Problems involving rolling without slipping or no sideways skidding, to name a few, introduce velocity-dependent constraints that can be efficiently treated by the method of Lagrange multipliers in the Lagrangian formulation of the classical equations of motion. In doing so, one finds, as a bonus, the constraint forces, which must be independent of the solution of the equations of motion and can only depend on the generalized coordinates and velocities, as well as time. In this paper, we establish conditions the Lagrangian and the constraints should obey in order to guarantee that the constraint forces can be obtained consistently.

1.
N. A.
Lemos
,
Analytical Mechanics
(
Cambridge U. P.
,
Cambridge
,
2018
).
2.
L. Q.
English
and
A.
Mareno
, “
Trajectories of rolling marbles on various funnels
,”
Am. J. Phys.
80
,
996
1000
(
2012
).
3.
G. D.
White
, “
On trajectories of rolling marbles in cones and other funnels
,”
Am. J. Phys.
81
,
890
898
(
2013
).
4.
Ju. I.
Neĭmark
and
N. A.
Fufaev
,
Dynamics of Nonholonomic Systems
(
American Mathematical Society
,
Providence, RI
,
1972
).
5.
I. R.
Gatland
, “
Nonholonomic constraints: A test case
,”
Am. J. Phys.
72
,
941
942
(
2004
).
6.
J.
Janová
and
J.
Musilová
, “
The streetboard rider: An appealing problem in non-holonomic mechanics
,”
Eur. J. Phys.
31
,
333
345
(
2010
).
7.
Thus we are excluding self-interaction forces that arise in electromagnetism, such as the Abraham-Lorentz force. See, for example,
A.
Zangwill
,
Modern Electrodynamics
(
Cambridge U. P
.,
Cambridge
,
2013
).
8.
H.
Rund
,
The Hamilton-Jacobi Theory in the Calculus of Variations
(
Van Nostrand
,
London
,
1966
).
9.
E. J.
Saletan
and
A. H.
Cromer
, “
A variational principle for nonholonomic systems
,”
Am. J. Phys.
38
,
892
897
(
1970
).
10.
M. R.
Flannery
, “
The enigma of nonholonomic constraints
,”
Am. J. Phys.
73
,
265
272
(
2005
).
11.
M. R.
Flannery
, “
d'Alembert-Lagrange analytical dynamics for nonholonomic systems
,”
J. Math. Phys.
52
,
032705
(
2011
).
12.
M. R.
Flannery
, “
The elusive d'Alembert-Lagrange dynamics of nonholonomic systems
,”
Am. J. Phys.
79
,
932
944
(
2011
).
13.
G. A.
Bliss
, “
The problem of Lagrange in the calculus of variations
,”
Am. J. Math.
52
,
673
744
(
1930
).
14.
J. R.
Ray
, “
Nonholonomic constraints
,”
Am. J. Phys.
34
,
406
408
(
1966
);
J. R.
Ray
,
Erratum
34
,
1202
1203
(
1966
).
15.
G. H.
Goedecke
, “
Undetermined multiplier treatments of the Lagrange problem
,”
Am. J. Phys.
34
,
571
574
(
1966
).
16.
V. V.
Kozlov
, “
Realization of nonintegrable constraints in classical mechanics
,”
Sov. Phys. Dokl.
28
,
735
737
(
1983
); available at http://www.mathnet.ru/links/fdf79b1a327182e29a80861324bb2629/dan46287.pdf.
17.
A. V.
Borisov
,
I. S.
Mamaev
, and
I. A.
Bizyaev
, “
Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics
,”
Russ. Math. Surv.
72
,
783
840
(
2017
).
18.
J. E.
Marsden
and
T. S.
Ratiu
,
Introduction to Mechanics and Symmetry
, 2nd ed. (
Springer
,
New York
,
1999
).
19.
E. T.
Whittaker
,
A History of the Theories of Aether and Electricity, Vol. I: The Classical Theories
(
Thomas Nelson and Sons
,
London
,
1951
), Chap. VII.
20.
E.
Wigner
, “
The unreasonable effectiveness of mathematics in the natural sciences
,”
Commun. Pure Appl. Math.
13
,
1
14
(
1960
).
21.
P. A. M.
Dirac
, “
The relation between mathematics and physics
,”
Proc. R. Soc. (Edinburgh)
59
,
122
129
(
1938
–39).
22.
V. I.
Arnold
, “
On teaching mathematics
,” address at the Palais de Découverte in Paris, March 7, 1997.
Russ. Math. Surv.
53
,
229
236
(
1998
); see <https://www.uni-muenster.de/Physik.TP/%7Emunsteg/arnold.html>.
23.
O. E.
Fernandez
and
A. M.
Bloch
, “
Equivalence of the dynamics of nonholonomic and variational nonholonomic systems for certain initial data
,”
J. Phys. A: Math. Theor.
41
,
344005
(
2008
).
24.
A. D.
Lewis
and
R. M.
Murray
, “
Variational principles for constrained systems: Theory and experiment
,”
Int. J. Non-Linear Mech.
30
,
793
815
(
1995
).
25.
T.
Kai
, “
Experimental comparison between nonholonomic and vakonomic mechanics in nonlinear constrained systems
,”
Nonlinear Theor. Appl.
4
,
482
499
(
2013
).
26.
H.
Goldstein
,
C. P.
Poole
, Jr.
, and
J. L.
Safko
,
Classical Mechanics
, 3rd ed. (
Addison Wesley
,
San Francisco
,
2001
).
27.
See <http://astro.physics.sc.edu/Goldstein/1-2-3To6.html> for errata, corrections, and comments on the third edition, co-author J. L. Safko states that Section 2.4 should be revised and cites, among others, Ref. 9 above.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.