Using the language of algebraic geometry, the phase space of the system is a partial compactification of the cotangent bundle to the moduli space of stable " G "-bundles for some reductive group " G ", on some compact algebraic curve.
42.
More abstractly, in classical mechanics phase space is the cotangent bundle of configuration space, and in this interpretation the procedure above expresses that a choice of local coordinates on configuration space induces a choice of natural local Darboux coordinates for the standard symplectic structure on a cotangent space.
43.
If the manifold M represents the set of possible positions in a dynamical system, then the cotangent bundle \ ! \, T ^ { * } \ ! M can be thought of as the set of possible " positions " and " momenta ".
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Although this is defined using a particular coordinate system, the transformation law relating the ? " i " and the " x " " i " ensures that ? " P " is a well-defined function on the cotangent bundle.
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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 \ mathbb { R } ^ n \ times \ mathbb { R } ^ n.
46.
Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article . ( Canonical transformations are a special case of a symplectomorphism . ) However, a brief introduction to the modern mathematical description is included at the end of this article.
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One of the main reasons the cotangent bundle rather than the tangent bundle is used in the construction of the exterior complex is that differential forms are capable of being pulled back by smooth maps, while vector fields cannot be pushed forward by smooth maps unless the map is, say, a diffeomorphism.
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In particular, if Q is the configuration manifold then the Lagrangian L is defined on the tangent bundle TQ and the Hamiltonian is defined on the cotangent bundle T ^ * Q the fiber derivative is a map \ mathbb { F } L : TQ \ rightarrow T ^ * Q such that
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The "'equivalence postulate "'states that all one-particle systems can be connected by a non-degenerate coordinate transformation, more precisely by a map over the cotangent bundle of the position manifold, so that there exists a quantum action function S ( q ) transforms as a scalar field.
50.
If I'm understanding Frankel right ( which is a big if ), it sounds like a tautological one-form is just the pullback of the projection from a cotangent bundle to the base manifold, or using \ theta and \ pi as defined in the article, \ theta = \ pi ^ *.
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