TY - JOUR
T1 - Differential geometry of curves in Lagrange Grassmannians with given Young diagram
JF - Differential Geom. Appl. 27 (2009) 723-742
Y1 - 2009
A1 - Igor Zelenko
A1 - Li Chengbo
AB - Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its kth column is equal to the rank of the kth derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.
PB - Elsevier
UR - http://hdl.handle.net/1963/3819
U1 - 508
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - RPRT
T1 - Jacobi Equations and Comparison Theorems for Corank 1 Sub-Riemannian structures with symmetries
Y1 - 2009
A1 - Li Chengbo
A1 - Igor Zelenko
AB - The Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical\\nbundle of moving frames and the complete system of symplectic invariants, called curvature maps, for\\nparametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural\\nequation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the\\nnormal form for the Jacobi equation along this extremal and the curvature maps can be seen as the\\n\\\"coefficients\\\"of this normal form. In the case of a Riemannian metric there is only one curvature map and it is naturally related to the Riemannian sectional curvature. In the present paper we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After the factorization by the integral foliation of this symmetry, such sub-Riemannian structure can be reduced to a Riemannian manifold equipped with a closed 2-form(a magnetic field). We obtain explicit expressions for the curvature maps of the original sub-Riemannian structure in terms of the curvature tensor of this Riemannian manifold and the magnetic field. We also estimate the number of conjugate points along the sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of an uniform magnetic field. The language developed for the calculation of the curvature maps can be applied to more general sub-Riemannian structures with symmetries, including sub-Riemmannian structures appearing naturally in Yang-Mills fields.
UR - http://hdl.handle.net/1963/3736
U1 - 581
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - On feedback classification of control-affine systems with one and two-dimensional inputs
JF - SIAM J. Control Optim. 46 (2007) 1431-1460
Y1 - 2007
A1 - Andrei A. Agrachev
A1 - Igor Zelenko
AB - The paper is devoted to the local classification of generic control-affine systems on an n-dimensional manifold with scalar input for any n>3 or with two inputs for n=4 and n=5, up to state-feedback transformations, preserving the affine structure. First using the Poincare series of moduli numbers we introduce the intrinsic numbers of functional moduli of each prescribed number of variables on which a classification problem depends. In order to classify affine systems with scalar input we associate with such a system the canonical frame by normalizing some structural functions in a commutative relation of the vector fields, which define our control system. Then, using this canonical frame, we introduce the canonical coordinates and find a complete system of state-feedback invariants of the system. It also gives automatically the micro-local (i.e. local in state-input space) classification of the generic non-affine n-dimensional control system with scalar input for n>2. Further we show how the problem of feedback-equivalence of affine systems with two-dimensional input in state space of dimensions 4 and 5 can be reduced to the same problem for affine systems with scalar input. In order to make this reduction we distinguish the subsystem of our control system, consisting of the directions of all extremals in dimension 4 and all abnormal extremals in dimension 5 of the time optimal problem, defined by the original control system. In each classification problem under consideration we find the intrinsic numbers of functional moduli of each prescribed number of variables according to its Poincare series.
UR - http://hdl.handle.net/1963/2186
U1 - 2058
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Parametrized curves in Lagrange Grassmannians
JF - C. R. Math. 345 (2007) 647-652
Y1 - 2007
A1 - Igor Zelenko
A1 - Li Chengbo
UR - http://hdl.handle.net/1963/2560
U1 - 1559
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - RPRT
T1 - A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class
Y1 - 2006
A1 - Boris Doubrov
A1 - Igor Zelenko
AB - In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in R5. We solve the analogous problems for rank 2 distributions in Rn for arbitrary n > 5. Our method is a kind of symplectification of the problem and it is completely different from the Cartan method of equivalence.
JF - C. R. Math. Acad. Sci. Paris 342 (2006) 589-594
UR - http://hdl.handle.net/1963/1712
U1 - 2439
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Fundamental form and Cartan tensor of (2,5)-distributions coincide
JF - J. Dyn. Control Syst. 12 (2006) 247-276
Y1 - 2006
A1 - Igor Zelenko
AB - In our previous paper for generic rank 2 vector distributions on n-dimensional manifold (n greater or equal to 5) we constructed a special differential invariant, the fundamental form. In the case n=5 this differential invariant has the same algebraic nature, as the covariant binary biquadratic form, constructed by E.Cartan in 1910, using his ``reduction- prolongation\\\'\\\' procedure (we call this form Cartan\\\'s tensor). In the present paper we prove that our fundamental form coincides (up to constant factor -35) with Cartan\\\'s tensor. This result explains geometric reason for existence of Cartan\\\'s tensor (originally this tensor was obtained by very sophisticated algebraic manipulations) and gives the true analogs of this tensor in Riemannian geometry. In addition, as a part of the proof, we obtain a new useful formula for Cartan\\\'s tensor in terms of structural functions of any frame naturally adapted to the distribution.
UR - http://hdl.handle.net/1963/2187
U1 - 2057
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1
JF - J. Math. Sci. 135 (2006) 3168-3194
Y1 - 2006
A1 - Igor Zelenko
AB - The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way first we obtain a new elementary proof of classical Levi-Civita\\\'s Theorem about the classification of all Riemannian geodesically equivalent metrics in a neighborhood of so-called regular (stable) point w.r.t. these metrics. Secondly we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally we make the classification of all pairs of geodesically equivalent Riemannian metrics on a surface, which proportional in an isolated point. This is the simplest case, which was not covered by Levi-Civita\\\'s Theorem.
UR - http://hdl.handle.net/1963/2205
U1 - 2039
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - On variational approach to differential invariants of rank two distributions
JF - Differential Geom. Appl. 24 (2006) 235-259
Y1 - 2006
A1 - Igor Zelenko
AB - n the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n=5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his \\\"reduction-prolongation\\\" procedure. After Cartan\\\'s work the following questions remained open: first the geometric reason for existence of Cartan\\\'s tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan\\\'s tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in our previous works with A. Agrachev . In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n greater than 4.\\nFor n=5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In our next paper we show that in the case n=5 our fundamental form coincides with Cartan\\\'s tensor.
UR - http://hdl.handle.net/1963/2188
U1 - 2056
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - CHAP
T1 - Complete systems of invariants for rank 1 curves in Lagrange Grassmannians
T2 - Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005
Y1 - 2005
A1 - Igor Zelenko
AB - Curves in Lagrange Grassmannians naturally appear when one studies intrinsically \\\"the Jacobi equations for extremals\\\", associated with control systems and geometric structures. In this way one reduces the problem of construction of the curvature-type invariants for these objects to the much more concrete problem of finding of invariants of curves in Lagrange Grassmannians w.r.t. the action of the linear Symplectic group. In the present paper we develop a new approach to differential geometry of so-called rank 1 curves in Lagrange Grassmannian, i.e., the curves with velocities being rank one linear mappings (under the standard identification of the tangent space to a point of the Lagrange Grassmannian with an appropriate space of linear mappings). The curves of this class are associated with \\\"the Jacobi equations for extremals\\\", corresponding to control systems with scalar control and to rank 2 vector distributions. In particular, we construct the tuple of m principal invariants, where m is equal to half of dimension of the ambient linear symplectic space, such that for a given tuple of arbitrary m smooth functions there exists the unique, up to a symplectic transformation, rank 1 curve having this tuple, as the tuple of the principal invariants. This approach extends and essentially simplifies some results of our previous paper (J. Dynamical and Control Systems, 8, 2002, No. 1, 93-140), where only the uniqueness part was proved and in rather cumbersome way. It is based on the construction of the new canonical moving frame with the most simple structural equation.
JF - Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005
UR - http://hdl.handle.net/1963/2310
N1 - Proceedings of 9th Conference on Differential Geometry and its Applications, Prague 2004
U1 - 1706
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - On curvatures and focal points of distributions of dynamical Lagrangian distributions and their reductions by first integrals
JF - J. Dyn. Control Syst. 11 (2005) 297-327
Y1 - 2005
A1 - Andrei A. Agrachev
A1 - Natalia N. Chtcherbakova
A1 - Igor Zelenko
AB - Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and the curvature form. These invariants can be seen as generalizations of the classical curvature tensor in Riemannian Geometry. In particular, in terms of these invariants one can localize the focal points along extremals of the corresponding variational problems. In the present paper we study the behavior of the curvature operator, the curvature form and the focal points of a dynamical Lagrangian distribution after its reduction by arbitrary first integrals in involution. The interesting phenomenon is that the curvature form of so-called monotone increasing Lagrangian dynamical distributions, which appear naturally in mechanical systems, does not decrease after reduction. It also turns out that the set of focal points to the given point w.r.t. the monotone increasing dynamical Lagrangian distribution and the corresponding set of focal points w.r.t. its reduction by one integral are alternating sets on the corresponding integral curve of the Hamiltonian system of the considered dynamical distributions. Moreover, the first focal point corresponding to the reduced Lagrangian distribution comes before any focal point related to the original dynamical distribution. We illustrate our results on the classical $N$-body problem.
UR - http://hdl.handle.net/1963/2254
U1 - 1993
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Geometry of Jacobi Curves I
JF - J. Dynam. Control Systems 8 (2002) 93-140
Y1 - 2002
A1 - Andrei A. Agrachev
A1 - Igor Zelenko
AB - Jacobi curves are deep generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions.\\nIn the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of \\\"flat curves\\\".
PB - Springer
UR - http://hdl.handle.net/1963/3110
U1 - 1223
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Geometry of Jacobi curves II
JF - J. Dynam. Control Systems 8 (2002), no. 2, 167--215
Y1 - 2002
A1 - Andrei A. Agrachev
A1 - Igor Zelenko
PB - SISSA Library
UR - http://hdl.handle.net/1963/1589
U1 - 2529
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - CHAP
T1 - Principal invariants of Jacobi curves
T2 - Nonlinear control in the Year 2000 / Alberto Isidori, Francoise Lamnabhi-Lagarrigue, Witold Respondek (eds.) - Springer : Berlin, 2001. - (Lecture notes in control and information sciences ; 258). - ISBN 1-85233-363-4 (v. 1). - p. 9-22.
Y1 - 2000
A1 - Andrei A. Agrachev
A1 - Igor Zelenko
AB - Jacobi curves are far going generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. Differential geometry of these curves provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. In the present paper we mainly discuss two principal invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmanian providing the curve with a natural projective structure, and a fundamental form, which is a 4-oder differential on the curve.
JF - Nonlinear control in the Year 2000 / Alberto Isidori, Francoise Lamnabhi-Lagarrigue, Witold Respondek (eds.) - Springer : Berlin, 2001. - (Lecture notes in control and information sciences ; 258). - ISBN 1-85233-363-4 (v. 1). - p. 9-22.
PB - Springer
UR - http://hdl.handle.net/1963/3825
U1 - 502
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -