International America's Cup Class

2013-09-21T09:43:11Z

70.24.249.39:

In [[mathematics]], the '''tautological one-form''' is a special [[1-form]] defined on the [[cotangent bundle]] ''T''*''Q'' of a [[manifold]] ''Q''. The [[exterior derivative]] of this form defines a [[symplectic form]] giving ''T''*''Q'' the structure of a [[symplectic manifold]]. The tautological one-form plays an important role in relating the formalism of [[Hamiltonian mechanics]] and [[Lagrangian mechanics]]. The tautological one-form is sometimes also called the '''Liouville one-form''', the '''Poincaré one-form''', the '''[[canonical (disambiguation)|canonical]] one-form''', or the '''symplectic potential'''. A similar object is the [[canonical vector field]] on the [[tangent bundle]]. In [[algebraic geometry]] and [[complex geometry]] the term "canonical" is discouraged, due to confusion with the [[canonical class]], and the term "tautological" is preferred, as in [[tautological bundle]].

In [[canonical coordinates]], the tautological one-form is given by

:<math>\theta = \sum_i p_i dq^i</math>

Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential ([[exact form]]), may be called canonical coordinates; transformations between different canonical coordinate systems are known as [[canonical transformation]]s.

The '''canonical symplectic form''' is given by

:<math>\omega = -d\theta = \sum_i dq^i \wedge dp_i</math>

The extension of this concept to general [[fibre bundle]]s is known as the [[solder form]].

==Coordinate-free definition==
The tautological 1-form can also be defined rather abstractly as a form on [[phase space]]. Let <math>Q</math> be a manifold and <math>M=T^*Q</math> be the [[cotangent bundle]] or [[phase space]]. Let

:<math>\pi:M\to Q</math>

be the canonical fiber bundle projection, and let

:<math>T_\pi:TM \to TQ </math>

be the induced [[Pushforward (differential)|tangent map]]. Let ''m'' be a point on ''M'', however, since ''M'' is the cotangent bundle, we can understand ''m'' to be a map of the tangent space at <math>q=\pi(m)</math>:

:<math>m:T_qQ \to \mathbb{R}</math>.

That is, we have that ''m'' is in the fiber of ''q''. The tautological one-form <math>\theta_m</math> at point ''m'' is then defined to be

:<math>\theta_m = m \circ T_\pi</math>

It is a linear map

:<math>\theta_m:T_mM \to \mathbb{R}</math>

and so

:<math>\theta:M \to T^*M</math>.

==Properties==
The tautological one-form is the unique [[horizontal form|horizontal one-form]] that "cancels" a [[pullback_(differential geometry)|pullback]]. That is, let

:<math>\beta:Q\to T^*Q</math>

be any 1-form on ''Q'', and (considering it as a map from ''Q'' to ''T*Q'' ) let <math>\beta^*</math> be its pullback. Then

:<math>\beta^*\theta = \beta</math>,

which can be most easily understood in terms of coordinates:

:<math>\beta^*\theta = \beta^*(\sum_i p_i\, dq^i) =
\sum_i \beta^*p_i\, dq^i = \sum_i \beta_i\, dq^i = \beta.</math>

So, by the commutation between the pull-back and the exterior derivative,

:<math>\beta^*\omega = -\beta^*d\theta = -d (\beta^*\theta) = -d\beta</math>.

==Action==
If ''H'' is a [[Hamiltonian mechanics|Hamiltonian]] on the [[cotangent bundle]] and <math>X_H</math> is its [[Hamiltonian flow]], then the corresponding [[action (physics)|action]] ''S'' is given by

:<math>S=\theta (X_H)</math>.

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]]:

:<math>S(E) = \sum_i \oint p_i\,dq^i</math>

with the integral understood to be taken over the manifold defined by holding the energy <math>E</math> constant: <math>H=E=const.</math> .

==On metric spaces==
If the manifold ''Q'' has a Riemannian or pseudo-Riemannian [[Metric (mathematics)|metric]] ''g'', then corresponding definitions can be made in terms of [[generalized coordinates]]. Specifically, if we take the metric to be a map

:<math>g:TQ\to T^*Q</math>,

then define

:<math>\Theta = g^*\theta</math>

and

:<math>\Omega = -d\Theta = g^*\omega</math>

In generalized coordinates <math>(q^1,\ldots,q^n,\dot q^1,\ldots,\dot q^n)</math> on ''TQ'', one has

:<math>\Theta=\sum_{ij} g_{ij} \dot q^i dq^j</math>

and

:<math>\Omega= \sum_{ij} g_{ij} \; dq^i \wedge d\dot q^j +
\sum_{ijk} \frac{\partial g_{ij}}{\partial q^k} \;
\dot q^i\, dq^j \wedge dq^k</math>

The metric allows one to define a unit-radius sphere in <math>T^*Q</math>. The canonical one-form restricted to this sphere forms a [[contact structure]]; the contact structure may be used to generate the [[geodesic flow]] for this metric.

==See also==
* [[fundamental class]]
* [[solder form]]

==References==
* [[Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X ''See section 3.2''.

[[Category:Symplectic geometry]]
[[Category:Hamiltonian mechanics]]
[[Category:Lagrangian mechanics]]

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