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 D may be viewed as nonconstrained Euler–Lagrange equations but on a new (generally not Lie) algebroid structure on D. 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.

1.
Arnold
,
V. I.
,
Dynamical Systems
(
Springer-Verlag
,
New York
,
1998
), Vol.
III
.
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.
Grabowski
,
J.
and
Rotkiewicz
,
M.
, e-print arXiv:math.DG/0702772.
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.
de León
,
M.
,
Marrero
,
J. C.
, and
Martín de Diego
,
D.
, e-print arXiv:math-ph/0801.4358v1.
18.
de León
,
M.
,
Marrero
,
J. C.
, and
Martínez
,
E.
, “
Lagrangian submanifolds and dynamics on Lie algebroids
,”
J. Phys. A
38
,
R241
(
2005
).
19.
Libermann
,
P.
, “
Lie algebroids and mechanics
,”
Arch. Math.
32
,
147
(
1996
).
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.
Yano
,
K.
and
Ishihara
,
S.
,
Tangent and Cotangent Bundles
(
Dekker
,
New York
,
1973
).
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