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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]].
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In [[canonical coordinates]], the tautological one-form is given by
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:<math>\theta = \sum_i p_i dq^i</math>
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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.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


The '''canonical symplectic form''', also known as the '''Poincaré two-form''', is given by
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:<math>\omega = -d\theta = \sum_i dq^i \wedge dp_i</math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The extension of this concept to general [[fibre bundle]]s is known as the [[solder form]].
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==Coordinate-free definition==
==Demos==
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>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


be the canonical fiber bundle projection, and let


:<math>T_\pi:TM \to TQ </math>
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
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** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


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


:<math>m:T_qQ \to \mathbb{R}</math>.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
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*[[Help:Formula]]


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
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>\theta_m = m \circ T_\pi</math>.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
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]]

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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