Cotangent space

The cotangent space at a point P on a smooth manifold M is formally defined as a quotient space of two vector spaces: it is the vector space of all infinitely differentiable functions which have the value 0 at P, divided by the subspace of all functions which also have derivative 0 at this point. As P varies, the cotangent spaces make up the cotangent bundle of M. If M represents the set of possible positions in a dynamical system, then the cotangent bundle can be thought of as the set of possible ''positions and speeds''. For example, this is an easy way to describe the (non-trivial) phase space of a three dimensional pendulum: a weighted ball able to move along a sphere.

A Riemannian metric on the manifold provides a (non-canonical) isomorphism between the cotangent space and the tangent space. Thus, they have the same smoothness properties. However, many definitions are more natural on the cotangent bundle.

For example, the cotangent bundle has a canonical symplectic two-form on it, as an exterior derivative of a one-form. The one-form assigns to a vector in the tangent bundle to the cotangent bundle the application of the element in the cotangent bundle (a linear functional) to the projection of the vector into the tangent bundle (the differential of the projection of the cotangent bundle to the original manifold). Proving 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 RnxRn. But there the one form defined is the sum of yidxi, and the differential is the canonical symplectic form, the sum of dyidxi.

The above symplectic construction, along with an appropriate energy function, gives a complete determination of the physics of systems, such as the pendulum example cited above.



In the News

Having The Climate Cake And Eating It, Too
Is it possible to solve climate change, reduce poverty and save biodiversity at a single stroke? It might seem like a dream, but this is exactly the issue that is being discussed at the United Nations Climate Change Conference. The key is to include reduced emissions from deforestation and degradation in the Kyoto Protocol so that developing countries can be compensated for saving their forests and woodlands.

Yoga stretches brain chemical

GABA yoga postureUS researchers have used a specialist brain scanning technique, magnetic resonance spectroscopic imaging, which is effectively an MRI scan carried out at the molecular level to reveal the effects of yoga practice on the brain. Specifically, they have investigated how concentrations of the feel-good compound gamma-aminobutyric (GABA), an inhibitory neurotransmitter, change after regular practice of yoga postures.

Eric Jensen and colleagues at Harvard Medical School looked at eight subjects prior to and after one hour of yoga as well as eleven control subjects who read a book rather than undertaking the yoga exercises. Although the samples are very small, they saw a marked difference in GABA levels in the yoga practitioners compared to the readers. Their findings suggest that yoga, and perhaps other forms of exercise, should be investigated as a complementary treatment for depression and anxiety disorders, which are commonly associated with low levels of GABA.

You can read moreon this in my write-up over on SpectroscopyNOW.com. Click here for the Sciencebase complementary medicine roundup

Engineers Discover Why Toucan Beaks Are Models Of Lightweight Strength
In a paper to be published Dec. 1 in Acta Materialia, researchers report that the secret to the toucan beak's lightweight strength is an unusual bio-composite. The interior of the beak is rigid "foam"made of bony fibers and drum-like membranes sandwiched between layers of keratin, the protein that makes up fingernails, hair, and horn.

Say 'Thank You': Learning How To Lie
Although honesty is generally taught as the best policy, around a child's birthday and holidays, the little white lie goes a long way. After all, kids are expected to grin and giggle at an itchy wool sweater as if it were the toy-of-the-moment they had been begging for. After a few years of awkward laughter and whispered scolding from parents, children tend to learn that a forced exclamation of joy earns them more smiles and hugs than the truth does.

Study Links Emotions In Play And Memories
Psychologists from Case Western Reserve University have found that the range of emotions that children use in play can be used as an indicator of how emotionally charged their memories will be.

Understanding The Oceans Microbes Is Key To The Earth's Future
Life on Earth may owe its existence to tiny microorganisms living in the oceans, but the effect of human-induced change on the vital services these microbes perform for the planet remains largely unstudied, says a report released today by the American Academy of Microbiology, entitled "Marine Microbial Diversity: The Key to Earth's Habitability."

[Odd] A Romanian couple has named their son Yahoo as a sign of gratitu
Daily Libertatea said on Thursday Cornelia and Nonu Dragoman, both from Transylvania, met and decided they were meant for each other following a three-month relationship over the net.They married and had a baby this Christmas, whom they decided to name after one of the worldwide web's most popular portals."We named him Lucian Yahoo after my father and the net, the main beacon of my life,"Cornelia Dragoman was quoted as saying.

Researchers Pioneer New Gene Therapy Technique Using Natural Repair Pr
Harnessing the strength of a natural process that repairs damage to the human genome, a researcher from UT Southwestern Medical Center has helped establish a method of gene therapy that can accurately and permanently correct mutations in disease-causing genes.

Examining Bhutto's Significance in Pakistan
Audio of October 2007 radio program and associated article about "who Benazir Bhutto is, why she's so popular, and why she's a target for assassination."The article describes how Bhutto "has twice been the prime minister of Pakistan, and twice been deposed amid allegations of corruption."Also includes links to related material. From National Public Radio (NPR).

[Scary] Driving instructor gives lesson in road rage
A Dutch driving instructor hit a knife-wielding fellow motorist with a baseball bat during a fight over his student's skills, police say."Words were exchanged between the instructor and the man who pulled a knife and then the other hit him with a baseball bat,"a police spokeswoman said on Wednesday.


MP3 Music Downloads

Preview songs, Download Free Music,Burn CDs at ITunes.com
iTunes_RGB_9mm

 


Google




InformationQuickFind.com - Find Information Fast

Links