1-jet Space Research Articles

Variational integrators for discrete Lagrange problems

A discrete Lagrange problem is defined as a discrete Lagrangian system endowed with a constraint submanifold in the space of 1-jets of the discretefibred manifold that configures the system. After defining the concepts of admissible section and infinitesimal admissible variation, the objective ofthese problems is to find admissible sections that are critical for the Lagrangian of the system with respect to the infinitesimal admissiblevariations. For admissible sections satisfying a certain regularity condition, we prove that critical sections are the solutions of an extendedunconstrained discrete variational problem canonically associated to the problem of Lagrange (discrete Lagrange multiplier rule). Next, we define theconcept of Cartan 1-form, establish a Noether theory for symmetries and introduce a notion of 'constrained variational integrator' that wecharacterize through a Cartan equation ensuring its symplecticity. Under a certain regularity condition of the problem of Lagrange, we prove theexistence and uniqueness of this kind of integrators in the neighborhood of a critical section, showing then that such integrators can be constructedfrom a generating function of the second class in the sense of symplectic geometry. Finally, the whole theory is illustrated with three elementaryexamples.

Jet Geometrical Objects Depending on a Relativistic Time

In this paper we study a collection of jet geometrical concepts, we refer to d-tensors, relativistic time dependent semisprays, harmonic curves and nonlinear connections on the 1-jet space J1(R;M), necessary to the construction of a Miron's-like geometrization for Lagrangians depending on a relativistic time. The geometrical relations between these jet geometrical objects are exposed.

On Implicit ODEs with Hexagonal Web of Solutions

Solutions of an implicit ODE form a web. Already for a cubic ODE the 3-web of solutions has a nontrivial local invariant, namely the curvature form. Thus, any local classification of implicit ODEs necessarily has functional moduli if no restriction on the class of ODEs is imposed. In this paper the most symmetric case of hexagonal 3-web of solutions is discussed, i.e. the curvature is supposed to vanish identically. A finite list of normal forms is established under some natural regularity assumptions. Geometric meaning of these assumptions is that the surface defined by an implicit ODE in the space of 1-jets is smooth as well as the criminant of the ODE, which is a critical set of surface projection into the plane of solutions.

Transformation of hyperbolic Monge-Ampére equations into linear equations with constant coefficients

.The class of Monge–Ampere equations is closedunder contact transformations, which is the propertythat singles out this class from the entire set of second-order equations. This fact was known to Sophus Lie,who studied Monge–Ampere equations by applyingcontact geometry methods [9, 10]. Lie also consideredclassification problems for Monge–Ampere equationspossessing intermediate integrals.In 1979, V.V. Lychagin proposed a geometricdescription for a wide class of second-order differentialequations on smooth manifolds. For a two-dimensionalmanifold, this class coincides with the class of Monge–Ampere equations (1). Lychagin’s basic idea [14] isthat Monge–Ampere equations and their multidimen-sional analogues are represented by differential formson the 1-jet space on a smooth manifold. An advantageof this approach over the classical one is an order reduc-tion of the jet space: the 2-jet space

Isotopies of Legendrian 1-knots and Legendrian 2-tori

We construct a Legendrian 2-torus in the 1-jet space of $S^1 x $\Bbb R$ (or of $\Bbb R^2$) from a loop of Legendrian knots in the 1-jet space of $\Bbb R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit any augmentation (linearization) but which still has non-trivial homology, as well as two Legendrian tori with isomorphic linearized contact homologies but with distinct contact homologies.

Morse flow trees and Legendrian contact homology in 1–jet spaces

Let $L\subset J^1(M)$ be a Legendrian submanifold of the 1-jet space of a Riemannian $n$-manifold $M$. A correspondence is established between rigid flow trees in $M$ determined by $L$ and boundary punctured rigid pseudo-holomorphic disks in $T^\ast M$, with boundary on the projection of $L$ and asymptotic to the double points of this projection at punctures, provided $n\le 2$, or provided $n>2$ and the front of $L$ has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact homology of $L$ in terms of Morse theory.

LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS

It is shown that Legendrian (respectively transverse) cable links in S3 with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J1(S1) with its standard tight contact structure.

Legendrian solid-torus links

Differential graded algebra invariants are constructed for Legendrian links in the 1-jet space of the circle. In parallel to the theory for R 3 , Poincare-Chekanov polynomials and characteristic al- gebras can be associated to such links. The theory is applied to dis- tinguish various knots, as well as links that are closures of Legendrian versions of rational tangles. For a large number of two-component links, the Poincare-Chekanov polynomials agree with the polynomials defined through the theory of generating functions. Examples are given of knots and links which differ by an even number of horizontal flypes that have the same polynomials but distinct characteristic algebras. Results ob- tainable from a Legendrian satellite construction are compared to results obtainable from the DGA and generating function techniques.

Generating function polynomials for legendrian links

It is shown that, in the 1-jet space of the circle, the swapping and the flyping procedures, which produce topologically equivalent links, can produce nonequivalent legendrian links. Each component of the links considered is legendrian isotopic to the 1-jet of the 0-function, and thus cannot be distinguished by the classical rotation number or Thurston-Bennequin invariants. The links are distinguished by calculating invariant polynomials defined via homology groups associated to the links through the theory of generating functions. The many calculations of these generating function polynomials support the belief that these polynomials carry the same information as a refined version of Chekanov's first order polynomials which are defined via the theory of holomorphic curves.

Bifurcations of simple umbilical points defined by vector fields normal to a surface immersed in ℝ4

Thev-principal configuration of an immersed surfaceM in ℝ4 is the set formed by the umbilical points and the lines of principal curvatures with respect to a unitary smooth vector fieldv normal toM. In this article we describe the bifurcation diagram ofv-principal configurations, wherev is parametrized in the space of 1-jets of normal vector fields which define an isolated umbilical point. Versal unfoldings of the nonlocally stable simple umbilical points are obtained.

Singularities of Second-Order Implicit Differential Equations: A Geometrical Approach

A second-order implicit differential equation R (x,y,\frac{dy}{dx},\frac{d^2y}{dx^2}) = 0 is an equation for which the second derivative cannot be written as a single-valued function of x, y, and the first derivative \frac{dy}{dx} = p. Such an equation defines a direction field in the space of 1-jets (x,y,p), but the existence and uniqueness of a direction at a point may not hold. Singularities can arise in a number of ways, and we examine the equation and its solution curves near the simplest of these singularities.

On the second law of thermodynamics and contact geometry

In this paper we consider the second law of thermodynamics for a dissipative system and its symmetry property in terms of contact geometry. We first show that the inaccessibility condition of Caratheodory and the assumption of semipositive definite property of the dissipative energy are equivalent to Clausius’ inequality. The inaccessibility condition then gives rise to a generalized Gibbs relation (GGR). By means of the GGR a 1-form ω can be defined such that the zero of ω reproduces the GGR. Such 1-form ω has the property ω∧(dω)n≠0 and ω∧(dω)n+1=0. The integral surface of the GGR is an n-dimensional 1-graph space G (Legendre submanifold) of a 1-jet space J1(En,R), where En is the base space of J1(En,R) with thermodynamic variables as its coordinates. The (2n+1)-dimensional J1(En,R) equipped with the 1-form ω is also called a contact bundle K, where the intensive thermodynamic variables are considered as the contact elements to K at every x. Next we construct an isovector field Xf such that the inaccessibility condition is invariant under the contact transformations generated by Xf. Finally, suppose under some specific assumptions the dynamical equations of the thermodynamic variables x can be approximated by the flow equations of a vector field XG on G. We can lift XG to Xf such that the 1-graph space G as well as the inaccessibility condition are preserved under the contact transformations generated by Xf.

Estimates for the Bennequin number of Legendrian links from state models for knot polynomials

A contact structure in a 3-dimensional manifold is a completely nonintegrable 2-dimensional distribution: if the distribution is the kernel of a (locally defined) 1-form λ then λ∧dλ � 0everywhere. The standard contact structure in 3-space, arising from the identification of R 3 with the space of 1-jets of functions on the line, is given by the contact form λ = dz − yd x; here x is a point on the line, z is the value of a function and y the value of the derivative at x. The same formula defines the standard contact structure in the space of 1-jets of functions on the circle (x being the cyclic coordinate), the space which is topologically the solid torus. In this paper we will be concerned only with these two contact manifolds. In a contact 3-dimensional manifold one considers two classes of knots and links: the transverse and the Legendrian ones. The former are everywhere transverse to the contact distribution, and the latter are everywhere tangent to it. Every topological isotopy class of knots in a contact 3-fold contains a transverse and a Legendrian knot. The main problem of contact knot theory (which has made, so far, only a few first steps) is to classify, up to contact isotopies, transverse and Legendrian knots within each topological isotopy class. In particular, one would like to have specifically contact invariants of transverse and Legendrian knots and links. One such invariant is easily defined. Legendrian and transverse knots in the standard contact space have natural framings given by the vector fields ∂z and ∂y, respectively. The corresponding self-linking number is called the Bennequin number; it is denoted by β(K) where K is a Legendrian or a transverse knot 1 . Similarly one defines the Bennequin number of an oriented Legendrian or transverse link. The study of knots in contact 3-dimensional manifolds was put forward by the seminal paper by D. Bennequin [Be]. One of the main results of this paper