Geodesic
Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems
Lobachevskii journal of mathematics, Tome 23 (2006), pp. 95-150
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper is a survey of the theory of Lagrangian systems with non-holonomic constraints in jet bundles. The subject of the paper are systems of second-order ordinary and partial differential equations that arise as extremals of variational functionals in fibered manifolds. A geometric setting for Euler–Lagrange and Hamilton equations, based on the concept of Lepage class is presented. A constraint is modeled in the underlying fibered manifold as a fibered submanifold endowed with a distribution (the canonical distribution). A constrained system is defined by means of a Lepage class on the constraint submanifold. Constrained EulerЧ-Lagrange equations and constrained Hamilton equations, and properties of the corresponding exterior differential systems, such as regularity, canonical form, or existence of a constraint Legendre transformation, are presented. The case of mechanics (ODEТs) and field theory (PDEТs) are investigated separately, however, stress is put on a unified exposition, so that a direct comparison of results and formulas is at hand.
Keywords: jet bundles, non-holonomic constraints, semiholonomic constraints, holonomic constraints, constrained Lagrangian systems, constrained Euler–Lagrange equations, Hamilton–De Donder equations, regularity of constrained systems, Hamiltonian
Mots-clés : momenta, Legendre transformation. 
@article{LJM_2006_23_a4,
     author = {O. Krupkov\'a and P. Voln\'y},
     title = {Differential equations with constraints in jet bundles: {Lagrangian} and {Hamiltonian} systems},
     journal = {Lobachevskii journal of mathematics},
     pages = {95--150},
     year = {2006},
     volume = {23},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/LJM_2006_23_a4/}
}
TY  - JOUR
AU  - O. Krupková
AU  - P. Volný
TI  - Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems
JO  - Lobachevskii journal of mathematics
PY  - 2006
SP  - 95
EP  - 150
VL  - 23
UR  - http://geodesic.mathdoc.fr/item/LJM_2006_23_a4/
LA  - en
ID  - LJM_2006_23_a4
ER  - 
%0 Journal Article
%A O. Krupková
%A P. Volný
%T Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems
%J Lobachevskii journal of mathematics
%D 2006
%P 95-150
%V 23
%U http://geodesic.mathdoc.fr/item/LJM_2006_23_a4/
%G en
%F LJM_2006_23_a4
O. Krupková; P. Volný. Differential equations with constraints in jet bundles: Lagrangian and Hamiltonian systems. Lobachevskii journal of mathematics, Tome 23 (2006), pp. 95-150. http://geodesic.mathdoc.fr/item/LJM_2006_23_a4/

[1] I. Anderson, T. Duchamp, “On the existence of global variational principles”, Am. J. Math., 102 (1980), 781–867 | DOI | MR

[2] E. Binz, M. de León, D. M. de Diego and D. Socolescu, “Nonholonomic constraints in classical field theories”, Reports on Math. Phys., 49 (2002), 151–166 | DOI | MR | Zbl

[3] F. Cantrijn, W. Sarlet W. and D. J. Saunders, “Regularity aspects and Hamiltonization of nonholonomic systems”, J. Phys. A: Math. Gen., 32 (1999), 6869–6890 | DOI | MR | Zbl

[4] N. G. Chetaev, “On the Gauss principle”, Izv. Kazan. Fiz.-Mat. Obsc., 6 (1932–33), 323–326 (in Russian)

[5] Th. De Donder, Théorie Invariantive du Calcul des Variations, Gauthier–Villars, Paris, 1930 | Zbl

[6] G. Giachetta, “Jet methods in nonholonomic mechanics”, J. Math. Phys., 33 (1992), 1652–1665 | DOI | MR | Zbl

[7] H. Goldschmidt and S. Sternberg, “The Hamilton–Cartan formalism in the calculus of variations”, Ann. Inst. Fourier, Grenoble, 23 (1973), 203–267 | MR | Zbl

[8] H. Helmholtz, “Über die physikalische Bedeutung des Prinzips der kleinsten Wirkung”, J. für die reine u. angewandte Math., 100 (1887), 137–166

[9] W. S. Koon and J. E. Marsden, “The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems”, Rep. Math. Phys., 40 (1997), 21–62 | DOI | MR | Zbl

[10] D. Krupka, “Some geometric aspects of variational problems in fibered manifolds”, Folia Fac. Sci. Nat. UJEP Brunensis, 14 (1973), 1–65; Electronic transcription: arXiv:math-ph/0110005

[11] D. Krupka, “On the local structure of the Euler–Lagrange mapping of the calculus of variations”, Proc. Conf. on Diff. Geom. and Its Appl. (1980), ed. O. Kowalski, Universita Karlova, Prague, 1981, 181–188 | MR

[12] D. Krupka, “Lepagean forms in higher order variational theory”, Modern Developments in Analytical Mechanics I: Geometrical Dynamics, Proc. IUTAM-ISIMM Symposium (Torino, Italy 1982), eds. S. Benenti, M. Francaviglia and A. Lichnerowicz, Accad. delle Scienze di Torino, Torino, 1983, 197–238 | MR

[13] D. Krupka and O. Štěpánková, “On the Hamilton form in second order calculus of variations”, Geometry and Physics, Proc. Int. Meeting (Florence, Italy, 1982), ed. M. Modugno, Pitagora Ed., Bologna, 1983, 85–101 | MR

[14] O. Krupková, “Lepagean 2-forms in higher order Hamiltonian mechanics, I. Regularity”, Arch. Math. (Brno), 22 (1986), 97–120 ; “Lepagean 2-forms in higher order Hamiltonian mechanics, II. Inverse problem”, Arch. Math. (Brno), 23 (1987), 155–170 | MR | Zbl | MR | Zbl

[15] O. Krupková, “Mechanical systems with nonholonomic constraints”, J. Math. Phys., 38 (1997), 5098–5126 | DOI | MR | Zbl

[16] O. Krupková, The Geometry of Ordinary Variational Equations, Lecture Notes in Mathematics, 1678, Springer, Berlin, 1997 | MR | Zbl

[17] O. Krupková, “On the geometry of non-holonomic mechanical systems”, Differential Geometry and Applications, Proc. Conf. (Brno, 1998), eds. O. Kowalski, I. Kolář, D. Krupka and J. Slovák, Masaryk University, Brno, Czech Republic, 1999, 533–546 | MR | Zbl

[18] O. Krupková, “Higher-order mechanical systems with constraints”, J. Math. Phys., 41 (2000), 5304–5324 | DOI | MR | Zbl

[19] O. Krupková, “Hamiltonian field theory”, J. Geom. Phys., 43 (2002), 93–132 | DOI | MR | Zbl

[20] O. Krupková, “Recent results in the geometry of constrained systems”, Reports on Math. Phys., 49 (2002), 269–278 | DOI | MR | Zbl

[21] O. Krupková, “The geometry of variational equations”, Global Analysis and Applied Mathematics, AIP Conference Proceedings 729, American Institute of Physics, 2004, 19–38 | MR | Zbl

[22] O. Krupková, “Partial differential equations with differential constraints”, J. Differential Equations, 220 (2006), 354–395 | DOI | MR | Zbl

[23] O. Krupková and D. Smetanová, “Legendre transformation for regularizable Lagrangians in field theory”, Letters in Math. Phys., 58 (2001), 189–204 | DOI | MR | Zbl

[24] O. Krupková, D. Smetanová, “On regularization of variational problems in first-order field theory”, The proceedings of the 20th winter school “Geometry and Physics” (Srní, Czech Republic, January 15–22, 2000), Suppl. Rend. Circ. Mat. Palermo, II. Ser., 66, eds. J. Slovák et al., Circolo Matematico di Palermo, Palermo, 2001, 133–140 | MR | Zbl

[25] O. Krupková, P. Volný, “Euler–Lagrange and Hamilton equations for non-holonomic systems in field theory”, J. Phys. A: Math. Gen., 38, 2005, 8715–8745 | MR | Zbl

[26] M. de León and D. M. de Diego, “On the geometry of non-holonomic Lagrangian systems”, J. Math. Phys., 37 (1996), 3389–3414 | DOI | MR | Zbl

[27] M. de León, J. C. Marrero and D. M. de Diego, “Non-holonomic Lagrangian systems in jet manifolds”, J. Phys. A: Math. Gen., 30 (1997), 1167–1190 | DOI | MR | Zbl

[28] E. Massa and E. Pagani, “A new look at classical mechanics of constrained systems”, Ann. Inst. Henri Poincaré, 66 (1997), 1–36 | MR | Zbl

[29] Ju. I. Neimark, N. A. Fufaev, Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, 33, American Mathematical Society, Rhode Island, 1972 | Zbl

[30] W. Sarlet, “A direct geometrical construction of the dynamics of non-holonomic Lagrangian systems”, Extracta Mathematicae, 11 (1996), 202–212 | MR

[31] W. Sarlet, F. Cantrijn and D. J. Saunders, “A geometrical framework for the study of non-holonomic Lagrangian systems”, J. Phys. A: Math. Gen., 28 (1995), 3253–3268 | DOI | MR | Zbl

[32] D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc. Lecture Notes Series, 142, Cambridge Univ. Press, 1989 | MR | Zbl

[33] D. J. Saunders, W. Sarlet and F. Cantrijn, “A geometrical framework for the study of non-holonomic Lagrangian systems: II”, J. Phys. A: Math. Gen., 29 (1996), 4265–4274 | DOI | MR | Zbl

[34] M. Swaczyna, “On the nonholonomic variational principle”, Global Analysis and Applied Mathematics, Proc. Internat. Workshop (Ankara, Turkey, April, 2004), AIP Conference Proceedings, 729, eds. K. Tas, D. Krupka, O. Krupkova and D. Baleanu, American Institute of Physics, New York, 2004, 297–306 | MR | Zbl

[35] F. Takens, “A global version of the inverse problem of the calculus of variations”, J. Diff. Geom., 14 (1979), 543–562 | MR | Zbl

[36] E. Tonti, “Variational formulation of nonlinear differential equations, I”, Bull. Acad. Roy. Belg. Cl. Sci., 55 (1969), 137–165 ; “II”, 262–278 | MR | Zbl | Zbl

[37] J. Vankerschaver, F. Cantrijn, M. de León, D. Martín de Diego, “Geometric aspects of nonholonomic field theories”, Rep. Math. Phys., 56:3 (2005), 387–411 | DOI | MR | Zbl

[38] P. Volný and O. Krupková, “Hamilton equations for non-holonomic mechanical systems”, Differential Geometry and Its Applications, Proc. Conf. (Opava, 2001), Mathematical Publications, 3, eds. O. Kowalski, D. Krupka and J. Slovák, Silesian University, Opava, Czech Republic, 2001, 369–380 | MR | Zbl