Abstract
In this chapter we consider the simplest nonholonomic variational problems. We study three-dimensional nonholonomic Lie groups, i.e. groups with a left-invariant nonholonomic distribution. Our main subject is the study of the nonholonomic geodesic flow (NG-flow), more precisely, of the nonholonomic sphere, of the wave front (Section 1), and of the general dynamical properties of the flow (Section 2). The mixed bundle for Lie groups is the direct product G × (V ⊕ V⊥). In Section 1.1 we show that the NG-flow on the mixed bundle is the semidirect product with base V ⊕ V⊥ and fiber G. In Section 1.2 we describe left-invariant metric tensors on Lie algebras; in Section 1.3 the normal forms for the equations of nonholonomic geodesics are obtained. In Section 1.4 we study the reduced flow on V ⊕ V ⊥. In the subsequent Sections (1.5–1.7) we describe local properties of the flow on the fiber; in Section 1.5 we describe the ε-wave front of the NG-flow and the ε-sphere of the nonholonomic metrics which appear to be manifolds with singularities, the same for all three-dimensional nonholonomic Lie groups. In Section 1.6 we describe their topology and in Section 1.7 their metric structure.
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References
Agrachev, A.A., Vakhrameev, S.A., Gamkrelidze, R.V. [1983]: Differential geometry and group-theoretical methods in optimal control theory. Itogi Nauki Tekh., Ser. Probl. Geom., 14, 3–56. English transl.: J. Sov. Math. 28, 145–182 (1985), Zbl.542.93045
Akhiezer, N.I. [1981]: The Calculus of Variations. Vishcha Shkola: Kharkov. English transl: Harwood Academic Publishers: London, 1988, Zbl.507.49001
Aleksandrov, A.D. [1947a, b]: Geometry and topology in the USSR. I–II. Usp. Mat. Nauk 2, No. 4, 3–58; 2, No. 5, 9–92 (Russian), Zbl.36, 379
Aleksandrov, A.D. [1958, 1959a, b]: On the principle of maximum. I–III. Izv. Vyssh. Uchebn. Zaved., Mat., 1958, No. 5, 126–157, Zbl.123,71; 1959, No. 3, 3–12; 195, No. 5, 16–32 (Russian), Zbl. 125, 59
Appell, P. [1953]: Traité de Mécanique Rationnelle, T.1, 2. Gauthier-Villars: Paris, Zbl.67, 132
Arnol’d, V.I. [1974]: Mathematical Methods of Classical Mechanics. Nauka: Moscow. English transl.: Graduate Texts in Math. 60. Springer-Verlag: New York-Berlin-Heidelberg, 1978, Zbl.386.70001
Arnol’d, V.I. [1983]: Singularities in the calculus of variations. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 22, 3–55. English transl.: J. Sov. Math. 27, 2679–2713 (1984), Zbl.537.58012
Arnol’d, V.I. [1988]: Geometrical Methods in the Theory of Ordinary Differential Equations. 2nd ed., Die Grundlehren der Math. Wissenschaften 250. Springer-Verlag: New York-Berlin-Heidelberg, Zbl.507.34003
Arnol’d, V.I., Givental, A.B. [1985]: Symplectic geometry. In: Dynamical Systems IV. Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 4, 7–139. English transl. in: Encyclopaedia of Math. Sciences 4, 1–136. Springer-Verlag: Berlin-Heidelberg-New York, 1988, Zbl.592.58038
Arnol’d, V.I., Kozlov, V.V., Nejshtadt, A.I. [1985]: Mathematical aspects of classical and celestial mechanics. In: Dynamical Systems III. Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 3, 5–304. English transl.: Encyclopaedia of Math. Sciences 3. Springer-Verlag: Berlin-Heidelberg-New York, 1988, Zbl.612.70002
Auslander, L., Green, L., Hahn, F., Markus, L. and Massey, W. [1963]: Flows on Homogeneous Spaces. Ann. Math. Stud. 53. Princeton University Press: Princeton, Zbl. 106, 368
Bennequin, D. [1983]: Entrelacement et équations de Pfaff. Astérisque 107/108, 87–161, Zbl.573.58022
Berezin, F.A. [1974]: Hamiltonian formalism in the general Lagrange problem. Usp. Mat. Nauk 29, No. 3, 183–184 (Russian), Zbl.307.70020
Birkhoff, G. [1967]: Lattice Theory. Am. Math. Soc. Colloq. Publ. 25. Am. Math. Soc: Providence, Zbl. 153, 25
Bishop, R.L., Crittenden, R.J. [1964]: Geometry of Manifolds. Academic Press: New York, Zbl.132, 160
Boothby, W.M., Wang, H.C. [1958]: On contact manifolds. Ann. Math., II. Ser. 68, 721–734, Zbl.84, 392
Bourbaki, N. [1972]: Groupes et Algèbres de Lie, Ch. 1–3. Hermann: Paris. English transl.: Lie Groups and Lie Algebras. Chap. 1–3, Springer-Verlag: Berlin-Heidelberg-New York, 1981, Zbl.244.22007
Brendelev, V.N. [1978]: On commutation relations in nonholonomic mechanics. Vestn. Mosk. Univ., Ser. I, 1978, No. 6, 47–54. English transl.: Mosc. Univ. Mech. Bull. 33, No. 5–6, 30–35 (1978), Zbl.401.70017
Bröcker, Th. [1975]: Differential Germs and Catastrophes. Cambridge University Press: Cambridge, Zbl.302.58006
Brockett, R. [1982]: Control theory and singular Riemannian geometry. In: New Directions in Applied Mathematics, Pap. present. 1980 on the ocass. of the Case centen. Celebr., (Hilton, P.J., Young, C.S. (Eds.)), 11–27, Zbl.483.49035
Brockett, R.W., Millman, R.S., and Sussmann, H.J. (eds) [1982]: Differential Geometric Control Theory. Progress in Math. 27, Birkhäuser: Boston-Basel-Stuttgart, Zbl.503.00014
Carathéodory, C. [1909]: Untersuchungen über die Grundlagen der Thermodynamik. Math. Ann. 67, 355–386
Cartan E. [1926]: Les groupes d’holonomie des espaces généralisés. Acta Math. 48, 1–42, Jbuch 52, 723
Cartan, E. [1935]: La Méthode du Repère Mobile, la Théorie des Groupes Continus et les Espaces Généralisés. Hermann: Paris, Zbl. 10, 395
Chaplygin S.A. [1949]: Papers on Dynamics of Nonholonomic Systems. GITTL: Moscow-Leningrad (Russian)
Chow, W.L. [1939]: Systeme von linearen partiellen Differential-gleichungen erster Ordnung. Math. Ann. 117, 98–105, Zbl.22, 23
Cornfel’d, I.P., Fomin, S.V., Sinai, Ya.G. [1980]: Ergodic Theory. Nauka: Moscow. English transl.: Die Grundlehren der Mathematischen Wissenschaften 245. Springer-Verlag: New York-Berlin-Heidelberg 1982, Zbl.493.28007
Davydov, A.A. [1985]: The attainability boundary is quasi-Hölderian. Tr. Semin. Vectorn. Tensorn. Anal., Prilozh. Geom. Mekh. Fiz. 22, 25–30. English transl.: Sel. Math. Sov. 9, No. 3, 229–234 (1990), Zbl.593.49028
Dirac, P.A.M. [1964]: Lectures on Quantum Mechanics. Yeshiva University: New York
Dobronravov, V.V. [1970]: Elements of Mechanics of Nonholonomic Systems. Vyssh. Shkola: Moscow (Russian), Zbl.216, 248
Dubrovin, B.A., Fomenko, A.T., Novikov, S.P. [1979]: Modern Geometry. Methods and Applications I. Nauka: Moscow. English transl.: Graduate Texts in Math. 93. Springer-Verlag: New York-Berlin-Heidelberg, 1984, Zbl.433.53001
Fajbusovich, L. [1988]: A Hamiltonian formalism for nonholonomic dynamical systems connected with generalized Toda flows. In: Nov. Global’nom Anal. 1988, 167–171 (Russian), Zbl.706.58058
Fefferman, C., Phong, D.H. [1979]: On the lowest eigenvalue of pseudodifferential operators. Proc. Natl. Acad. Sci. USA 76, 6055–6056, Zbl.434.35071
Fefferman, C., Sanchez-Calle, A. [1986]: Fundamental solution for second order subelliptic operators. Ann. Math., II. Ser. 124, 247–272, Zbl.613.35002
Filippov, A.F. [1959]: On certain questions of the optimal control theory. Vestn. Mosk. Univ., Ser. Mat. Mech. Astron. Fiz. Chim. 14, No. 2, 25–32 English transl.: J. Soc. Ind. Appl. Math., Ser. A: Control 1 (1962), 76–84 (1964), Zbl.90, 69
Folland, G.E. [1973]: A fundamental solution for a subelliptic operator. Bull. Am. Math. Soc. 79, 373–376, Zbl.256.35020
Folland, G.E. [1975]: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13, 161–207, Zbl.312.35026
Folland, G.E., Stein, E.M. [1974]: Estimates for ?̄-complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522, Zbl.293.35012
Franklin, P., Moore, C.L.E. [1931]: Geodesics of Pfaffians. J. Math. Phys. 10, 157–190, Zbl.2, 412
Gardner, R.B. [1967]: Invariants of Pfaffian systems. Trans. Am. Math. Soc. 126, 514–533, Zbl.161, 413
Gardner, R.B. [1983]: Differential geometric methods interfacing control theory. In: Differential Geometric Control Theory. Proc. Conf. Mich. Technol. Univ. 1982, Prog. Math. 27, 117–180, Zbl.522.49028
Gaveau, B. [1976]: Systèmes dynamiques associés à certains opérateurs hypoelliptiques. Bull. Sci. Math. II. Ser. 102, 203–229, Zbl.391.35019
Gaveau, B. [1977]: Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139, 95–153, Zbl.366.22010
Gaveau, B., Greiner, P., Vantier, J. [1985]: Intégrales de Fourier quadratiques et calcul symbolique exact sur le groupe d’Heisenberg. J. Funt. Anal. 68, 248–272, Zbl.609.43007
Gaveau, B., and Vauthier, J. [1986]: The Dirichlet problem on the Heisenberg group III. Can. J. Math. 38, No. 3, 666–671, Zbl.603.22003
Gel’fand, I.M., Fomin, S.V. [1961]: Variational Calculus. Fizmatgiz: Moscow. English transl.: Prentice Hall: Englewood Cliffs, N.J., 1965, 3rd ed., Zbl.127, 54
Gershkovich, V.Ya. [1984a]: Two-sided estimates of metrics generated by absolutely non-holonomic distributions on Riemannian manifolds. Dokl. Akad. Nauk SSSR 278, No. 5, 1040–1044. English transl.: Sov. Math., Dokl. 30, 506–510 (1984), Zbl.591.53033
Gershkovich, V.Ya. [1984b]: A variational problem with nonholonomic constraints on SO(3). Geometriya i Topologiya v Global’nykh Nelinejnykh Zadachakh, Nov. Global’nom Anal. 1984, 149–152, Zbl.554.58021
Gershkovich, V.Ya. [1985]: The method of penalty metrics in problems with nonholonomic constraints. In: Applications of Topology in Modern Analysis, 138–145. Voronezh University: Voronezh (Russian)
Gershkovich, V. [1988]: On normal form of distribution jets. In: Topology and Geometry, Rokhlin Semin. 1984–1986, Lect. Notes Math. 1346 (Viro, O.Ya. (Ed.)), 77–98. Zbl.649.58004
Gershkovich, V. [1989]: Short time hypoelliptic diffusion. Proc. of the 5th Internat. Conf. on Probability Theory, Vilnius, Vol. 3, 130–131
Gershkovich, V. [1990]: Estimates of nonholonomic ε-balls. In: Lect. Notes Math. 1543 (Borisovich, Yu.G. Gliklikh, Yu.E. (Eds.))
Gershkovich, V., Vershik, A. [1988]: Nonholonomic manifolds and nilpotent analysis. J. Geom. Phys. 5, No. 3, 407–452, Zbl.693.53006
Gibbs, J.W. [1873]: Graphical methods in the thermodynamics of fluids. Trans. Connec. Acad. 2, 309–342, Jbuch 5, 585
Godbillon, C. [1969]: Géométrie Différentielle et Mécanique Analytique. Collection Méthodes. Hermann: Paris, Zbl. 174, 246
Golubitsky, M., Guillmin, V. [1973]: Stable Mappings and their Singularities. Graduate Texts in Mathematics 14. Springer-Verlag: New York, Berlin-Heidelberg Zbl.294.58004
Gorbatenko, E.M. [1985]: Differential geometry of nonholonomic manifolds (after V.V. Vagner). Geom. Sbornik, 31–43, Tomsk University: Tomsk
Goursat, E. [1922]: Leçons sur le problème de Pfaff. Hermann: Paris, Jbuch 48, 538
Gray, J.W. [1959]: Some global properties of contact structures. Ann. Math., II. Ser. 69, No. 2, 421–450, Zbl.92, 393
Griffiths, P.A. [1983]: Exterior Differential Systems and the Calculus of Variations. Progress in Math. 25. Birkhäuser: Boston-Basel-Berlin, Zbl.512.49003
Grigoryan, A.T., Fradlin, B.N. [1977]: Mechanics in USSR. Nauka: Moscow (Russian)
Grigoryan, A.T., Fradlin, B.N. [1982]: A History of Mechanics of the Rigid Body. Nauka: Moscow (Russian)
Gromoll, D., Klingenberg, W., Meyer, W. [1968]: Riemannsche Geometrie im Grossen. Lect. Notes Math. 55, Zbl. 155, 307
Gromov, M. [1981a]: Groups of polynomial growth and expanding maps. Publ. Math., Inst. Hautes Etud. Sci. 53, 53–78, Zbl.474.20018
Gromov, M. [1981b]: Structures métriques pour les variétés Riemanniennes. Redigé par J. Lafontaine et P. Pansu. Textes Mathématiques, 1. Cedic/Nathan. VII: Paris, Zbl.509.53034
Günter, N.M. [1941]: Lectures on the Calculus of Variations. ONTI: Leningrad-Moscow (Russian)
Hartman, P. [1964]: Ordinary Differential Equations. J. Wiley & Sons: New York-London-Sydney, Zbl.125, 321
Haynes, G.W., Hermes, H. [1970]: Nonlinear controllability via Lie theory. SIAM J. Control Optimization, 8, 450–460, Zbl.229.93012
Helffer, B., Nourrigat, J. [1985]: Hypoellipticité Maximale pour des Opérateurs Polynômes de Champs de Vecteurs. Progress in Math. 58. Birkhäuser: Basel-Boston-Stuttgart, Zbl.568.35003
Helgason, S. [1978]: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press: New York-London, Zbl.451.53038
Hertz, H.R. [1894] Die Prinzipien der Mechanik. Gesammelte Werke, Bd. 2. J.A. Barth: Leipzig, Jbuch 25, 1310
Hörmander, L. [1967]: Hypoelliptic second order differential equations. Acta Math. 119, 147–171, Zbl.156, 107
Hörmander, L., Melin, A. [1978]: Free systems of vector fields. Ark. Mat. 16, 83–88, Zbl.383.35013
Hueber, H., Müller, D. [1989]: Asymptotics for some Green kernels of the Heisenberg group and the Martin boundary. Math. Ann. 283, 97–119, Zbl.639.31005
Hulanicki, A. [1976]: The distribution of energy in the Brownian motion in Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group. Stud. Math. 56, 165–173, Zbl.336.22007
Jakubczyk, B., Przytycki, F. [1984]: Singularities of k-tuples of vector fields. Diss. Math. 213, Zbl.565.58007
Judjevic V., Sussmann, H.J. [1972]: Controllability of nonlinear systems. J. Differ. Equations 12, 95–116, Zbl.242.49040
Kannai, Y. [1977]: Off diagonal short time asymptotics for fundamental solutions of diffusion equations. Commun. Partial Differ. Equations 2, 781–830, Zbl.381.35039
Karapetyan, A.V. [1981]: On realization of nonholonomic constraints by viscous friction forces and celtic stones stability. Prikl. Mat. Mekh. 45, 542–51. English transl.: J. Appl. Math. Mech. 45, 30–36 (1982), Zbl.493.70008
Kobyashi, S., Nomizu, K. [1963, 1969]: Foundations of Differential Geometry, Vol. 1, 2. Interscience Publ.: New York-London, Zbl.119, 375, Zbl.175, 485
Kozlov, V.V. [1982a]: Dynamics of systems with non-integrable constraints. I. Vestn. Mosk. Univ., Ser. I, No. 3, 92–100. English transl., Mosc. Univ. Mech. Bull. 37, No. 3–4, 27–34 (1982), Zbl.501.70016
Kozlov, V.V. [1982b]: Dynamics of systems with non-integrable constraints. II. Vestn. Mosk. Univ., Ser. I, No. 4, 70–76. English transl. Mosc. Univ. Mech. Bull. 37, No. 3–4, 74–80 (1982), Zbl.508.70012
Kozlov, V.V. [1983]: Dynamics of systems with constraints. III. Vestn. Mosk. Univ., Ser. I, No. 3, 102–111. English transl.: Mosc. Univ. Mech. Bull. 38, No. 3, 40–51 (1983), Zbl.516.70017
Kumpera, A., Ruisz, C. [1982]: Sur l’équivalence locale des systèmes de Pfaffen drapeau. In: Monge-Ampère Equations and Related Topics. Proc. Semin., Firenze, 1980, 201–248, Zbl.516.58004
Lanczos, C. [1949]: The Variational Principles of Mechanics. Univ. Toronto Press: Toronto, Zbl.37, 399
Laptev, G.F. [1953]: Differential geometry of embedded manifolds. Tr. Mosk. Mat. O-va 2, 275–382 (Russian), Zbl.53, 428
Lefshetz, S. [1957]: Differential Equations. Geometric Theory. Interscience Publishers: New York-London-Sydney, Zbl.80, 64
Levitt, N., Sussmann, H.J. [1975]: On controllability by means of two vector fields. SIAM J. Control Optimization, 13, No. 6, 1271–1281, Zbl.313.93006
Lobry, C. [1973]: Dynamical polysystems and control theory. In: Geom. Methods Syst. Theory, Proc. NATO Adv. Study Inst. London, 1–42, Zbl.279.93012
Lumiste Yu.G. [1966]: Connections in homogeneous bundles. Mat. Sb., Nov. Ser. 69 (111), 434–469. English transl.: Am. Math. Soc., Transl., II. Ser. 92, 231–274 (1970), Zbl.143, 446
Lutz, R. [1974]: Structures de contact en codimension quelconque. In: Géométrie Différentielle, Colloque, Santiago de compostella, Lect. Notes Math. 392 (Vidal, E. (Ed.)), 23–29, Zbl.294.5303
Manin, Yu.I. [1984]: Gauge Field Theory and Complex Geometry. Nauka: Moscow. English transl.: Die Grundlehren der Mathematischen Wissenschaften 289. Springer-Verlag: New York-Berlin-Heidelberg, 1988, Zbl.576.53002
Martinet, I. [1971]: Formes de contact sur les variétés de dimension 3. In: Proc. Liverpool Singularities-Symp. II, Lect. Notes Math. 209 (Wall, C.T.C. (Ed.)), 142–163, Zbl.215, 230
Melin, A. [1983]: Parametrix constructions for right invariant differential operators on nilpotent groups. Ann. Global Anal. Geom. 1, No. 1, 79–130, Zbl.524.58044
Menikoff, A., Sjøstrand, J. [1978]: On the eigenvalues of a class of hypoelliptic operators. Math. Ann. 235, 55–85, Zbl.375.35014
Métivier, G. [1976]: Fonction spectrale et valeurs propres d’opérateurs non elliptiques. C. R. Acad. Sci. Paris, Ser. A283, 453–456, Zbl.337.35057
Milnor, J. [1963]: Morse Theory. Ann. Math. Studies 51. Princeton Univ. Press: Princeton N. J., Zbl.108, 104
Mitchell, J. [1985]: On Carnot-Caratheodory metrics. J. Differ. Geom. 21, 35–45, Zbl.554.53023
Mormul, P. [1988]: Singularities of vector fields in ℝ 4. Institute of Math. of the Polish Academy of Sciences, Preprint No. 420: Warszawa
Nachman, A. [1982]: The wave equation on the Heisenberg group. Commun. Partial Differ. Equations 7, 675–714, Zbl.524.35065
Nagano, T. [1966]: Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Japan. 18, No. 4, 398–404, Zbl.147,235
Nagel, A., Stein, E.M., Wainger, S. [1985]: Balls and metrics defined by vector field. I. Acta Math. 155, No. 1–2, 103–147, Zbl.578.32044
Nejmark, Yu.L., Fufaev, N.A. [1967]: Dynamics of Nonholonomic Systems. Nauka: Moscow (Russian), Zbl. 171, 455
Pansu, P. [1983]: Croissance des boules et des géodésiques fermées dans les nilvariétés. Ergodic Theory Dyn. Syst. 3, 415–445, Zbl.509.53040
Rashevsky, P.K. [1938]: Any two points of a totally nonholonomic space may be connected by an admissible line. Uch. Zap. Ped. Inst. im. Liebknechta. Ser. Phys. Math., Vol. 2, 83–94 (Russian)
Rashevsky, P.K. [1947]: A Geometric Theory of Partial Differential Equations. ONTI: Moscow-Leningrad (Russian), Zbl. 36, 64
Rashevsky, P.K. [1954]: Linear differential geometrical objects. Dokl. Akad. Nauk SSSR 97, 606–611 (Russian), Zbl.57, 134
Rotschild, L., Stein, E.M. [1976]: Hypoelliptic differential operators and nilpotent groups. Acta Math. 137, 247–320, Zbl.346.35030
Routh, E.G. [1905]: Dynamics of a System of Rigid Bodies, Vol. 1. McMillan & Co: London, Jbuch. 36, 749
Rund, H. [1959]: Differential Geometry of Finsler Spaces. Die Grundlehren der Math. Wissenschaften 101. Springer-Verlag: New York-Berlin-Heidelberg, Zbl.87, 366
Sanchez-Calle, A. [1984]: Fundamental solutions and geometry of the sum of squares of vector fields. Invent. Math. 78, 143–160, Zbl.582.58004
Shcherbakov, R.N. [1980]: Twenty issues of Geometricheski Sbornik (a bibliography). In: Geom. Sb. 21, 79–90 (Russian)
Schoenberg, I. [1931]: Notes on geodesics in higher spaces. Bull. Am. Math. Soc. 37, No. 5, 345, Jbuch 57, 967
Schouten, J.A. [1928]: On nonholonomic connections. Proc. Amsterdam, Nederl. Akad. Wetensch., Ser. A, 31, No. 3, 291–299, Jbuch 54, 758
Schouten, J.A., van Kampen, E.R. [1930]: Zur Einbettung und Krümmungstheorie nichtholonomer Gebilde. Math. Ann. 103, 752–783, Jbuch 56, 635
Schouten, J.A., Kulk, V.D. [1949]: Pfaff’s Problem and its Generalization. Clarendon Press: Oxford, Zbl.33, 369
Sidorov, A.F., Shapeev, V.P., Yanenko, N.N. [1984]: Method of Differential Constraints and its Applications to Gas Dynamics. Nauka: Moscow (Russian), Zbl.604.76062
Sintsov, D.M. [1972]: Papers on Nonholonomic Geometry. Vischa Shkola: Kiev (Russian)
Smirnov, V.I. [1957]: Lectures on Higher Mathematics, Vol. 4 (3rd edition). GITTL: Moscow. German transl.: Berlin (1958), Zbl.44, 320
Solov’ev, A. [1982]: Second fundamental form of a distribution. Math. Zametki 31, No. 1, 139–146. English transl.: Math. Notes 31, 71–75 (1982), Zbl.492.53028
Solov’ev, A. [1984]: Curvature of a distribution. Mat. Zametki 35, 111–124. English transl.: Math. Notes 35, 61–68 (1984), Zbl.553.53017
Solov’ev, A. [1988]: On Pontryagin classes of completely geodesic subbundle. Sib. Mat. Zh. 29, No. 3, 216–219. English transl.: Sib. Math. J. 29, 510–512 (1988), Zbl.654.57012
Stein E.M. [1987]: Problems in harmonic analysis related to curvature and oscillatory integrals. Proc. Int. Congr. Math., Berkeley, USA 1986, Vol. 1, 196–221, Zbl.718.42012
Sternberg, S. [1964]: Lectures on Differential Geometry. Prentice Hall: Englewood Cliffs, Zbl. 129, 131
Strichartz, R.S. [1978]: Sub-Riemannian geometry. J. Differ. Geom. 24, 221–263, Zbl.609.53021
Suslov, D.K. [1946]: Theoretical Mechanics. ONTI: Moscow-Leningrad (Russian)
Sussmann, H.J. [1973]: Orbits of families of vector fields and integrability of distributions. Trans. Am. Math. Soc. 280, 171–188, Zbl.274.58002
Sussmann, H.J. [1983]: Lie brackets and local controllability: a sufficient condition for scalar-input systems. SIAM J. Control, Optimization 21, 686–713, Zbl.523.49026
Synge, J.L. [1927]: On geometry of dynamics. Philos. Trans. R. Soc. Lond., Ser. A, 226, No. 2, 31–107
Synge, J.L. [1928]: Geodesics in non-holonomic geometry. Math. Ann. 99, 738–751, Jbuch 54, 758
Synge, J.L. [1936]: Tensorial Methods in Dynamics. Toronto University Press: Toronto, Zbl. 17, 41
Taylor, M. [1984]: Noncommutative Microlocal Analysis. Part I. Mem. Am. Math. Soc. No. 313. Am. Math. Soc: Providence, Zbl.554.35025
Taylor, M. [1986]: Noncommutative Harmonic Analysis. Mathematical Surveys Monographs, No. 22. Am. Math. Soc: Providence, Zbl.604.43001
Taylor, T.J.S. [1989a]: Some aspects of differential geometry associated with hypoelliptic second order operators. Pac. J. Math. 136, No. 2, 355–378, Zbl.698.35041
Taylor, T.J.S. [1989b]: Off diagonal asymptotics of hypoelliptic diffusion equations and singular Riemannian geometry. Pac. J. Math. 136 (2), 379–399, Zbl.692.35011
Treves, F. [1980]: Introduction to Pseudodifferential and Fourier Integral Operators. Plenum Press: New York-London, Zbl.453.47027
Ulam, S.M. [1960]: A Collection of Mathematical Problems. Interscience Publ.: New York-London, Zbl.86, 241
Vagner, V.V. [1935]: Differential geometry of nonholonomic manifolds. Tr. Semin. Vectora. Tenzorn. Anal. 213, 269–314 (Russian), Zbl.12, 318
Vagner, V.V. [1940]: Differential geometry of nonholonomic manifolds. In: The VIII-th International Competition for the N.I. Lobatschewski Prize. Report. The Kazan Physico-Mathematical Society: Kazan
Vagner, V.V. [1941]: A geometric interpretation of nonholonomic dynamical systems. Tr. Semin. Vectora. Tenzorn. Anal. 5, 301–327
Vagner, V.V. [1965]: Geometria del calcolo delle variazioni, Vol. 2. C.I.M.E.: Roma
Varchenko, A.N. [1981]: On obstructions to local equivalence of distributions. Mat. Zametki 29, No. 6, 939–947. English transl.: Math. Notes 29, 479–484 (1981), Zbl.471.58004
Varopoulos, N.Th. [1986]: Analysis on nilpotent Lie groups. J. Funct. Anal. 66, No. 3, 406–431, Zbl.595.22008
Varopoulos, N.Th. [1986]: Analysis on Lie groups. J. Funct. Anal. 76, No. 2, 346–410, Zbl.634.22008
Varopoulos, N.Th. [1988]: Estimations du noyau de la chaleur sur une variété. C. R. Acad. Sci., Paris, Ser. I 307, 527–529, Zbl.649.58032
Varopoulos, N.T. [1988]: Estimations gaussiennes du noyau de la chaleur sur les variétés. C. R. Acad. Sci., Paris, Ser. I 307, 861–863, Zbl.671.58039
Vasil’ev, A.M., Efimov, N.N., Rashevsky, P.K. [1967]: The studies of differential geometry in Moscow University in the Soviet period. Vestn. Mosk. Univ., Ser I 22, No. 5, 12–23 (Russian), Zbl. 155, 9
Vershik, A.M. [1984]: Classical and nonclassical dynamics with constraints. In: Geometriya: Topologiya v Global’nykh Nelinejnykh Zadachakh, Nov. Global’nom Anal. 1984, 23–48. English transl.: Lect. Notes Math. 1108, 278–301 (1984), Zbl.538.58012
Vershik, A., Berestouski, V. [1992]: Manifolds with intrinsic metrics and non-holonomic spaces. Advances in Soviet Math. 9, 255–267
Vershik, A.M., Chernyakov, A.G. [1982]: Fields of convex polyhedra and the Pareto-Smale optimum. Optimizatsiya 28 (45), 112–146 (Russian), Zbl.498.90065
Vershik, A.M., Faddeev, L.D. [1972]: Differential geometry and Lagrangian mechanics with constraints. Dokl. Akad Nauk SSSR 202, No. 3, 555–557. English transl.: Sov. Phys. Dokl. 17, 34–36 (1972), Zbl.243.70014
Vershik, A.M., Faddeev, L.D. [1975]: Lagrangian mechanics in invariant form. In: Probl. Theor. Phys., Vol. II, 129–141. English transí.: Sel. Math. Sov. 1, 339–350, Zbl.518.58015
Vershik, A.M., Gershkovich, V.Ya. [1986a]: Nonholonomic problems and geometry of distributions. In: Griffiths, P.A. Exterior differential systems and the calculus of variations (Russian translation), 317–349. MIR: Moscow. English transl.: Acta Appl. Math. 12, No. 2, 181–209, 1988, Zbl.666.58004
Vershik, A.M., Gershkovich, V.Ya. [1986b]: Geodesic flows on SL2ℝ with nonholonomic constraints. Zap. Nauchn. Semin. Leningr., Otd. Mat. Inst. Steklova 155, 7–17. English transl.: J. Sov. Math. 41, No. 2, 891–898 (1988), Zbl.673.58034
Vershik, A., Gershkovich, V. [1987]: The geometry of the nonholonomic sphere of three-dimensional Lie groups. In: Geometriya i Teoriya Osobennosej v Nelinnejnykh Uravneniyakh, Nov. Global’nom Anal. 1987, 61–75. English transl.: Lect. Notes Math. 1334 (Borisovich, Yu.G., Gliklikh, Yu.E. (Eds.)), 309–331 (1988), Abl.637.53061
Vershik, A., Gershkovich, V. [1988]. Determination of the functional dimension of the orbit space of generic distributions. Mat. Zametki 44, 596–603. English transl.: Math. Notes 44, 806–810 (1988), Zbl.694.58005
Vershik, A., Gershkovich, V. [1989]: A nilpotent Lie algebra bundle over a nonholonomic manifold (Nilpotentization). Zap. Nauchn. Semin. Leningr., Otd. Mat. Inst. Steklova 172, 21–40 (Russian), Zbl.705.58004
Vershik, A., Granichina, O. [1991]: Reduction of nonholonomic variational problems to the iso-perimetric problems and connections in principal bundles. Mat. Zametki 49, 37–44
Veselov, A.P., Veselova, L.E. [1986]: Flows on Lie groups with nonholonomic constraints and integrable non-Hamiltonian systems Funkts. Anal. Prilozh. 20, No. 4, 65–66. English transl: Funct. Anal. Appl. 20, 308–309, Zbl.621.58024
Veselov, A., Veselova, L. [1988]: Integrable nonholonomic systems on Lie groups. Math. Zametki 44, No. 5, 604–619. English transl.: Math. Notes 44, No. 5, 810–819 (1988), Zbl.662.58020
Vinogradov, A.M., Krasil’shchik, I.S., Lychagin, V.V. [1981]: Introduction to the Geometry of Nonlinear Differential Equations. Nauka: Moscow. English transl.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Gordon and Breach Publ.: New York, 1986, Zbl.592.35002, Zbl.722.35001
Vosilyus, R.V. [1983]: A contravariant theory of differential prolongation in a model of a space with connection. Itogi Nauki Tekh. Ser. Probl. Geom. 14, 101–176. English transl.: J. Sov. Math. 28, 208–256 (1985), Zbl.528.58039
Vranceanu, G. [1931]: Parallèlisme et courbure dans une variété non holonome. Atti del Congresso Internaz. dei Matematici, Bologna 1928, No. 5., 63–75. Zanichelli: Bologna
Whittaker, E.T. [1937]: A Treatise on the Analytical Dynamics of Rigid Bodies. Cambridge Univ. Press: Cambridge, Zbl.61, 418
Young, L.C. [1969]: Lectures on the Calculus of Variations and Optimal Control Theory. W.B. Saunders Co: Philadelphia-London-Toronto, Zbl. 177, 378
Zhitomirsky, M. (= Zhitomirskij, M.) [1990]: Normal forms of two-dimensional distribution germs in ℝ 4. Mat. Zametki 48 (Russian)
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Arnol’d, V.I., Novikov, S.P. (1994). Nonholonomic Variational Problems on Three-Dimensional Lie Groups. In: Arnol’d, V.I., Novikov, S.P. (eds) Dynamical Systems VII. Encyclopaedia of Mathematical Sciences, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06796-3_4
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