dF ¼
@F
@u
du þ
@F
@u 0 du
0
þ
@F
@u 00 du
00
þ Á Á Á þ
@F
@u ðnÞ du
ðnÞ
:
ð1:76Þ
Also, variation and differentiation are interchangeable
2 : d du=dx
ð
Þdu
0
¼ dðduÞ=dx, as are variation and integration: d
R
Fdx ¼
R
dFdx. Variations of
independent variables are not allowed, that is, for the functional F(x, u(x), u′(x), ÁÁÁ)
one has dx 0 by definition. Stationarity of the functional F requires dF = 0 for
arbitrary admissible variations du.
Expressions for strain energy are required when using Hamilton’s principle for
elastic structures. For one-dimensional bending of beams, the strain energy per unit
beam length is 1/2M
2
/EI, where M is the internal bending moment and EI the
bending stiffness. For one-dimensional extension of rods the similar expression is
1/2N
2 /EA, where N is the internal extensional force and EA the longitudinal stiffness
(e.g., El Naschie 1990a).
With non-conservative systems, when calculating L one can subtract from V the
work done by non-potential forces, just as with Lagrange’s equations (Sect. 1.5.2).
Example Using Hamilton’s principle to derive the equation of motion governing
transverse vibrations u(x, t) of the beam in Fig. 1.6. The beam is simply supported
at x = 0 and x = l, has length l and bending stiffness EI, distributed mass qA per
unit length, a concentrated mass m at x = x 0 , and is subjected to a point load P(t) at
x = x 0 . The kinetic and potential energies, Lagrangian and action integral are,
respectively:
T ¼
Z l
0
1
2
qA _
u
2 dx þ
1
2
m _
uðx 0 ; tÞ
ð
Þ
2 ¼
Z l
0
1
2
qA þ
1
2
m ~ dðx À x 0 Þ
_
u
2 dx;
V ¼
Z l
0
1
2
M
2
EI
dx À ÀPuðx 0 ; tÞ
ð
Þ¼
Z l
0
1
2
EIðu
00
Þ
2 þ PðtÞ ~ dðx À x 0 Þu
dx;
L ¼ T À V ¼
Z l
0
hðx; t; u; _
u; u
00
Þdx; H ¼
Z t 2
t 1
Ldt ;
ð1:77Þ
Fig. 1.6 Example system for applying Hamilton’s principle
2
Here the assumption on holonomic constraints is important. With non-holonomic systems
Hamilton’s principle takes the form
R t2
t1 dLdt ¼ 0 (Greenwood 2003), which is not a statement of
stationarity, and generally not the same as (1.75).
1.5 Energy Methods for Setting up Equations of Motion
23
@F
@u
du þ
@F
@u 0 du
0
þ
@F
@u 00 du
00
þ Á Á Á þ
@F
@u ðnÞ du
ðnÞ
:
ð1:76Þ
Also, variation and differentiation are interchangeable
2 : d du=dx
ð
Þdu
0
¼ dðduÞ=dx, as are variation and integration: d
R
Fdx ¼
R
dFdx. Variations of
independent variables are not allowed, that is, for the functional F(x, u(x), u′(x), ÁÁÁ)
one has dx 0 by definition. Stationarity of the functional F requires dF = 0 for
arbitrary admissible variations du.
Expressions for strain energy are required when using Hamilton’s principle for
elastic structures. For one-dimensional bending of beams, the strain energy per unit
beam length is 1/2M
2
/EI, where M is the internal bending moment and EI the
bending stiffness. For one-dimensional extension of rods the similar expression is
1/2N
2 /EA, where N is the internal extensional force and EA the longitudinal stiffness
(e.g., El Naschie 1990a).
With non-conservative systems, when calculating L one can subtract from V the
work done by non-potential forces, just as with Lagrange’s equations (Sect. 1.5.2).
Example Using Hamilton’s principle to derive the equation of motion governing
transverse vibrations u(x, t) of the beam in Fig. 1.6. The beam is simply supported
at x = 0 and x = l, has length l and bending stiffness EI, distributed mass qA per
unit length, a concentrated mass m at x = x 0 , and is subjected to a point load P(t) at
x = x 0 . The kinetic and potential energies, Lagrangian and action integral are,
respectively:
T ¼
Z l
0
1
2
qA _
u
2 dx þ
1
2
m _
uðx 0 ; tÞ
ð
Þ
2 ¼
Z l
0
1
2
qA þ
1
2
m ~ dðx À x 0 Þ
_
u
2 dx;
V ¼
Z l
0
1
2
M
2
EI
dx À ÀPuðx 0 ; tÞ
ð
Þ¼
Z l
0
1
2
EIðu
00
Þ
2 þ PðtÞ ~ dðx À x 0 Þu
dx;
L ¼ T À V ¼
Z l
0
hðx; t; u; _
u; u
00
Þdx; H ¼
Z t 2
t 1
Ldt ;
ð1:77Þ
Fig. 1.6 Example system for applying Hamilton’s principle
2
Here the assumption on holonomic constraints is important. With non-holonomic systems
Hamilton’s principle takes the form
R t2
t1 dLdt ¼ 0 (Greenwood 2003), which is not a statement of
stationarity, and generally not the same as (1.75).
1.5 Energy Methods for Setting up Equations of Motion
23