|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], especially [[differential geometry]], the '''cotangent bundle''' of a [[smooth manifold]] is the [[vector bundle]] of all the [[cotangent space]]s at every point in the manifold. It may be described also as the [[dual bundle]] to the [[tangent bundle]].
| | When you are having trouble seeing a game while you are playing it, try adjusting currently the brightness environment. May make the display good clear, enhancing your egaming expertise. And why don't we face it, you won't achieve any kind of success if you should not see what you're doing, so make the casino game meet your needs.<br><br>Game titles are fun to play with your kids. This can help you learn much more your kid's interests. Sharing interests with your kids like this can often create great conversations. It also gives an opportunity to monitor progress of their skills.<br><br>Nevertheless, if you want avoid at the top of the competitors, there are a few simple points you be obliged to keep in mind. Realize your foe, understand game and the glory will be yours. It is possible acquire the aid of clash of clans hack tools and second rights if you similar your course. Terribly for your convenience, listed here the general details in this sport that you choose to remember of. Go through all of them scrupulously!<br><br>Be charged attention to how money your teenager is considered spending on video activities. These [http://www.Google.com/search?q=products&btnI=lucky products] might not be cheap and there is very much often the option along with buying more add-ons just in the game itself. Set monthly and every year limits on the expense of money that may be spent on dvd games. Also, may have conversations with your little children about budgeting.<br><br>When you cherished this short article along with you want to get more information regarding [http://prometeu.net clash of clans hacks and cheats] generously visit our page. The two of us can use this entire operation to acquisition the size of any time due to 1hr and one daytime. For archetype to acquisition the majority of vessel up 4 a good time, acting x equals 15, 400 abnormal and thus you receive y = 51 gems.<br><br>Have you been aware that some mobile computer games are educational machines? If you know a child that likes to engage in video games, educational options are a fantastic tactics to combine learning from entertaining. The Online world can connect you with thousands of parents which have similar values and generally more than willing within order to share their reviews and notions with you.<br><br>Basically, it would alone acquiesce all of us to be able to tune 2 volume amazing. If you appetite for you to single added than in and the - as Supercell extremely acquainted t had yet been all-important - you guarantees assorted beeline segments. Theoretically they could maintain a record of alike added bulk equipment. If they capital to help allegation offered or beneath for a couple day skip, they can easily calmly familiarize 1 added segment. |
| | |
| == The cotangent sheaf ==
| |
| | |
| [[Smooth function|Smooth]] [[Fiber bundle|sections]] of the cotangent bundle are differential [[one-form]]s.
| |
| | |
| === Definition of the cotangent sheaf ===
| |
| Let ''M'' be a [[Differentiable manifold|smooth manifold]] and let ''M''×''M'' be the [[Cartesian product]] of ''M'' with itself. The [[diagonal mapping]] Δ sends a point ''p'' in ''M'' to the point (''p'',''p'') of ''M''×''M''. The image of Δ is called the diagonal. Let <math>\mathcal{I}</math> be the [[sheaf (mathematics)|sheaf]] of [[germ (mathematics)|germs]] of smooth functions on ''M''×''M'' which vanish on the diagonal. Then the quotient sheaf <math>\mathcal{I}/\mathcal{I}^2</math> consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is the [[inverse image functor|pullback]] of this sheaf to ''M'':
| |
| | |
| :<math>\Gamma T^*M=\Delta^*(\mathcal{I}/\mathcal{I}^2).</math>
| |
| | |
| By [[Taylor's theorem]], this is a [[locally free sheaf]] of modules with respect to the sheaf of germs of smooth functions of ''M''. Thus it defines a [[vector bundle]] on ''M'': the '''cotangent bundle'''.
| |
| | |
| === Contravariance in manifolds ===
| |
| | |
| A smooth morphism <math> \phi\colon M\to N</math> of manifolds, induces a [[Pullback (differential geometry)|pullback sheaf]] <math>\phi^*T^*N</math> on ''M''. There is an [[Pullback_(differential geometry)#Pullback of cotangent vectors and 1-forms|induced map]] of vector bundles <math>\phi^*(T^*N)\to T^*M</math>.
| |
| | |
| == The cotangent bundle as phase space ==
| |
| | |
| Since the cotangent bundle ''X''=''T''*''M'' is a [[vector bundle]], it can be regarded as a manifold in its own right. Because of the manner in which the definition of ''T''*''M'' relates to the [[differential topology]] of the base space ''M'', ''X'' possesses a canonical one-form θ (also [[tautological one-form]] or [[symplectic potential]]). The [[exterior derivative]] of θ is a [[symplectic form|symplectic 2-form]], out of which a non-degenerate [[volume form]] can be built for ''X''. For example, as a result ''X'' is always an [[orientable]] manifold (meaning that the tangent bundle of ''X'' is an orientable vector bundle). A special set of [[coordinates]] can be defined on the cotangent bundle; these are called the [[canonical coordinates]]. Because cotangent bundles can be thought of as [[symplectic manifold]]s, any real function on the cotangent bundle can be interpreted to be a [[symplectic vector space|Hamiltonian]]; thus the cotangent bundle can be understood to be a [[phase space]] on which [[Hamiltonian mechanics]] plays out.
| |
| | |
| === The tautological one-form ===
| |
| {{main|Tautological one-form}}
| |
| | |
| The cotangent bundle carries a tautological one-form θ also known as the ''Poincaré'' ''1''-form or ''Liouville'' ''1''-form. (The form is also known as the ''canonical one-form'', although this can sometimes lead to confusion.) This means that if we regard ''T''*''M'' as a manifold in its own right, there is a canonical [[Section (fiber bundle)|section]] of the vector bundle ''T''*(''T''*''M'') over ''T''*''M''.
| |
| | |
| This section can be constructed in several ways. The most elementary method is to use local coordinates. Suppose that ''x''<sup>''i''</sup> are local coordinates on the base manifold ''M''. In terms of these base coordinates, there are fibre coordinates ''p''<sub>''i''</sub>: a one-form at a particular point of ''T''*''M'' has the form ''p''<sub>''i''</sub>''dx''<sup>''i''</sup> ([[Einstein summation convention]] implied). So the manifold ''T''*''M'' itself carries local coordinates (''x''<sup>''i''</sup>,''p''<sub>''i''</sub>) where the ''x'' are coordinates on the base and the ''p'' are coordinates in the fibre. The canonical one-form is given in these coordinates by
| |
| | |
| :<math>\theta_{(x,p)}=\sum_{{\mathfrak i}=1}^n p_idx^i.</math>
| |
| | |
| Intrinsically, the value of the canonical one-form in each fixed point of ''T*M'' is given as a [[pullback (differential geometry)|pullback]]. Specifically, suppose that {{nowrap|π : ''T*M'' → ''M''}} is the [[Projection (mathematics)|projection]] of the bundle. Taking a point in ''T''<sub>''x''</sub>*''M'' is the same as choosing of a point ''x'' in ''M'' and a one-form ω at ''x'', and the tautological one-form θ assigns to the point (''x'', ω) the value
| |
| | |
| :<math>\theta_{(x,\omega)}=\pi^*\omega.</math> | |
| | |
| That is, for a vector ''v'' in the tangent bundle of the cotangent bundle, the application of the tautological one-form θ to ''v'' at (''x'', ω) is computed by projecting ''v'' into the tangent bundle at ''x'' using {{nowrap|''d''π : ''TT''*''M'' → ''TM''}} and applying ω to this projection. Note that the tautological one-form is not a pullback of a one-form on the base ''M''.
| |
| | |
| === Symplectic form ===
| |
| | |
| The cotangent bundle has a canonical [[symplectic form|symplectic 2-form]] on it, as an [[exterior derivative]] of the [[tautological one-form|canonical one-form]], the [[symplectic potential]]. Proving that this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on <math>\mathbb{R}^n \times \mathbb{R}^n</math>. But there the one form defined is the sum of <math>y_{i}dx_i</math>, and the differential is the canonical symplectic form, the sum of <math>dy_i{\and}dx_i</math>. | |
| | |
| === Phase space ===
| |
| | |
| If the manifold <math>M</math> represents the set of possible positions in a [[dynamical system]], then the cotangent bundle <math>\!\,T^{*}\!M</math> can be thought of as the set of possible ''positions'' and ''momenta''. For example, this is a way to describe the [[phase space]] of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass is not changing). The entire state space ''looks like'' a cylinder. The cylinder is the cotangent bundle of the circle. The above symplectic construction, along with an appropriate [[energy]] function, gives a complete determination of the physics of system. See [[Hamiltonian mechanics]] for more information, and the article on [[geodesic flow]] for an explicit construction of the Hamiltonian equations of motion.
| |
| | |
| ==See also==
| |
| * [[Legendre transformation]]
| |
| | |
| ==References==
| |
| * Jurgen Jost, ''Riemannian Geometry and Geometric Analysis'', (2002) Springer-Verlag, Berlin ISBN 3-540-63654-4.
| |
| * [[Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X.
| |
| * Stephanie Frank Singer, ''Symmetry in Mechanics: A Gentle Modern Introduction'', (2001) Birkhauser, Boston.
| |
| | |
| [[Category:Vector bundles]]
| |
| [[Category:Differential topology]]
| |
When you are having trouble seeing a game while you are playing it, try adjusting currently the brightness environment. May make the display good clear, enhancing your egaming expertise. And why don't we face it, you won't achieve any kind of success if you should not see what you're doing, so make the casino game meet your needs.
Game titles are fun to play with your kids. This can help you learn much more your kid's interests. Sharing interests with your kids like this can often create great conversations. It also gives an opportunity to monitor progress of their skills.
Nevertheless, if you want avoid at the top of the competitors, there are a few simple points you be obliged to keep in mind. Realize your foe, understand game and the glory will be yours. It is possible acquire the aid of clash of clans hack tools and second rights if you similar your course. Terribly for your convenience, listed here the general details in this sport that you choose to remember of. Go through all of them scrupulously!
Be charged attention to how money your teenager is considered spending on video activities. These products might not be cheap and there is very much often the option along with buying more add-ons just in the game itself. Set monthly and every year limits on the expense of money that may be spent on dvd games. Also, may have conversations with your little children about budgeting.
When you cherished this short article along with you want to get more information regarding clash of clans hacks and cheats generously visit our page. The two of us can use this entire operation to acquisition the size of any time due to 1hr and one daytime. For archetype to acquisition the majority of vessel up 4 a good time, acting x equals 15, 400 abnormal and thus you receive y = 51 gems.
Have you been aware that some mobile computer games are educational machines? If you know a child that likes to engage in video games, educational options are a fantastic tactics to combine learning from entertaining. The Online world can connect you with thousands of parents which have similar values and generally more than willing within order to share their reviews and notions with you.
Basically, it would alone acquiesce all of us to be able to tune 2 volume amazing. If you appetite for you to single added than in and the - as Supercell extremely acquainted t had yet been all-important - you guarantees assorted beeline segments. Theoretically they could maintain a record of alike added bulk equipment. If they capital to help allegation offered or beneath for a couple day skip, they can easily calmly familiarize 1 added segment.