scholarly journals Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Z. Ok Bayrakdar ◽  
T. Bayrakdar
Keyword(s):  
Vector Fields ◽  
Integral Manifold ◽  
Integrability Condition ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Jet Bundle ◽  
First Order ◽  
Burgers Equations ◽  
Integral Curves ◽  
Metric Connection

We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt=u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

scholarly journals Towards Noncommutative Linking Numbers via the Seiberg-Witten Map

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
H. García-Compeán ◽  
O. Obregón ◽  
R. Santos-Silva
Keyword(s):  
Three Dimensional ◽  
Simons Theory ◽  
Dimensional Manifold ◽  
First Order ◽  
Linking Number ◽  
Linking Numbers ◽  
Bohm Effect ◽  
Aharonov Bohm ◽  
Chern Simons ◽  
Noncommutative Parameter

Some geometric and topological implications of noncommutative Wilson loops are explored via the Seiberg-Witten map. In the abelian Chern-Simons theory on a three-dimensional manifold, it is shown that the effect of noncommutativity is the appearance of6nnew knots at thenth order of the Seiberg-Witten expansion. These knots are trivial homology cycles which are Poincaré dual to the higher-order Seiberg-Witten potentials. Moreover the linking number of a standard 1-cycle with the Poincaré dual of the gauge field is shown to be written as an expansion of the linking number of this 1-cycle with the Poincaré dual of the Seiberg-Witten gauge fields. In the process we explicitly compute the noncommutative “Jones-Witten” invariants up to first order in the noncommutative parameter. Finally in order to exhibit a physical example, we apply these ideas explicitly to the Aharonov-Bohm effect. It is explicitly displayed at first order in the noncommutative parameter; we also show the relation to the noncommutative Landau levels.

When is an Anosov flow geodesic?

1992 ◽  
Vol 12 (2) ◽  
pp. 227-232
Author(s):  
Leon W. Green
Keyword(s):  
Flow Field ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Vector Fields ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Anosov Flow ◽  
Unit Tangent Bundle

AbstractLet X, H+, H− be vector fields tangent, respectively, to an Anosov flow and its expanding and contracting foliations in a compact three-dimensional manifold, with γ, α+, α− one forms dual to them. If α+([H+, H−]) = α−([H+, H−]) and γ([H+, H−]) = α−([X, H−]) − α+([X, H+]), then the manifold has the structure of the unit tangent bundle of a Riemannian orbifold with geodesic flow field X.

scholarly journals Complete lift of vector fields and sprays to T∞M

2015 ◽  
Vol 12 (10) ◽  
pp. 1550113 ◽  
Author(s):  
Ali Suri ◽  
Somaye Rastegarzadeh
Keyword(s):  
Vector Field ◽  
Differential Equations ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Dimensional Manifold ◽  
Closed Curves ◽  
Infinite Dimensional ◽  
Integral Curves ◽  
Complete Lift ◽  
Infinite Dimensional Manifold

In this paper for a given Banach, possibly infinite dimensional, manifold M we focus on the geometry of its iterated tangent bundle TrM, r ∈ ℕ ∪ {∞}. First we endow TrM with a canonical atlas using that of M. Then the concepts of vertical and complete lifts for functions and vector fields on TrM are defined which they will play a pivotal role in our next studies i.e. complete lift of (semi)sprays. Afterward we supply T∞M with a generalized Fréchet manifold structure and we will show that any vector field or (semi)spray on M, can be lifted to a vector field or (semi)spray on T∞M. Then, despite of the natural difficulties with non-Banach modeled manifolds, we will discuss about the ordinary differential equations on T∞M including integral curves, flows and geodesics. Finally, as an example, we apply our results to the infinite-dimensional case of manifold of closed curves.

scholarly journals PRE-MULTISYMPLECTIC CONSTRAINT ALGORITHM FOR FIELD THEORIES

2005 ◽  
Vol 02 (05) ◽  
pp. 839-871 ◽  
Author(s):  
MANUEL DE LEÓN ◽  
JESÚS MARÍN-SOLANO ◽  
JUAN CARLOS MARRERO ◽  
MIGUEL C. MUÑOZ-LECANDA ◽  
NARCISO ROMÁN-ROY
Keyword(s):  
Fiber Bundle ◽  
General Framework ◽  
Vector Fields ◽  
Classical Field ◽  
Field Theories ◽  
Second Step ◽  
Field Equations ◽  
Jet Bundle ◽  
First Order ◽  
Classical Field Theories

We present a geometric algorithm for obtaining consistent solutions to systems of partial differential equations, mainly arising from singular covariant first-order classical field theories. This algorithm gives an intrinsic description of all the constraint submanifolds. The field equations are stated geometrically, either representing their solutions by integrable connections or, what is equivalent, by certain kinds of integrable m-vector fields. First, we consider the problem of finding connections or multivector fields solutions to the field equations in a general framework: a pre-multisymplectic fiber bundle (which will be identified with the first-order jet bundle and the multi-momentum bundle when Lagrangian and Hamiltonian field theories are considered). Then, the problem is stated and solved in a linear context, and a pointwise application of the results leads to the algorithm for the general case. In a second step, the integrability of the solutions is also studied. Finally, the method is applied to Lagrangian and Hamiltonian field theories and, for the former, the problem of finding holonomic solutions is also analyzed.

scholarly journals Steady Euler flows and Beltrami fields in high dimensions

2020 ◽  
pp. 1-24
Author(s):  
ROBERT CARDONA
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
High Dimensions ◽  
Beltrami Fields ◽  
Open Books ◽  
Euler Flows ◽  
Reeb Field ◽  
Beltrami Field ◽  
Volume Preserving

Abstract Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three-dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.

scholarly journals Helicity of vector fields preserving a regular contact form and topologically conjugate smooth dynamical systems

2012 ◽  
Vol 33 (5) ◽  
pp. 1550-1583 ◽  
Author(s):  
STEFAN MÜLLER ◽  
PETER SPAETH
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Contact Dynamics ◽  
Dimensional Manifold ◽  
Contact Form ◽  
Hamiltonian Flows ◽  
Higher Dimensional ◽  
Calabi Invariant ◽  
Regular Contact ◽  
Volume Preserving

AbstractWe compute the helicity of a vector field preserving a regular contact form on a closed three-dimensional manifold, and improve results of Gambaudo and Ghys [Enlacements asymptotiques. Topology 36(6) (1997), 1355–1379] relating the helicity of the suspension of a surface isotopy to the Calabi invariant of the isotopy. Based on these results, we provide positive answers to two questions posed by Arnold in [The asymptotic Hopf invariant and its applications. Selecta Math. Soviet. 5(4) (1986), 327–345]. In the presence of a regular contact form that is also preserved, the helicity extends to an invariant of an isotopy of volume-preserving homeomorphisms, and is invariant under conjugation by volume-preserving homeomorphisms. A similar statement also holds for suspensions of surface isotopies and surface diffeomorphisms. This requires the techniques of topological Hamiltonian and contact dynamics developed by Banyaga and Spaeth [On the uniqueness of generating Hamiltonians for topological strictly contact isotopies.Preprint, 2012], Buhovsky and Seyfaddini [Uniqueness of generating Hamiltonians for continuous Hamiltonian flows. J. Symplectic Geom. to appear, arXiv:1003.2612v2], Müller [The group of Hamiltonian homeomorphisms in the$L^\infty $-norm. J. Korean Math. Soc.45(6) (2008), 1769–1784], Müller and Oh [The group of Hamiltonian homeomorphisms and$C^0$-symplectic topology. J. Symplectic Geom. 5(2) (2007), 167–219], Müller and Spaeth [Topological contact dynamics I: symplectization and applications of the energy-capacity inequality.Preprint, 2011, arXiv:1110.6705v2] and Viterbo [On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows. Int. Math. Res. Not. (2006), 34028; Erratum,Int. Math. Res. Not.(2006), 38748]. Moreover, we generalize an example of Furstenberg [Strict ergodicity and transformation of the torus. Amer. J. Math. 83(1961), 573–601] on topologically conjugate but not$C^1$-conjugate area-preserving diffeomorphisms of the two-torus to trivial$T^2$-bundles, and construct examples of Hamiltonian and contact vector fields that are topologically conjugate but not$C^1$-conjugate. Higher-dimensional helicities are considered briefly at the end of the paper.

Vector Fields

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fixed Point ◽  
Vector Field ◽  
Scalar Field ◽  
Vector Fields ◽  
Integral Curve ◽  
Jacobi Matrix ◽  
Partial Differential Operator ◽  
Material Derivative ◽  
First Order ◽  
Integral Curves

The velocity of a fluid at each point of space-time is a vector field (or flow). It is best to think of it in terms of the effect of fluid flow on some scalar field. A vector field is thus a first order partial differential operator, called the material derivative in fluid mechanics. The path of a speck of dust carried along (advected) by the fluid is the integral curve of the velocity field. Even simple vector fields can have quite complicated integral curves: a manifestation of chaos. Of special interest are incompressible (with zero divergence) and irrotational (with zero curl) flows. A fixed point of a vector field is a point at which it vanishes. The derivative of a vector field at a fixed point is a matrix (the Jacobi matrix) whose spectrum is independent of the choice of coordinates.

scholarly journals An analysis of the first-order form of gauge theories

2012 ◽  
Vol 90 (2) ◽  
pp. 165-174 ◽  
Author(s):  
N. Kiriushcheva ◽  
S.V. Kuzmin ◽  
D.G.C. McKeon
Keyword(s):  
Gauge Theory ◽  
Vector Fields ◽  
Three Dimensional ◽  
Mass Term ◽  
Maxwell Theory ◽  
First Order ◽  
Order Form ◽  
Constraint Structure ◽  
The Relationship ◽  
Hilbert Action

The first-order form of a Maxwell theory and U(1) gauge theory in which a gauge invariant mass term appears is analyzed using the Dirac procedure. The form of the gauge transformation that leaves the action invariant is derived from the constraints present. A nonabelian generalization is similarly analyzed. This first-order three dimensional massive gauge theory is rewritten in terms of two interacting vector fields. The constraint structure when using light-cone coordinates is considered. The relationship between first- and second-order forms of the two-dimensional Einstein–Hilbert action is explored where a Lagrange multiplier is used to ensure their equivalence.

scholarly journals Approximate Ad Hoc Parametric Solutions for Nonlinear First-Order PDEs Governing Two-Dimensional Steady Vector Fields

2010 ◽  
Vol 2010 ◽  
pp. 1-23
Author(s):  
M. P. Markakis
Keyword(s):  
Vector Fields ◽  
Ad Hoc ◽  
Three Dimensional ◽  
Solid Surfaces ◽  
Two Dimensional ◽  
Nonlinear Odes ◽  
First Order ◽  
Parabolic Geometry ◽  
Abel Differential Equation ◽  
Analytical Tools

Through a suitable ad hoc assumption, a nonlinear PDE governing a three-dimensional weak, irrotational, steady vector field is reduced to a system of two nonlinear ODEs: the first of which corresponds to the two-dimensional case, while the second involves also the third field component. By using several analytical tools as well as linear approximations based on the weakness of the field, the first equation is transformed to an Abel differential equation which is solved parametrically. Thus, we obtain the two components of the field as explicit functions of a parameter. The derived solution is applied to the two-dimensional small perturbation frictionless flow past solid surfaces with either sinusoidal or parabolic geometry, where the plane velocities are evaluated over the body's surface in the case of a subsonic flow.

Fefferman-type metrics and the projective geometry of sprays in two dimensions

2007 ◽  
Vol 142 (3) ◽  
pp. 509-523 ◽  
Author(s):  
M. CRAMPIN ◽  
D. J. SAUNDERS
Keyword(s):  
Three Dimensional ◽  
Differentiable Manifold ◽  
Simple Expression ◽  
Two Dimensions ◽  
Second Order ◽  
Simple Extension ◽  
Dimensional Manifold ◽  
Conformal Curvature ◽  
Projective Curvature ◽  
Integral Curves

AbstractA spray is a second-order differential equation field on the slit tangent bundle of a differentiable manifold, which is homogeneous of degree 1 in the fibre coordinates in an appropriate sense; two sprays which are projectively equivalent have the same base-integral curves up to reparametrization. We show how, when the base manifold is two-dimensional, to construct from any projective equivalence class of sprays a conformal class of metrics on a four-dimensional manifold, of signature (2, 2); the Weyl conformal curvature of these metrics is simply related to the projective curvature of the sprays, and the geodesics of the sprays determine null geodesics of the metrics. The metrics in question have previously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a different method involving the exploitation of a close analogy between the Cartan geometry of second-order ordinary differential equations and of three-dimensional Cauchy–Riemann structures. From this perspective the derived metrics are seen to be analoguous to those defined by Fefferman in the CR theory, and are therefore said to be of Fefferman type. Our version of the construction reveals that the Fefferman-type metrics are derivable from the scalar triple product, both directly in the flat case (which we discuss in some detail) and by a simple extension in general. There is accordingly in our formulation a very simple expression for a representative metric of the class in suitable coordinates.

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