Cotangent Bundle

ONE-FORMS (THE COTANGENT SHEAF)

Smooth Sections of the cotangent bundle are differential One-form s.

Definition of the cotangent sheaf

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

\mathcal{I}

be the Sheaf of Germs of smooth functions on ''M''×''M'' which vanish on the diagonal. Then the quotient sheaf

\mathcal{I}/\mathcal{I}^2

consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is the Pullback of this sheaf to ''M'':

M=\Delta^---(\mathcal{I}/\mathcal{I}^2).

By Taylor's Theorem , this is a Locally Finite 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.

THE COTANGENT BUNDLE AS PHASE SPACE

''M'', since it is a Vector Bundle , 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'' possess a canonical one-form θ (also Tautological one-form or Symplectic Potential ). The Exterior Derivative of θ is a 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 Hamiltonian ; thus the cotangent bundle can be understood to be a Phase Space on which Hamiltonian Mechanics plays out.

The canonical one-form

''M'' as a manifold in its own right, there is a canonical 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''^''i'' are local coordinates on the base manifold ''M''. In terms of these base coordinates, there are fibre coordinates ''p''_''i'': a one-form at a particular point of ''T''---''M'' has the form ''p''_''i''''dx''^''i'' ( Einstein Summation Convention implied). So the manifold ''T''---''M'' itself caries local coordinates (''x''^''i'',''p''_''i'') 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

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	CATEGORIES ABOUT COTANGENT BUNDLE

	vector bundles

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Information About

Cotangent Bundle