# Differential-geometric aspects of adapted contact structures.

http://hdl.handle.net/10150/185532
Title:
Differential-geometric aspects of adapted contact structures.
Author:
Pitucco, Anthony Peter.
Issue Date:
1991
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Let M denote a 2n-dimensional globally defined orientable manifold from which is constructed the product space N = M x R. It is assumed that N is endowed with a set of 2n independent smooth 1-forms {π(h),πʰ:h = 1,..,n}. Certain conditions are imposed on {π(h),πʰ} which imply the existence of local coordinates {qʰ,p(h)} on M and a function H(qʰ,p(h),t) on N, where t is the single coordinate on R, such that dπ = π(h) ∧ πʰ, where π has the structure of a Cartan form on N. The assumption that the function h = p(h)∂H/∂p(h)-H is non-zero on a region D ⊂ N, implies that π has maximal class on D. This construction gives rise to a local adapted contact structure on N and a local symplectic structure on M. A vector field X on N is said to be a contact field if there exists a smooth function σ : N → R such that ₤ₓπ = σπ. A vector field Z on N is called a canonical vector field if it admits the representation Z = ∂/∂t + (H, ) where (,) denotes the Poisson bracket on M. For a given function σ, a prescription is given for the construction of the space c(σ)(N) of contact fields in terms of solutions F of the p.d.e. Z<F> = σh. The vector space (UNFORMATTED EQUATION FOLLOWS) c(N) = ∪ (σ∊C)(∞)c(σ)(N) (END UNFORMATTED EQUATION) is shown to have the structure of a Lie sub-algebra of the Lie algebra of vector fields on N. It is shown that the associated subspace A(π) = {X:X˩π = 0} is such that c(σ)(N) ∩ A(π) = {0}. This implies that there is an X in c(σ)(N) such that X˩π ≠ 0. Thus, if the function H that appears in the Cartan form π is such that H = X˩π for any X ∊ c(σ)(N) it is possible to deduce that ∂H/∂t ≠ 0, which shows that such vector fields may be of relevance to the theory of non-conservative systems. A different set of 2n 1-forms {π(h),πʰ} is introduced on N that are subject to analogous conditions which ensure the existence of local coordinates (qʰ,p(h)) on M and a function K(qʰ,p(h),t) that gives rise to a new Cartan form π on N such that dπ= π(h) ∧ πʰ. By definition, the fundamental invariant of a parameter-dependent canonical transformation on N is dπ = dπ. In this new setting a contact field X satisfies the ₤ₓπ = σπ for some function σ: N to R. The relationship between the contact vector fields X and X is investigated in depth.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Geometry, Differential.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Degree Grantor:
University of Arizona
Rund, Hanno

DC FieldValue Language
dc.language.isoenen_US
dc.titleDifferential-geometric aspects of adapted contact structures.en_US
dc.creatorPitucco, Anthony Peter.en_US
dc.contributor.authorPitucco, Anthony Peter.en_US
dc.date.issued1991en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractLet M denote a 2n-dimensional globally defined orientable manifold from which is constructed the product space N = M x R. It is assumed that N is endowed with a set of 2n independent smooth 1-forms {π(h),πʰ:h = 1,..,n}. Certain conditions are imposed on {π(h),πʰ} which imply the existence of local coordinates {qʰ,p(h)} on M and a function H(qʰ,p(h),t) on N, where t is the single coordinate on R, such that dπ = π(h) ∧ πʰ, where π has the structure of a Cartan form on N. The assumption that the function h = p(h)∂H/∂p(h)-H is non-zero on a region D ⊂ N, implies that π has maximal class on D. This construction gives rise to a local adapted contact structure on N and a local symplectic structure on M. A vector field X on N is said to be a contact field if there exists a smooth function σ : N → R such that ₤ₓπ = σπ. A vector field Z on N is called a canonical vector field if it admits the representation Z = ∂/∂t + (H, ) where (,) denotes the Poisson bracket on M. For a given function σ, a prescription is given for the construction of the space c(σ)(N) of contact fields in terms of solutions F of the p.d.e. Z<F> = σh. The vector space (UNFORMATTED EQUATION FOLLOWS) c(N) = ∪ (σ∊C)(∞)c(σ)(N) (END UNFORMATTED EQUATION) is shown to have the structure of a Lie sub-algebra of the Lie algebra of vector fields on N. It is shown that the associated subspace A(π) = {X:X˩π = 0} is such that c(σ)(N) ∩ A(π) = {0}. This implies that there is an X in c(σ)(N) such that X˩π ≠ 0. Thus, if the function H that appears in the Cartan form π is such that H = X˩π for any X ∊ c(σ)(N) it is possible to deduce that ∂H/∂t ≠ 0, which shows that such vector fields may be of relevance to the theory of non-conservative systems. A different set of 2n 1-forms {π(h),πʰ} is introduced on N that are subject to analogous conditions which ensure the existence of local coordinates (qʰ,p(h)) on M and a function K(qʰ,p(h),t) that gives rise to a new Cartan form π on N such that dπ= π(h) ∧ πʰ. By definition, the fundamental invariant of a parameter-dependent canonical transformation on N is dπ = dπ. In this new setting a contact field X satisfies the ₤ₓπ = σπ for some function σ: N to R. The relationship between the contact vector fields X and X is investigated in depth.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectGeometry, Differential.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US