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Файл:Additional Chapters of Higher Mathematics for Masters in Civill and Geotechnical Engineering. Учебное пособие по дополнительным разделам высшей матема
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Chapter 4. Ordinary differential equations
X
G
H
H
x
G
G
171
__________________________________________________________________________
4.2.2. Hamiltonian formalism for a second-order linear
ODE
Herein, we consider a second-order differential equation with constant coefficients,
apply the outlined procedure of reducing it to a system of two first-order equations, and
then construct the Hamiltonian and Rayleigh dissipative function.
Basic concepts 4.2.1 (Hamiltonian formalism)
Let the second-order differential equation be
with
Eq. (4.2.7) can be reduced to the following system of the first-order ODE’s:
The obtained system can be rewritten in a more concise form:
where
Now we relate the obtained system (4.2.9) with Hamiltonian formalism.
Definition 4.2.3 (Hamiltonian)
In the considered case of differential equations with constant coefficients, the Hamiltonian can
be defined as a differentiable scalar function
. Introducing a new variable
0a ≠
⎛⎞ ⎛ ⎞⎛⎞ ⎛ ⎞
⎜⎟ ⎜ ⎟⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠⎝⎠ ⎝ ⎠
GG
⎛⎞ ⎛ ⎞ ⎛ ⎞
XP
== =
⎜⎟ ⎜ ⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠ ⎝ ⎠
(4.2.8)
vx=
xx
vcabavpta
x
;;
vcabapta
01 0
=⋅+
// ()/
−−
GG
XP=⋅+G
G
01 0
// ()/
−−
(4.2.7)
()ax bx cx p t++=
. (4.2.9)
, (4.2.10)
. (4.2.11)
, such that
(,)
xv
∂∂
xv
==−
or, by introducing a 2-dimensional vector
Eqs. (4.2.12) can be rewritten in a form
where
;
vx
∂∂
⎛⎞
X
=
⎜⎟
⎝⎠
d
XH
=⋅∇J
dt
H
. (4.2.12)
(4.2.13)
v
, (4.2.14)
Chapter 4. Ordinary differential equations
H
H
H
R
R
x
X
G
G
H
R
H
172
__________________________________________________________________________
01
⎛⎞
=
J
⎜⎟
10
−
⎝⎠
(4.2.15)
is sometimes called as the Poisson matrix, and gradient
Definition 4.2.4 (Rayleigh dissipation function)
In the considered case, the Rayleigh dissipation function (or Rayleigh viscous dissipation
function to emphasize that it is associated with viscous friction) can be defined as a
differentiable scalar function
Definition 4.2.5 (Hamiltonian dynamical system)
The Hamiltonian dynamical system (or Hamiltonian dynamical system with dissipation to
emphasize presence of the dissipation term) is a system, whose governing equation of motion
can be written in a form
∇=
(,)
xv
∂∂
∂∂
d
⎛⎞
H
⎜⎟
⎝⎠
, such that
0;
==−
=⋅∇ −∇+J
dt
/
x
∂∂
∂∂
v
. (4.2.16)
/
v
R
. (4.2.17)
v
G
HRP
, (4.2.18)
∇
is
where XG is defined by (4.2.13) and P
Proposition 4.2.2 (constructing Hamiltonian dynamical system)
Let functions
and the loading vector
functions (4.2.19) define a Hamiltonian dynamical system with viscous dissipation.
Proof
The proof of coincidence of Eqs. (4.2.9) and (4.2.18) at functions (4.2.19) is straightforward.
Example 4.2.1
Let the second-order differential equation be
(,)
xv
and
(,)
xv
(,) /
xv c a x v
(,) /
Rxv b a v
G
be defined by (4.2.11)3, then Eqs. (4.2.9) and (4.2.18) coincide, i.e.
P
is an arbitrary loading vector.
be
11
=+
()
22
1
=
()
2
22
2
(4.2.19)
Chapter 4. Ordinary differential equations
x
H
x
173
__________________________________________________________________________
The outlined procedure of reducing this equation to a first order-system gives:
xx
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
vv
⎝⎠ ⎝⎠
Performing multiplication in the right-hand side of (4.2.21), yields
235cos()
xx t++= ω
01
⎛⎞
=⋅+
⎜⎟
53
−−
22
⎝⎠
. (4.2.20)
0
⎛⎞
⎜⎟
cos( )
⎜⎟
⎝⎠
. (4.2.21)
t
ω
2
⎛⎞
x
⎛⎞
=+
⎜⎟
⎜⎟
⎜⎟
−−
v
⎝⎠
⎝⎠
v
53
xv
() ()
22
0
⎛⎞
⎜⎟
cos( )
⎜⎟
⎝⎠
. (4.2.22)
t
ω
2
According to (4.2.19) Hamiltonian and the Rayleigh dissipation functions for the considered
equation are:
15 1
(,)
xv x v
=⋅ +
(,)
Rxv v
13
=⋅
22
22 2
2
. (4.2.23)
22
Direct verification shows
H
∂
⎛⎞
H
∇= =
⎜⎟
⎝⎠
⎛⎞
R
∇= =
⎜⎟
⎝⎠
x
∂
H
∂
v
∂
R
∂
x
∂
R
∂
v
∂
5
⎛⎞
2
⎜⎟
1
⎝⎠
0
⎛⎞
⎜⎟
3
2
⎝⎠
. (4.2.24)
In terms of gradients (4.2.24), the Hamiltonian dynamical system (4.2.18) takes the form
5
G
01
d
dt
⎛⎞
X
=⋅−+ =
⎜⎟
−
10
⎝⎠
⎛⎞⎛⎞⎛ ⎞
x
()
2
⎜⎟⎜⎟⎜ ⎟
v
⎛⎞
⎜⎟
⎜⎟
−−
⎝⎠
v
55
xv
() ()
22
00
3
()
2
0
⎛⎞
+
⎜⎟
cos( )
ω
⎜⎟
⎝⎠
cos( )
⎜⎟⎜⎟⎜⎟
v
⎝⎠⎝⎠⎝⎠
t
2
t
ω
2
. (4.2.25)
Now, it is clear that (4.2.25) coincides with (4.2.22). Thus, functions (4.2.23) define a
Hamiltonian dynamical system with (viscous) dissipation.
Remark 4.2.4 (Generalized Hamiltonian formalism)
While Hamiltonian formalism plays a very important role in theoretical studies of mechanical
systems, we will not construct Hamiltonians and Rayleigh dissipation functions explicitly for
the considered differential equations, but confine ourselves to the first step of Hamiltonian
approach that relates to reduction of the higher-order differential equation(s) to a system of
differential equations of the first order. Such a reduction will be called as the generalized
Hamiltonian formalism, since for the ODE’s considered later, possibly no Hamiltonians or
Rayleigh dissipation functions that satisfy Eqs. (4.2.12), (4.2.17), and (4.2.18) exist.
Chapter 4. Ordinary differential equations
p
p
p
p
174
__________________________________________________________________________
4.2.3. The generalized Hamiltonian formalism for
a linear higher-order ODE
Herein, we consider a higher-order differential equation with constant coefficients and
apply the outlined procedure of reducing it to a system of n first-order equations.
Basic concepts 4.2.2
Applying procedure similar to one described in Remark 4.1.6, we arrive at the following
reduction of Eq. (4.2.1) to a system of equations of the first order
xx
≡
0
xx
=
10
.....
xx
xx x
=
12
nn
−−
aa
10
n
−
+++=
11 0
nn
−−
aaa
nnn
...
. (4.2.26)
t
()
Regrouping terms in Eqs. (4.2.26), yields
xx
=
01
.....
=− − − +
xx x
nn
−−
10 1
xx
=
nn
−−
21
aa
01
...
aaa
nnn
n
−
Equations (4.2.27) can be represented in a matrix form:
01
xx
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
... ...
d
⎜⎟ ⎜⎟
⎜⎟ ⎜⎟
xx
dt
nn
⎜⎟ ⎜⎟
xx
nn
⎝⎠ ⎝⎠
Now, by introducing two
⎛⎞
00
⎜⎟
⎜⎟
⎜⎟
=⋅+
⎜⎟
−−
22
−−
11
aa
01
⎜⎟
−− ⎜⎟
⎜⎟
aa
nn
⎝⎠
-dimensional vectors
n
x
⎛⎞
⎜⎟
GG
...
⎜⎟
==
XP
⎜⎟
x
n
⎜⎟
x
n
⎝⎠
... ...
0
−
2
−
1
...
1
n
−
0
⎛⎞
⎜⎟
...
;
⎜⎟
⎜⎟
0
⎜⎟
()/
ta
⎝⎠
. (4.2.27)
t
()
⎛⎞
⎜⎟
0
⎜⎟
...
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
. (4.2.28)
0
t
()
a
n
(4.2.29)
n
and denoting
Chapter 4. Ordinary differential equations
X
G
175
__________________________________________________________________________
01
⎛⎞
⎜⎟
⎜⎟
⎜⎟
=
G , (4.2.30)
⎜⎟
aa
01
⎜⎟
−−
⎜⎟
aa
nn
⎝⎠
we can rewrite system (4.2.28) in a more concise form:
...
... ...
1
n
−
d
=⋅+G
dt
Definition 4.2.6
The procedure of obtaining system (4.2.31) composed of
(4.2.27) related to the initial single differential equation (4.2.1) of n -th order will be called the
generalized Hamiltonian formalism.
Proposition 4.2.3 (non-singularity of Jacobian G )
The Jacobian (4.2.30) is non-singular at any coefficients
0& 0
aa≠≠
0
Proof
The proof flows out from computing determinant of
a
det( ) 1 ... 1 0
where we have used expansion of det( )
0
=− ×× × ≠G
(
a
n
G by elements in the first row.
GG
XP
n
)
. (4.2.31)
first-order differential equations
n
, 1, ...,
ak n=
k
, provided
. (4.2.32)
:
G
, (4.2.33)
Remark 4.2.5
It is interesting to note that for constructing solutions of linear ODE’s, the Jacobian need not be
non-singular. However, from henceforth it will be assumed that conditions (4.2.32) are satisfied
ensuring non-singularity of the Jacobian.
4.2.4. Solution of the initial-value (Cauchy) problem
Herein we construct the solution for the initial-value (Cauchy) problem (4.1.11). The
corresponding differential equation will be represented as a system of differential
equations of the first-order (4.2.28) with the zero loading vector.
Chapter 4. Ordinary differential equations
X
G
X
G
G
X
X
G
X
G
G
X
176
__________________________________________________________________________
Proposition 4.2.4 (The main property of the fundamental matrix)
Let G be the Jacobian of the linear system (4.2.28), or the same system represented in the form
Proof
(4.2.31), then the fundamental matrix
exp tG
Substituting into homogeneous Eq. (4.2.31) instead of vector
exp tG
, yields
()
satisfies the homogeneous Eq. (4.2.31).
()
the fundamental matrix
d
exp( ) exp( )
tt
dt
But, performing differentiation in the left-hand side of (4.2.34) gives exactly what stands in the
right-hand side of (4.2.34). We should also note, that matrix
Corollary (Constructing the general solution)
G
Let
be an arbitrary n-dimensional vector, then a vector
C
() exp( )
ttC=⋅G
is the general solution of the homogeneous Eq. (4.2.31).
Proof
The proof is analogous to the preceding one and relies on direct substitution of (4.2.35) into
both sides of homogeneous Eq. (4.2.31).
Proposition 4.2.5 (Solution of Cauchy problem)
G
Let
be a vector, specifying initial conditions at
0
()
tX=
00
=⋅GG G
. (4.2.34)
(4.2.35)
tt=
0
G
, (4.2.36)
G commutes with exp( )tG .
:
Proof
then the following vector field
GG
( ) exp( ( ))
tttX=−⋅G
C
is the solution of Cauchy problem (4.2.36) at
00
tt=
0
(4.2.37)
for homogeneous Eq. (4.2.31).
In view of the preceding corollary, the vector field (4.2.37) represents a solution of the
considered homogeneous equation. That is since this vector field coincides with (4.2.35), where
exp( )CtX=−⋅G
00
. (4.2.38)
G
Now, it remains to observe that at
, the vector field (4.2.37) equals to
tt=
0
.
0
Chapter 4. Ordinary differential equations
X
G
177
__________________________________________________________________________
Remark 4.2.6
Quite often in applications
Example 4.2.2
Herein we consider the same second-order differential equation, as in Example 4.2.1, but with
vanishing loading vector. The corresponding Jacobian
the initial boundary conditions at
To construct the fundamental solution
normal form
. In such a case the solution of Cauchy problem takes the form
0t =
0
GG
( ) exp( )
ttX=⋅G
C
01
⎛⎞
=
G
⎜⎟
53
−−
22
⎝⎠
will be as follows
0t =
10
⎛⎞
X
=
0
⎜⎟
20
⎝⎠
. (4.2.39)
0
is
G
. (4.2.40)
. (4.2.41)
exp( )tG we need to reduce matrix G to its Jordan
01
⎛⎞
⎜⎟
53
−−
22
⎝⎠
1
−
=⋅Λ⋅
WW
, (4.2.42)
G
where
⎛⎞
−+
i
331
⎜⎟
4
⎜⎟
Λ= =
G
⎜⎟
01
⎜⎟
⎝⎠
0
;
−− +
ii
331 331
410
⎛⎞
−
i
331
1
⎜⎟
⎜⎟
W . (4.2.43)
10
⎜⎟
⎜⎟
⎝⎠
Decomposition (4.2.42) allows us to construct the fundamental matrix
1
exp exptt
GW W
()
−
=⋅Λ⋅
()
G
. (4.2.44)
From (4.2.44) with account of (4.2.43), we get
3
i
⎛⎞
(
⎜⎟
exp
()
⎜⎟
t
G ,
=
⎜⎟
⎜⎟
⎝⎠
1 (11 3 31) 20
+− α+β
31
−
i
()
)
40
3
i
5
β−α
()
31
(
2
i
β−α
()
31
i
++ α+β
1(113 31) 20
31
i
()
)
40
(4.2.45)
where
⎛⎞⎛⎞
331 331
ii
α= β=
−+ +
exp ; exp
⎜⎟⎜⎟
⎝⎠⎝⎠
tt
. (4.2.46)
44
Chapter 4. Ordinary differential equations
⎡
⎢⎥⎣
⎡
⎢⎥⎣
G
G
178
__________________________________________________________________________
Accounting the constructed fundamental solution (4.2.45) and the initial conditions (4.2.41),
allows us to represent the desired solution in a form of (4.2.39):
⎛⎞
11 11
ii
−+α+ +β
51 1
with
α and β defined by (4.2.46).
The plot on Fig. 4.2.1 shows the corresponding phase trajectory for the real part of the vector
field (4.2.47). The starting point is at
trajectory was done by use of analytical solution (4.2.47), and then plotting a parametric plot.
GG
Xt tX
() exp( )
C
G
=⋅=
-2 2 4 6 8 10 12 14
Figure 4.2.1. Phase trajectory for equation (4.2.20)
0
20
15
10
5
0
-5
-10
(
⎜⎟
⎜⎟
88
⎜⎟
⎝⎠
ii
10 1 1
(
31 31
10;20
()
)
31 31
+α+− +β
)
. In view of Remark 4.1.9 plotting phase
(
(
⎤
)
⎦
(4.2.47)
⎤
)
⎦
4.2.5. Partial solution for inhomogeneous equation
with arbitrary loading
Proposition 4.2.6 (An auxiliary result for constructing partial solution)
Let PG be a loading vector, then the equation
G
(4.2.48)
GG
te−G
, we arrive at the initial equation (4.2.31).
(4.2.49)
Proof
is equivalent to Eq. (4.2.31).
Performing differentiation in the left-hand-side of (4.2.48), yields
Now, multiplying both sides of (4.2.49) by a (non-singular at any
into account commutativity of matrices
−⋅ ⋅ + ⋅ = ⋅
d
dt
−− −
GG G
eXe XeP
G
tt
−−
GG
eXeP
⋅= ⋅
()
G
d
dt
and
tt t
) matrix
G
t
G
and taking
e
Chapter 4. Ordinary differential equations
X
G
G
X
G
X
G
X
X
G
G
X
179
__________________________________________________________________________
Proposition 4.2.7 (Constructing partial solution)
Let PG be a loading vector assumed to be a (locally) integrable vector-function of parameter t ,
then partial solution of Eq. (4.2.31) can be represented in a form
() ( ) ()
te e Pd e Pd
= ⋅ ⋅ττ= ⋅ττ
p
GGG
tt
GG G
−τ −τ
tt
∫∫
tt
00
()
. (4.2.50)
Proof
Integrating both sides of (4.2.48) and multiplying them by a non-singular matrix
t
eG, we arrive
at (4.2.50).
Remark 4.2.7
Thus, the constructed partial solution (4.2.50) generally does not satisfy initial conditions. More
precisely, solution (4.2.50) satisfies only homogeneous initial conditions. Actually, it flows out
from (4.2.50) that
() 0
Xt=
0
p
. (4.2.51)
Proposition 4.2.8 (Constructing solution that satisfies initial conditions and inhomogeneous equation
(4.2.31))
G
Let
loading vector, then the solution of Eq. (4.2.31) that satisfies initial conditions can be
represented as the sum of two solutions:
be a vector, specifying initial conditions at
0
given by (4.2.37) and
t
()
C
tt=
, and
0
be a (locally) integrable
P
G
given by (4.2.50).
()
t
p
Thus,
Proof
The proof is obvious due to Remark 4.2.7.
Corollary
If initial conditions are homogeneous (
Proof
The proof flows out from representation (4.2.52) and Remark 4.2.7.
−τ
t
()
G
−τ
0
t
∫
t
0
. (4.2.53)
GG G G G
() () () ( )
tXtXte X e Pd
=+= ⋅+ ⋅ττ
Cp
GG G
() () ()
tXt e Pd
== ⋅ττ
p
()
−
G
tt
0
0
t
G
∫
t
0
), then
0X =
()
t
. (4.2.52)
Chapter 4. Ordinary differential equations
G
X
G
180
__________________________________________________________________________
Remark 4.2.8
It is interesting to note, that the partial solution (4.2.50), or the complete solution (4.2.52) does
not impose any restrictions on the loading vector, except a condition of its local integrability.
Example 4.2.3
Herein we consider the same second-order differential equation, as in Examples 4.2.1 and 4.2.2,
but this time with loading vector corresponding to the instantaneous impulse applied at
0
Pt
⎛⎞
=
()
⎜⎟
()
tt
δ−
⎝⎠
, (4.2.54)
0
tt=
:
0
where
()ttδ−
homogeneous:
is Dirak’s delta-function. Initial conditions at
0
G
G
.
0X =
0
are assumed to be
tt=
0
Multiplying matrix (4.2.45) at
⎛⎞⎛⎞
−+ +
331 331
α= −τ β= −τ
exp ; exp
⎜⎟⎜⎟
⎝⎠⎝⎠
ii
tt
() ()
(4.2.55)
44
by vector (4.2.54)
3
i
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎜⎟
⎝⎠
1(113 31) 20
+− α+β
()
31
−
i
()
40
3
5
i
β−α
()
31
2
⎛⎞
i
⎜⎟
⎜⎟
3
i
++ α+βδτ−
1(113 31) 20 ( )
()
⎜⎟
⎝⎠
()
31
()
β−α δ τ−
()
31
it
2
i
β−α
()
31
i
1(113 31) 20
++ α+β
31
()
i
()
40
t
0
0
40
0
⎛⎞
⋅=
⎜⎟
δτ−
t
()
⎝⎠
0
(4.2.56)
and performing integration yields:
⎛⎞
⎛⎞
⎜⎟
()
Xt
G
⎜⎟
1
=
⎜⎟
2
⎜⎟
3
()
⎜⎟
⎜⎟
⎝⎠
⎛⎞
i
1 (11 3 31)exp 20exp
++ −+ −
⎜⎟
⎜⎟
31
⎝⎠
⎛⎞⎛ ⎞
331 331
ii
2exp exp
itt tt
+−+
⎜⎟
⎜⎟⎜ ⎟
⎜⎟⎜ ⎟
⎜⎟
⎝⎠⎝ ⎠
⎝⎠
itt tt
−− −
() ()
44
00
31
⎛⎞⎛⎞
331 331
ii
−+ +
⎜⎟⎜⎟
⎜⎟⎜⎟
⎝⎠⎝⎠
() ()
44
00
40
(4.2.57)
the multiplier
in the last expression appeared because of integration starting from
2
(only
t
0
1
“half” of the delta impulse accounts in such integration).
On Fig. 4.2.2 the phase portrait of
Re( )
is plotted. It is based on the analytical solution
(4.2.57) and a parametric plot.
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