The standard form of Hamilton's principle is only applicable to material control volumes. There exist specialized formulations of Hamilton's principle that are tailored to nonmaterial (open) control volumes. In case of continuous mechanical systems, these formulations contain extra terms for the virtual shift of kinetic energy and the net transport of a product of the virtual displacement and the momentum across the system boundaries. This raises the theoretically and practically relevant question whether there is also a virtual shift of potential energy across the boundary of open systems. To answer this question from a theoretical perspective, we derive various formulations of Hamilton's principle applicable to material and nonmaterial control volumes. We explore the roots and consequences of (virtual) transport terms if nonmaterial control volumes are considered and show that these transport terms can be derived by Reynolds transport theorem. The equations are deduced for both the Lagrangian and the Eulerian description of the particle motion. This reveals that the (virtual) transport terms have a different form depending on the respective description of the particle motion. To demonstrate the practical relevance of these results, we solve an example problem where the obtained formulations of Hamilton's principle are used to deduce the equations of motion of an axially moving elastic tension bar.

References

References
1.
Trusdell
,
C.
, and
Toupin
,
A.
,
1960
, “
The Classical Field Theories
,”
Handbuch Der Physik
(Prinzipien der klassischen Mechanik und Feldtheorie, Vol. III/1),
Springer
,
Berlin
, pp.
226
793
.
2.
Holzapfel
,
G.
,
2000
,
Nonlinear Solid Mechanics: A Continuum Approach for Engineering
,
Wiley
,
New York
.
3.
Kundu
,
P.
,
Cohen
,
I.
, and
Dowling
,
D.
,
2012
,
Fluid Mechanics
,
5th ed.
,
Academic Press
,
Amsterdam, The Netherlands
.
4.
Irschik
,
H.
, and
Holl
,
H.
,
2004
, “
Mechanics of Variable-Mass Systems—Part 1: Balance of Mass and Linear Momentum
,”
ASME Appl. Mech. Rev.
,
57
(
2
), pp.
145
160
.
5.
Irschik
,
H
., and
Belyaev
,
A
., eds.,
2014
,
Dynamics of Mechanical Systems With Variable Mass
(CISM International Centre for Mechanical Sciences, Vol.
557
),
Springer
,
Wien, Austria
.
6.
Dost
,
S.
, and
Tabarrok
,
B.
,
1979
, “
Application of Hamilton's Principle to Large Deformation and Flow Problems
,”
ASME J. Appl. Mech.
,
46
(
2
), pp.
285
290
.
7.
Penfield
,
P.
,
1966
, “
Hamilton's Principle for Fluids
,”
Phys. Fluids
,
9
(
6
), pp.
1184
1194
.
8.
Bampi
,
F.
, and
Morro
,
A.
,
1984
, “
The Connection Between Variational Principles in Eulerian and Lagrangian Descriptions
,”
J. Math. Phys.
,
25
(
8
), pp.
2418
2421
.
9.
Benjamin
,
T.
,
1961
, “
Dynamics of a System of Articulated Pipes Conveying Fluid—I: Theory
,”
Proc. R. Soc. London A
,
261
(
1307
), pp.
457
486
.
10.
McIver
,
D.
,
1973
, “
Hamilton's Principle for Systems of Changing Mass
,”
J. Eng. Math.
,
7
(
3
), pp.
249
261
.
11.
Irschik
,
H.
, and
Holl
,
H.
,
2002
, “
The Equations of Lagrange Written for a Non-Material Volume
,”
Acta Mech.
,
153
(
3–4
), pp.
231
248
.
12.
Irschik
,
H.
, and
Holl
,
H.
,
2015
, “
Lagrange's Equations for Open Systems, Derived Via the Method of Fictitious Particles, and Written in the Lagrange Description of Continuum Mechanics
,”
Acta Mech.
,
226
(
1
), pp.
63
79
.
13.
Casetta
,
L.
, and
Pesce
,
C.
,
2013
, “
The Generalized Hamilton's Principle for a Non-Material Volume
,”
Acta Mech.
,
224
(
4
), pp.
919
924
.
14.
Kheiri
,
M.
, and
Païdoussis
,
M.
,
2014
, “
On the Use of Generalized Hamilton's Principle for the Derivation of the Equation of Motion of a Pipe Conveying Fluid
,”
J. Fluids Struct.
,
50
, pp.
18
24
.
15.
Batra
,
G.
,
1987
, “
On Hamilton's Principle for Thermo-Elastic Fluids and Solids, and Internal Constraints in Thermo-Elasticity
,”
Arch. Ration. Mech. Anal.
,
99
(
1
), pp.
37
59
.
16.
Meirovitch
,
L.
,
1970
,
Methods of Analytical Dynamics. Advanced Engineering Series
,
McGraw-Hill
,
New York
.
17.
Atanacković
,
T.
,
Konjik
,
S.
,
Oparnica
,
L.
, and
Pilipović
,
S.
,
2010
, “
Generalized Hamilton's Principle With Fractional Derivatives
,”
J. Phys. A
,
43
(
25
), p.
255203
.
18.
Haupt
,
P.
,
2002
,
Continuum Mechanics and Theory of Materials
,
2nd ed.
,
Springer
,
Berlin
.
19.
Basar
,
Y.
, and
Weichert
,
D.
,
2000
,
Nonlinear Continuum Mechanics of Solids
,
Springer
,
Berlin
.
20.
Wu
,
H.-C.
,
2005
,
Continuum Mechanics and Plasticity
(Modern Mechanics and Mathematics),
Chapman & Hall/CRC
,
Boca Raton, FL
.
21.
Benaroya
,
H.
, and
Wei
,
T.
,
2000
, “
Hamilton's Principle for External Viscous Fluid-Structure Interaction
,”
J. Sound Vib.
,
238
(
1
), pp.
113
145
.
22.
Rosof
,
B.
,
1971
, “
Hamilton's Principle and Nonequilibrium Thermodynamics
,”
Phys. Rev. A
,
4
(
3
), pp.
1268
1274
.
You do not currently have access to this content.