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NationMaster - Encyclopedia: Cotangent space (828 words) |
 | All cotangent spaces have the same dimension, equal to the dimension of the manifold. |
| Note that since the tangent space and the cotangent space at a point are both real vector spaces of the same dimension, they are isomorphic to each other. |
| All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold. |
| Reference.com/Encyclopedia/Cotangent bundle (834 words) |
| Because cotangent bundles can be thought of as symplectic manifolds, any real function on the cotangent bundle can be interpreted to be a Hamiltonian; thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out. |
| That is: The one-form assigns to a vector in the tangent bundle of 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). |
| The cylinder is the cotangent bundle of the circle. |
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