In this expository article, a simple concise treatment of Lagrange's prescription for constraint forces and constraint moments in the dynamics of rigid bodies is presented. The treatment is suited to both Newton–Euler and Lagrangian treatments of rigid body dynamics and is illuminated with a range of examples from classical mechanics and orthopedic biomechanics.

References

1.
Lagrange
,
J. L.
,
1997
,
Analytical Mechanics (Boston Studies in the Philosophy of Science)
, Vol.
191
,
Kluwer Academic Publishers Group
,
Dordrecht
[Translated from the 1811 French original, with an introduction and edited by
A.
,
Boissonnade
, and
V. N.
,
Vagliente
, and with a preface by
C. G.
,
Fraser
].
2.
Baruh
,
H.
,
1999
,
Analytical Dynamics
,
McGraw-Hill
,
Boston
.
3.
Papastavridis
,
J. G.
,
2002
,
Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; for Engineers, Physicists, and Mathematicians
,
Oxford University Press
,
Oxford
.
4.
Synge
,
J. L.
, and
Griffith
,
B. A.
,
1959
,
Principles of Mechanics
, 3rd ed.,
McGraw-Hill
,
New York
.
5.
Kane
,
T. R.
,
Likins
,
P. W.
, and
Levinson
,
D. A.
,
1983
,
Spacecraft Dynamics
,
McGraw-Hill
,
New York
.
6.
Papastavridis
,
J. G.
,
1998
, “
Panoramic Overview of the Principles and Equations of Motion of Advanced Engineering Dynamics
,”
ASME Appl. Mech. Rev.
,
51
(
4
), pp.
239
265
.10.1115/1.3099003
7.
Udwadia
,
F. E.
, and
Kalaba
,
R. E.
,
1996
,
Analytical Dynamics
,
Cambridge University Press
,
Cambridge
.
8.
Gantmacher
,
F.
,
1970
,
Lectures in Analytical Mechanics
,
McGraw-Hill
,
Moscow
[Translated from the Russian by G. Yankovsky].
9.
Planck
,
M.
,
1949
,
Introduction to Theoretical Physics
(General Mechanics), Vol.
1
,
The McMillan Company
,
New York
[Translated from the German by Henry L. Brose].
10.
Huston
,
R. L.
,
1990
,
Multibody Dynamics
,
Butterworth-Heinemann
,
Boston
.
11.
O'Reilly
,
O. M.
,
2008
,
Intermediate Engineering Dynamics: A Unified Approach to Newton-Euler and Lagrangian Mechanics
,
Cambridge University Press
,
New York
.
12.
Shabana
,
A. A.
,
2001
,
Computational Dynamics
, 2nd ed.,
Wiley-Interscience
,
New York
.
13.
Casey
,
J.
, and
O'Reilly
,
O. M.
,
2006
, “
Geometrical Derivation of Lagrange's Equations for a System of Rigid Bodies
,”
Math. Mech. Solids
,
11
(
4
), pp.
401
422
.10.1177/1081286505044137
14.
O'Reilly
,
O. M.
,
2007
, “
The Dual Euler Basis: Constraints, Potentials, and Lagrange's Equations in Rigid Body Dynamics
,”
ASME J. Appl. Mech.
,
74
(
2
), pp.
256
258
.10.1115/1.2190231
15.
O'Reilly
,
O. M.
, and
Srinivasa
,
A. R.
,
2002
, “
On Potential Energies and Constraints in the Dynamics of Rigid Bodies and Particles
,”
Math. Prob. Eng.
,
8
(
3
), pp.
169
180
.10.1080/10241230215286
16.
Shuster
,
M. D.
,
1993
, “
A Survey of Attitude Representations
,”
J. Astronaut. Sci.
,
41
(
4
), pp.
439
517
.
17.
Desroches
,
G.
,
Chèze
,
L.
, and
Dumas
,
R.
,
2010
, “
Expression of Joint Moment in the Joint Coordinate System
,”
ASME J. Biomech. Eng.
,
132
(
11
), p.
114503
.10.1115/1.4002537
18.
O'Reilly
,
O. M.
,
Sena
,
M.
,
Feeley
,
B. T.
, and
Lotz
,
J. C.
,
2013
, “
On Representations for Joint Moments Using a Joint Coordinate System
,”
ASME J. Biomech. Eng.
,
135
(
11
), p.
114504
.10.1115/1.4025327
19.
O'Reilly
,
O. M.
, and
Srinivasa
,
A. R.
,
2001
, “
On a Decomposition of Generalized Constraint Forces
,”
Proc. R. Soc., London, Ser. A
,
457
(
2010
), pp.
1307
1313
and 3052.10.1098/rspa.2000.0717
20.
O'Reilly
,
O. M.
,
2010
,
Engineering Dynamics: A Primer
, 2nd ed.,
Springer-Verlag
,
New York
.
21.
Kessler
,
P.
, and
O'Reilly
,
O. M.
,
2002
, “
The Ringing of Euler's Disk
,”
Regul. Chaotic Dyn.
,
7
(
1
), pp.
49
60
.10.1070/RD2002v007n01ABEH000195
22.
Le Saux
,
C.
,
Leine
,
R. I.
, and
Glocker
,
C.
,
2005
, “
Dynamics of a Rolling Disk in the Presence of Dry Friction
,”
J. Nonlinear Sci.
,
15
(
1
), pp.
27
61
.10.1007/s00332-004-0655-4
23.
Leine
,
R.
,
2009
, “
Experimental and Theoretical Investigation of the Energy Dissipation of a Rolling Disk During Its Final Stage of Motion
,”
Arch. Appl. Mech.
,
79
(
11
), pp.
1063
1082
.10.1007/s00419-008-0278-6
24.
Casey
,
J.
,
1995
, “
On the Advantages of a Geometrical Viewpoint in the Derivation of Lagrange's Equations for a Rigid Continuum
,”
Z. Angew. Math. Phys.
,
46
(
Special Issue 1995
), pp.
S805
S847
.10.1007/BF00917882
25.
Fox
,
E.
,
1967
,
Mechanics
,
Harper & Row
,
New York, Evanston, London
.
26.
Dankowicz
,
H.
,
1998
,
Mechanics Problems, and Their Solutions
,
Royal Institute of Technology (KTH)
,
Stockholm, Sweden
.
27.
Fried
,
E.
,
2010
, “
New Insights Into the Classical Mechanics of Particle Systems
,”
Discrete Continuous Dyn. Syst.
,
28
(
4
), pp.
1469
1504
.10.3934/dcds.2010.28.1469
28.
Moon
,
F. C.
,
2008
,
Applied Dynamics With Applications to Multibody and Mechatronic Systems
, 2nd ed.,
Wiley-VCH
,
Weinheim, Germany
.
29.
Noll
,
W.
,
2004
, “
On Material-Frame Indifference
,”
Five Contributions to Natural Philosophy
,
Department of Mathematics, Carnegie-Mellon University
,
Pittsburgh, PA
, Chap. 2, pp.
13
22
.
30.
Grood
,
E. S.
, and
Suntay
,
W. J.
,
1983
, “
A Joint Coordinate System for the Clinical Description of Three-Dimensional Motions: Application to the Knee
,”
ASME J. Biomech. Eng.
,
105
(
2
), pp.
136
144
.10.1115/1.3138397
31.
Gauss
,
C. F.
,
2009
, “
Üher ein neues allgemeines Grundgesetz der Mechanik
,”
Crelle J. Reine Angew. Math.
,
1829
(
4
), pp.
232
235
.10.1515/crll.1829.4.232
32.
Cardin
,
F.
, and
Favretti
,
M.
,
1996
, “
On Nonholonomic and Vakonomic Dynamics of Mechanical Systems With Nonintegrable Constraints
,”
J. Geom. Phys.
,
18
(
4
), pp.
295
325
.10.1016/0393-0440(95)00016-X
33.
Lewis
,
A. D.
, and
Murray
,
R. M.
,
1995
, “
Variational Principles for Constrained Systems: Theory and Experiment
,”
Int. J. Non-Linear Mech.
,
30
(
6
), pp.
793
815
.10.1016/0020-7462(95)00024-0
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