The purpose of this paper is to show that, at least for Lagrangians of mechanical type, nonholonomic Euler–Lagrange equations for a nonholonomic linear constraint may be viewed as nonconstrained Euler–Lagrange equations but on a new (generally not Lie) algebroid structure on . The proposed novel formalism allows us to treat in a unified way a variety of situations in nonholonomic mechanics and gives rise to a version of Noether theorem producing actual first integrals in the case of symmetries.
REFERENCES
1.
2.
Bloch
, A. M.
, Krishnaprasad
, P. S.
, Marsden
, J. E.
, and Murray
, R. M.
, “Nonholonomic mechanical systems with symmetry
,” Arch. Ration. Mech. Anal.
136
, 21
(1996
).3.
Bullo
, F.
and Žefran
, M.
, “On mechanical control systems with nonholonomic constraints and symmetries
,” Syst. Control Lett.
45
, 133
(2002
).4.
Cantrijn
, F.
, de León
, M.
, and Martín de Diego
, D.
, “On almost-Poisson structures in nonholonomic mechanics
,” Nonlinearity
12
, 721
(1999
).5.
Chaplygin
, S. A.
, “On the theory of the motion of nonholonomic systems. Theorem on the reducing multiplier
,” Math. USSR Sb.
28
, 303
(1911
).6.
Cortés
, J.
, de León
, M.
, Marrero
, J. C.
, Martín de Diego
, D.
, and Martínez
, E.
, “A survey on Lagrangian Mechanics and control on Lie algebroids and groupoids
,” Int. J. Geom. Methods Mod. Phys.
3
, 509
(2006
).7.
Cortés
, J.
, de León
, M.
, Marrero
, J. C.
, and Martínez
, E.
, “Nonholonomic Lagrangian systems on Lie algebroids
,” Discrete Contin. Dyn. Syst.
(to be published).8.
Grabowska
, K.
and Grabowski
, J.
, “Variational calculus with constraints on general algebroids
,” J. Phys. A
41
, 175204
(2008
).9.
Grabowska
, K.
, Grabowski
, J.
, and Urbański
, P.
, “Geometrical mechanics on algebroids
,” Int. J. Geom. Methods Mod. Phys.
3
, 559
(2006
).10.
11.
Grabowski
, J.
and Urbański
, P.
, “Tangent lifts of Poisson and related structures
,” J. Phys. A
28
, 6743
(1995
).12.
Grabowski
, J.
and Urbański
, P.
, “Algebroids—general differential calculi on vector bundles
,” J. Geom. Phys.
31
, 111
(1999
).13.
Grabowski
, J.
and Urbański
, P.
, “Lie algebroids and Poisson-Nijenhuis structures
,” Rep. Math. Phys.
40
, 195
(1997
).14.
Ibort
, A.
, de León
, M.
, Marrero
, J. C.
, and Martín de Diego
, D.
, “Dirac brackets in constrained dynamics
,” Fortschr. Phys.
47
, 459
(1999
).15.
Konieczna
, K.
and Urbański
, P.
, “Double vector bundles and duality
,” Arch. Math.
35
, 59
(1999
).16.
Koon
, W. S.
and Marsden
, J. E.
, “Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction
,” SIAM J. Control Optim.
35
, 901
(1997
).17.
18.
de León
, M.
, Marrero
, J. C.
, and Martínez
, E.
, “Lagrangian submanifolds and dynamics on Lie algebroids
,” J. Phys. A
38
, R241
(2005
).19.
20.
Mackenzie
, K. C. H.
, General Theory of Lie Groupoids and Lie Algebroids
,London Mathematical Society Lecture Note
Series No. 213 (Cambridge University Press
, Cambridge
, 2005
).21.
Marle
, C. -M.
, “Various approaches to conservative and nonconservative nonholonomic systems
,” Rep. Math. Phys.
42
, 211
(1998
).22.
Marmo
, G.
, Mendella
, G.
, and Tulczyjew
, W. M.
, “Integrability of implicit differential equations
,” J. Phys. A
30
, 149
(1995
).23.
Marmo
, G.
, Mendella
, G.
, and Tulczyjew
, W. M.
, “Constrained Hamiltonian systems as implicit differential equations
,” J. Phys. A
30
, 277
(1997
).24.
Martínez
, E.
, “Lagrangian Mechanics on Lie Algebroids
,” Acta Appl. Math.
67
, 295
(2001
).25.
Martínez
, E.
, “Geometric formulation of Mechanics on Lie algebroids
,” Proceedings of the VIII Fall Workshop on Geometry and Physics
, Medina del Campo, 1999
, (RSME
, Madrid
, 2001
), Vol. 2
, pp. 209
–222
.26.
Martínez
, E.
, “Variational calculus on Lie algebroids, ESAIM: Control
,” COCV
14
, 356
(2008
).27.
Neimark
, J.
and Fufaev
, N.
, Dynamics of Nonholonomic Systems
, Translation of Mathematics Monographs
, Vol. 33
(AMS
, Providence, RI
, 1972
).28.
Tulczyjew
, W. M.
, “Les sous-variétés Lagrangiennes et la dynamique Hamiltonienne
,” C.R. Acad. Sci. Paris Sér. A-B
283
, 15
(1976
).29.
Tulczyjew
, W. M.
, “Les sous-variétés Lagrangiennes et la dynamique Lagrangienne
,” C.R. Acad. Sci. Paris Sér. A-B
283
, 675
(1976
).30.
Tulczyjew
, W. M.
, “A note on holonomic constraint
,” Boston Stud. Philos. Sci.
234
, 403
(2003
).31.
Tulczyjew
, W. M.
and Urbański
, P.
, “A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting (Warsaw, 1998)
,” Acta Phys. Pol. B
30
, 2909
(1999
).32.
Pradines
, J.
, “Fibrés vectoriels doubles et calcul des jets non holonomes
,” Notes Polycopiées
, Amiens, 1974
(unpublished) (in French).33.
J.
Pradines
, “Représentation des jets non holonomes par des morphismes vectoriels doubles soudés
,” C.R. Acad. Sci. Paris Sér. A
278
, 1523
(1974
).34.
van der Schaft
, A. J.
and Maschke
, B.
, “On the Hamiltonian formulation of nonholonomic machanical systems
,” Rep. Math. Phys.
34
, 225
(1994
).35.
Weinstein
, A.
, “Lagrangian Mechanics and groupoids
,” Fields Inst. Commun.
7
, 207
(1996
).36.
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2009
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