
be the
cotangent bundle or
phase space. Let
be the canonical fiber bundle projection, and let
be the
induced tangent map
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of ''φ, d\varphi_x,'' at a point ''x'' is, in some sense, the bes ...
. Let
be a point on
Since
is the cotangent bundle, we can understand
to be a map of the tangent space at
:
That is, we have that
is in the fiber of
The tautological one-form
at point
is then defined to be
It is a linear map
and so
Symplectic potential
The symplectic potential is generally defined a bit more freely, and also only defined locally: it is any one-form
such that
; in effect, symplectic potentials differ from the canonical 1-form by a
closed form.
Properties
The tautological one-form is the unique one-form that "cancels"
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
. That is, let
be a 1-form on
is a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
For an arbitrary 1-form
on
the pullback of
by
is, by definition,
Here,
is the
pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
of
Like
is a 1-form on
The tautological one-form
is the only form with the property that
for every 1-form
on
So, by the commutation between the pull-back and the exterior derivative,
Action
If
is a
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
on the
cotangent bundle and
is its
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 ...
, then the corresponding
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
is given by
In more prosaic terms, the Hamiltonian flow represents the classical trajectory of a mechanical system obeying the
Hamilton-Jacobi equations of motion. The Hamiltonian flow is the integral of the Hamiltonian vector field, and so one writes, using traditional notation for
action-angle variables
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving ...
:
with the integral understood to be taken over the manifold defined by holding the energy
constant:
On Riemannian and Pseudo-Riemannian Manifolds
If the manifold
has a Riemannian or pseudo-Riemannian
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
then corresponding definitions can be made in terms of
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
. Specifically, if we take the metric to be a map
then define
and
In generalized coordinates
on
one has
and
The metric allows one to define a unit-radius sphere in
The canonical one-form restricted to this sphere forms a
contact structure
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. ...
; the contact structure may be used to generate the
geodesic flow
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
for this metric.
References
*
Ralph Abraham and
Jerrold E. Marsden, ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ''See section 3.2''.
{{Manifolds
Symplectic geometry
Hamiltonian mechanics
Lagrangian mechanics