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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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2012-12-21T05:59:23Z

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Talk:Epsilon

2011-11-15T11:15:00Z

70.24.248.23: /* Disambiguation? */

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For a complete list of functions and a side by side contrast of all hosting strategies you need to go to Hostgator's website below. Below is a testimonial of the most essential functions of the "Child" plan, this is probably the most appropriate bundle for the majority of users, and it is our favorite plan. Disk area 100GB - This amount has actually been recently upgraded by Hostgator from 5GB to a massive 100GB. All users are most likely to discover it [http://Www.google.com/search?q=impossible&btnI=lucky impossible] to exhaust this amount of disk area. Bandwidth 1000GB/month - Also enhanced is the bandwidth allocation, from 75GB to an apparently excessive 1000GB. Again virtually zero opportunity of using all that, however it's nice to understand that you absolutely will not be facing additional costs for going over your limit, even if you have an extremely hectic internet site(s). 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