On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints
In this article we consider the affinely-rigid body moving in the three-dimensional physical space and subject to the Kirchhoff-Love constraints, i.e., while it deforms homogeneously in the two-dimensional central plane of the body it simultaneously performs one-dimensional oscillations orthogonal t...
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Kovalchuk, V. 2019-02-09T08:54:17Z 2019-02-09T08:54:17Z 2010 On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints / V. Kovalchuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37N15; 70E15; 70H33; 74A99 DOI:10.3842/SIGMA.2010.031 https://nasplib.isofts.kiev.ua/handle/123456789/146341 In this article we consider the affinely-rigid body moving in the three-dimensional physical space and subject to the Kirchhoff-Love constraints, i.e., while it deforms homogeneously in the two-dimensional central plane of the body it simultaneously performs one-dimensional oscillations orthogonal to this central plane. For the polar decomposition we obtain the stationary ellipsoids as special solutions of the general, strongly nonlinear equations of motion. It is also shown that these solutions are conceptually different from those obtained earlier for the two-polar (singular value) decomposition. This paper is a contribution to the Proceedings of the Eighth International Conference “Symmetry in Nonlinear Mathematical Physics” (June 21–27, 2009, Kyiv, Ukraine). The full collection is available at http://www.emis.de/journals/SIGMA/symmetry2009.html. ts This paper contains results obtained within the framework of the research project 501 018 32/1992 financed from the Scientific Research Support Fund in 2007–2010. The author is greatly indebted to the Ministry of Science and Higher Education for this financial support. The author is also very grateful to the referees for their valuable remarks and comments concerning this article and some propositions of the further investigation of the subject. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints Article published earlier |
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On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints |
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On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints Kovalchuk, V. |
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On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints |
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On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints |
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On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints |
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on classical dynamics of affinely-rigid bodies subject to the kirchhoff-love constraints |
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Kovalchuk, V. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України |
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In this article we consider the affinely-rigid body moving in the three-dimensional physical space and subject to the Kirchhoff-Love constraints, i.e., while it deforms homogeneously in the two-dimensional central plane of the body it simultaneously performs one-dimensional oscillations orthogonal to this central plane. For the polar decomposition we obtain the stationary ellipsoids as special solutions of the general, strongly nonlinear equations of motion. It is also shown that these solutions are conceptually different from those obtained earlier for the two-polar (singular value) decomposition.
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1815-0659 |
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https://nasplib.isofts.kiev.ua/handle/123456789/146341 |
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On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints / V. Kovalchuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 18 назв. — англ. |
| work_keys_str_mv |
AT kovalchukv onclassicaldynamicsofaffinelyrigidbodiessubjecttothekirchhoffloveconstraints |
| first_indexed |
2025-11-24T04:38:18Z |
| last_indexed |
2025-11-24T04:38:18Z |
| _version_ |
1850842227020922880 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 6 (2010), 031, 12 pages
On Classical Dynamics of Affinely-Rigid Bodies
Subject to the Kirchhoff–Love Constraints?
Vasyl KOVALCHUK
Institute of Fundamental Technological Research, Polish Academy of Sciences,
5B Pawińskiego Str., 02-106 Warsaw, Poland
E-mail: vkoval@ippt.gov.pl
Received November 13, 2009, in final form March 31, 2010; Published online April 08, 2010
doi:10.3842/SIGMA.2010.031
Abstract. In this article we consider the affinely-rigid body moving in the three-dimen-
sional physical space and subject to the Kirchhoff–Love constraints, i.e., while it deforms
homogeneously in the two-dimensional central plane of the body it simultaneously performs
one-dimensional oscillations orthogonal to this central plane. For the polar decomposition
we obtain the stationary ellipsoids as special solutions of the general, strongly nonlinear
equations of motion. It is also shown that these solutions are conceptually different from
those obtained earlier for the two-polar (singular value) decomposition.
Key words: affinely-rigid bodies with degenerate dimension; Kirchhoff–Love constraints;
polar decomposition; Green deformation tensor; deformation invariants; stationary ellipsoids
as special solutions
2010 Mathematics Subject Classification: 37N15; 70E15; 70H33; 74A99
1 Introduction
The special interest in the present work is devoted to the classical description of an affinely-rigid
(homogeneously deforming) mechanical system subject to the Kirchhoff–Love constraints. We
know that the standard continuum theory as well as some fundamental theories deal with such
objects as membranes, plates, discs, etc. So, the main contribution of this work is to present
a toy model for the analytical description of the above-mentioned objects.
The structure of this article is as follows: Firstly, we will present the main notions about the
concept of the affinely-rigid body, as a generalization of the metrically-rigid one, and of its special
case, i.e., the affinely-rigid body with degenerate dimension. Secondly, for convenience of the
Reader the main results obtained earlier for the case of two-polar (singular value) decomposition
are remembered. Thirdly, an alternative (polar) decomposition is introduced and the equations
of motion for our toy model are obtained for the general form of the inertial tensor, i.e., when
J1 6= J2 6= J3. And finally, three main branches of special solutions (stationary ellipsoids) for our
strongly nonlinear equations of motion are gathered in the form of Proposition 1. Additionally
some remarks about the complementarity of the obtained results to those described in our
previous work [6] are presented in the Summary.
So, let us remind some basic facts generally concerning the notion of affinely-rigid bodies [14,
15, 17, 18].
Let (M,V,→) be an affine space and (M,V,→, g) be the corresponding Euclidean one, where
M is a physical space in which the classical system of material points (discrete or continuous)
is placed, V is a linear space of translations (free vectors) in M , and g ∈ V ∗ ⊗ V ∗ is the metric
?This paper is a contribution to the Proceedings of the Eighth International Conference “Symmetry in
Nonlinear Mathematical Physics” (June 21–27, 2009, Kyiv, Ukraine). The full collection is available at
http://www.emis.de/journals/SIGMA/symmetry2009.html
mailto:vkoval@ippt.gov.pl
http://dx.doi.org/10.3842/SIGMA.2010.031
http://www.emis.de/journals/SIGMA/symmetry2009.html
2 V. Kovalchuk
tensor. Also let us introduce an affine (N,U,→) and the corresponding Euclidean (N,U,→, η)
spaces, where N is the material space of labels which are assigned to every material point of our
body in some way, U is the corresponding linear space of translations in N , and η ∈ U∗ ⊗ U∗
is the metric tensor. Then the position of the a-th material point at the time instant t will be
denoted by x(t, a) (x ∈ M, a ∈ N) and an affine mapping from the material space into the
physical one is as follows:
xi(t, a) = ri(t) + ϕi
A(t)aA,
where ϕ(t) is a linear part of the affine mapping (ϕ is non-singular for any time instant t), i.e.,
ϕ(t) ∈ LI(U, V ), where LI(U, V ) is a manifold of linear isomorphisms from the linear space U
into the linear space V , r(t) is the radius-vector of the centre of mass of our body if in the
material space the position of the centre of mass is aA = 0. If the system is continuous, then the
label a becomes the Lagrangian radius-vector (material variables) and x becomes the Eulerian
radius-vector (physical variables). Thus, at any fixed t ∈ R the configuration space Q of our
problem is given by the following expression:
Q = AfI(N,M) = Qtr ×Qint = M × LI(U, V ),
where “tr” and “int” refer to the translational (spatial translations) and internal (rotations and
homogeneous deformations) motions respectively.
The considered system is called an affinely-rigid body [14, 15, 16, 17, 18], i.e., during any
admissible motion all affine relations between constituents of the body are invariant (the ma-
terial straight lines remain straight lines, their parallelism is conserved, and all mutual ratios
of segments placed on the same straight lines are constant). The conception of the affinely-
rigid body is a generalization of the usual metrically-rigid body, in which during any admissible
motion all distances (metric relations) between constituents of the body are constant (see, for
example, [1, 4]).
In this article we concentrate mainly on the case of such an affinely-rigid body that is subject
to the additional constraints, i.e., it can deform homogeneously in the two-dimensional central
plane of the body and simultaneously performs one-dimensional oscillations orthogonal to this
central plane. Then the material space N is presented as the Cartesian product R+ × R2 and
the group of material transformations has the form R+ ×GL (2, R), where R+ is the dilatation
group in the third dimension and the material transformations in R2 act as in the case of the
usual affinely-rigid body with degenerate dimension [11, 12].
We can identify configurations Φ : R3 → R3 with the pairs (%, ϕ), where ϕ describes the
immersion of the central plane in the physical space, i.e., analytically ϕi
A is the 3 × 2 matrix.
An element (k, B) acts on (%, ϕ) as follows:
(k, B) ∈ R+ ×GL(2, R) : (%, ϕ) 7→ (k%, ϕB).
The conservation of orthogonality of the direction of dilatations to the central plane means that
the matrix
Φ =
Φ1
1 Φ1
2 Φ1
3
Φ2
1 Φ2
2 Φ2
3
Φ3
1 Φ3
2 Φ3
3
fulfils the condition that third column has to be proportional to the vector product of first
and second ones. If we consider Φa
1, Φb
2, a, b = 1, 2, 3, as independent and arbitrary, then
Φa
3 = ` εa
bcΦb
1Φc
2, where εabc is the completely antisymmetrical Levi-Civita (permutation)
symbol, ` is the parameter which depends both on the variable describing one-dimensional
On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff–Love Constraints 3
oscillations orthogonal to the central plane of the body and on the ones describing the state of
deformation in this central plane, e.g., for the two-polar (singular value) (1) and the polar (4)
decompositions we have respectively that
`two−polar =
%
λµ
, `polar =
%
ξζ − α2
,
where the meaning of variables λ, µ, α, ξ, ζ, % is clear from the expressions (2) and (5) below.
The above-described orthogonality is well known in the theory of plates and shells as the
Kirchhoff–Love condition [8].
2 Two-polar decomposition
In [6] we discussed the language of the two-polar (singular value) decomposition:
Φ
(
k;λ, µ, %; θ
)
= R
(
k
)
D (λ, µ, %) U (θ)−1 , λ, µ, % > 0, (1)
where R,U ∈ SO(3, R) are proper orthogonal matrices (whereas k is a rotation vector, i.e.,
a non-normalized vector codirectional with the rotation axis whose magnitude is equal to the
rotation angle) and D is diagonal, i.e.,
D(λ, µ, %) =
λ 0 0
0 µ 0
0 0 %
, U(θ)−1 =
cos θ sin θ 0
− sin θ cos θ 0
0 0 1
. (2)
Then the co-moving angular velocities for R- and U -tops [17, 18] are as follows:
ω = R−1Ṙ = RT Ṙ =
0 ω3 −ω2
−ω3 0 ω1
ω2 −ω1 0
, ωT = −ω,
and
ϑ = U−1U̇ = UT U̇ = θ̇
0 −1 0
1 0 0
0 0 0
, ϑT = −ϑ.
For Φ̇ and Φ̇T we have the following expressions:
Φ̇ = R
(
Ḋ + ωD −Dϑ
)
U−1, Φ̇T = U
(
Ḋ + ϑD −Dω
)
RT .
The kinetic energy is assumed to have the usual form (we have only to substitute the con-
straints):
T = 1
2Tr
(
JΦ̇T Φ̇
)
= 1
2Tr
(
U−1JU
[
Ḋ + ϑD −Dω
][
Ḋ + ωD −Dϑ
])
,
where J ∈ U ⊗ U is the twice contravariant, symmetric, non-singular, positively-definite tensor
describing the inertial properties of our affinely-rigid body. If we take J in the diagonal form
J = Diag (J1, J2, J3), then the above kinetic energy can be rewritten as follows:
T =
J1 cos2 θ + J2 sin2 θ
2
(
dλ
dt
)2
+
J1 sin2 θ + J2 cos2 θ
2
(
dµ
dt
)2
+
J3
2
(
d%
dt
)2
+
(
J1 sin2 θ + J2 cos2 θ
)
µ2 + J3%
2
2
ω2
1
4 V. Kovalchuk
+
(
J1 cos2 θ + J2 sin2 θ
)
λ2 + J3%
2
2
ω2
2 + (J1 + J2) λµω3
dθ
dt
+ (J1 − J2) sin 2θ
[(
µ
dµ
dt
− λ
dλ
dt
)
dθ
dt
+
(
λ
dµ
dt
− µ
dλ
dt
)
ω3 + λµω1ω2
]
+
(
J1 cos2 θ + J2 sin2 θ
)
λ2 +
(
J1 sin2 θ + J2 cos2 θ
)
µ2
2
ω2
3
+
(
J1 sin2 θ + J2 cos2 θ
)
λ2 +
(
J1 cos2 θ + J2 sin2 θ
)
µ2
2
(
dθ
dt
)2
. (3)
The above expressions significantly simplify when we consider the isotropic case in the central
plane of the body, i.e., when we have J1 = J2 = J . Then
T =
J
2
[(
dλ
dt
)2
+
(
dµ
dt
)2
]
+
J3
2
(
d%
dt
)2
+
Jµ2 + J3%
2
2
ω2
1
+
Jλ2 + J3%
2
2
ω2
2 + 2Jλµω3
dθ
dt
+
J
2
(
λ2 + µ2
) [
ω2
3 +
(
dθ
dt
)2
]
.
We also remind here that the corresponding expression for the kinetic energy in the canonical
variables has the following form:
T =
s2
1
2 (Jµ2 + J3%2)
+
s2
2
2 (Jλ2 + J3%2)
+
(
λ2 + µ2
) (
s2
3 + p2
θ
)
− 4λµpθs3
2J (λ2 − µ2)2
+
p2
λ + p2
µ
2J
+
p2
%
2J3
.
Then introducing some modelled potentials in [6] we obtained the Hamiltonian (total energy)
and calculated the corresponding equations of motion for the isotropic case with the help of the
Poisson brackets. In the present article we concentrate mainly on the alternative decomposition,
i.e., the polar one. The main advantages of this decomposition are the more physically intuitive
division on three main terms in the kinetic energy expression (see the formulas (6)–(9) below)
and the possibility to obtain the equations of motion in the quite simple form (see the expressions
(11)–(17) below) even for the general case, when the inertial tensor is not isotropic in the central
plane (J1 6= J2).
3 Polar decomposition
Instead of (1) we can also use the language of the polar decomposition, i.e.,
Φ (κ;α, ξ, ζ, %) = L (κ) S (α, ξ, ζ, %) , (4)
where L ∈ SO(3, R) is a proper orthogonal matrix and S ∈ Sym(3, R) is symmetrical. The
connection between the polar and two-polar decompositions is given by the following expressions:
L = RU−1,
ν = L−1L̇ = −νT =
0 ν3 −ν2
−ν3 0 ν1
ν2 −ν1 0
= U (ω − ϑ) U−1
=
0 ω3 + θ̇ −ω1 sin θ − ω2 cos θ
−ω3 − θ̇ 0 ω1 cos θ − ω2 sin θ
ω1 sin θ + ω2 cos θ ω2 sin θ − ω1 cos θ 0
,
On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff–Love Constraints 5
S =
ξ α 0
α ζ 0
0 0 %
= UDU−1 =
λ cos2 θ + µ sin2 θ (λ− µ) sin θ cos θ 0
(λ− µ) sin θ cos θ λ sin2 θ + µ cos2 θ 0
0 0 %
, (5)
and then the Green deformation tensor, which is not sensitive with respect to the left orthogonal
mappings, is as follows:
G = ΦT Φ = S2 =
ξ2 + α2 (ξ + ζ) α 0
(ξ + ζ) α ζ2 + α2 0
0 0 %2
= UD2U−1
=
λ2 cos2 θ + µ2 sin2 θ
(
λ2 − µ2
)
sin θ cos θ 0(
λ2 − µ2
)
sin θ cos θ λ2 sin2 θ + µ2 cos2 θ 0
0 0 %2
,
where for the positive definiteness the parameters have to fulfil the conditions
ξ = λ cos2 θ + µ sin2 θ > 0, ζ = λ sin2 θ + µ cos2 θ > 0,
ξζ − α2 = λµ > 0, % > 0.
For the polar decomposition we can as well introduce the concept of deformation invariants Ka,
a = 1, 2, 3, which may be chosen, e.g., as the eigenvalues of the symmetric matrix G:
det [G−KI3] = 0,
where I3 is the 3× 3 identity matrix, and the solutions are as follows:
K1,2 = 1
2
(
ξ2 + ζ2 + 2α2 ± (ξ + ζ)
√
(ξ − ζ)2 + 4α2
)
, K3 = %2.
The above deformation invariants are not sensitive with respect to both the spatial and material
rigid rotations (isometries).
Let us consider the Lagrangian L = T − V (Φ) and then the Hamiltonian H = T + V (Φ),
where the kinetic energy (3) can be rewritten for the polar decomposition as follows:
T = Trot + Trot−def + Tdef , (6)
where
Trot =
J1α
2 + J2ζ
2 + J3%
2
2
ν2
1 +
J1ξ
2 + J2α
2 + J3%
2
2
ν2
2
+
J1ξ
2 + J2ζ
2 + (J1 + J2) α2
2
ν2
3 − (J1ξ + J2ζ) αν1ν2 (7)
describes the coupling between the angular velocity ν of the L-top and deformation matrix S,
Trot−def =
(
J1α
dξ
dt
− J2α
dζ
dt
− (J1ξ − J2ζ)
dα
dt
)
ν3 (8)
describes the connection between the angular and deformation velocities, and finally
Tdef =
J1 + J2
2
(
dα
dt
)2
+
J1
2
(
dξ
dt
)2
+
J2
2
(
dζ
dt
)2
+
J3
2
(
d%
dt
)2
(9)
describes the kinetic energy of the deformation oscillations, whereas the potential term V (Φ)
depends on Φ only through the Green deformation tensor G = S2, i.e., the potential term
adapted to the polar decomposition is a function only of α, ξ, ζ, and %.
6 V. Kovalchuk
Performing the Legendre transformation we obtain that
π1 =
∂T
∂ν1
=
(
J1α
2 + J2ζ
2 + J3%
2
)
ν1 − (J1ξ + J2ζ) αν2,
π2 =
∂T
∂ν2
=
(
J1ξ
2 + J2α
2 + J3%
2
)
ν2 − (J1ξ + J2ζ) αν1,
π3 =
∂T
∂ν3
=
(
J1ξ
2 + J2ζ
2 + (J1 + J2) α2
)
ν3 + J1αξ̇ − J2αζ̇ − (J1ξ − J2ζ) α̇,
pα =
∂T
∂α̇
= (J1 + J2) α̇− (J1ξ − J2ζ) ν3,
pξ =
∂T
∂ξ̇
= J1
(
ξ̇ + αν3
)
,
pζ =
∂T
∂ζ̇
= J2
(
ζ̇ − αν3
)
,
p% =
∂T
∂%̇
= J3%̇,
where πi are canonical “spin” variables conjugate to angular velocities νi.
Therefore after inverting the above dependencies, i.e.,
ν1 =
(
J1ξ
2 + J2α
2 + J3%
2
)
π1 + (J1ξ + J2ζ) απ2
J1J2 (α2 − ξζ)2 + [J1ξ2 + J2ζ2 + (J1 + J2) α2]J3%2 + J2
3%4
,
ν2 =
(J1ξ + J2ζ) απ1 +
(
J1α
2 + J2ζ
2 + J3%
2
)
π2
J1J2 (α2 − ξζ)2 + [J1ξ2 + J2ζ2 + (J1 + J2) α2]J3%2 + J2
3%4
,
ν3 =
(J1 + J2) [π3 + α (pζ − pξ)] + (J1ξ − J2ζ) pα
J1J2 (ξ + ζ)2
,
dα
dt
=
(J1ξ − J2ζ) [π3 + α (pζ − pξ)] +
(
J1ξ
2 + J2ζ
2
)
pα
J1J2 (ξ + ζ)2
,
dξ
dt
=
pξ
J1
− α
(J1 + J2) [π3 + α (pζ − pξ)] + (J1ξ − J2ζ) pα
J1J2 (ξ + ζ)2
,
dζ
dt
=
pζ
J2
+ α
(J1 + J2) [π3 + α (pζ − pξ)] + (J1ξ − J2ζ) pα
J1J2 (ξ + ζ)2
,
d%
dt
=
p%
J3
,
we obtain the kinetic energy in the canonical variables as follows:
T =
(
J1ξ
2 + J2α
2 + J3%
2
)
π2
1 +
(
J1α
2 + J2ζ
2 + J3%
2
)
π2
2
2
(
J1J2 (α2 − ξζ)2 + [J1ξ2 + J2ζ2 + (J1 + J2) α2]J3%2 + J2
3%4
)
+
(J1ξ + J2ζ) απ1π2
J1J2 (α2 − ξζ)2 + [J1ξ2 + J2ζ2 + (J1 + J2) α2]J3%2 + J2
3%4
+
J1 + J2
2J1J2 (ξ + ζ)2
[π3 + α (pζ − pξ)]
2 +
J1ξ
2 + J2ζ
2
2J1J2 (ξ + ζ)2
p2
α
+
J1ξ − J2ζ
J1J2 (ξ + ζ)2
[π3 + α (pζ − pξ)] pα +
p2
ξ
2J1
+
p2
ζ
2J2
+
p2
%
2J3
. (10)
From the above kinetic energy expressions (6)–(9) one can see that the generalized velocities α̇,
ξ̇, ζ̇ corresponding to α, ξ, ζ and other variables describing the motion in the central plane of the
On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff–Love Constraints 7
body are separated from the generalized velocity %̇ describing the one-dimensional oscillations
orthogonal to this central plane. The same can be said also about the above expression in
the canonical variables (10), i.e., the momentum p% conjugated to % is orthogonal (in the sense
of metrics encoded in the kinetic energy expression) to the other canonical momenta. Hence,
the most simple are those dynamical models in which also the isotropic potential will have the
separated form:
V (α, ξ, ζ, %) = Vplane (α, ξ, ζ) + V% (%) ,
where as the potential V% we can take, e.g., the following potential which describes the nonlinear
oscillations and is in accordance with the main demands of the elasticity theory, i.e.,
V%(%) =
a
%
+
b
2
%2, a, b > 0,
where the first term prevents from the unlimited compressing of the body, whereas the second
one restricts the motion for large values of %, i.e., prevents from the non-physical unlimited
stretching of the body.
So, the Hamiltonian (total energy) can be written as follows:
H = T + Vplane (α, ξ, ζ) + V%(%),
where T is taken in the form of (10). Then the equations of motion can be calculated with the
help of the following Poisson brackets:
dπi
dt
= {πi,H} ,
dpα
dt
= {pα,H} ,
dpξ
dt
= {pξ,H} ,
dpζ
dt
= {pζ ,H} ,
dp%
dt
= {p%,H} .
The only non-zero basic Poisson brackets are
{α, pα} = {ξ, pξ} = {ζ, pζ} = {%, p%} = 1, {πi, πj} = −εij
kπk,
where the former expressions follow directly from the definition of the Poisson bracket and the
latter ones are based on the structure constants of the special orthogonal group SO (3, R).
First of all, let us rewrite the kinetic energy (10) in a more symbolic way, i.e.,
T =
Ω (π1, π2)
2Ξ
+
Υ(π3 + α (pζ − pξ) , pα)
2J1J2 (ξ + ζ)2
+
p2
ξ
2J1
+
p2
ζ
2J2
+
p2
%
2J3
,
where
Ξ = J1J2
(
α2 − ξζ
)2 +
[
J1ξ
2 + J2ζ
2 + (J1 + J2) α2
]
J3%
2 + J2
3%4,
and two expressions built of the canonical momenta are as follows:
Ω (π1, π2) =
(
J1ξ
2 + J2α
2 + J3ρ
2
)
π2
1 + 2 (J1ξ + J2ζ) απ1π2
+
(
J1α
2 + J2ζ
2 + J3ρ
2
)
π2
2,
Υ(π3 + α (pζ − pξ) , pα) = (J1 + J2) [π3 + α (pζ − pξ)]
2 +
(
J1ξ
2 + J2ζ
2
)
p2
α
+ 2 (J1ξ − J2ζ) [π3 + α (pζ − pξ)] pα.
Then we obtain the following equations of motion:
dπ1
dt
= −
[
(J1ξ + J2ζ) απ1 +
(
J1α
2 + J2ζ
2 + J3%
2
)
π2
]
π3
Ξ
8 V. Kovalchuk
+
π2 [(J1 + J2) [π3 + α (pζ − pξ)] + (J1ξ − J2ζ) pα]
J1J2 (ξ + ζ)2
, (11)
dπ2
dt
=
[(
J1ξ
2 + J2α
2 + J3%
2
)
π1 + (J1ξ + J2ζ) απ2
]
π3
Ξ
−
π1 [(J1 + J2) [π3 + α (pζ − pξ)] + (J1ξ − J2ζ) pα]
J1J2 (ξ + ζ)2
, (12)
dπ3
dt
=
(J1ξ + J2ζ) α
(
π2
1 − π2
2
)
+
[
J1
(
α2 − ξ2
)
+ J2
(
ζ2 − α2
)]
π1π2
Ξ
, (13)
dα
dt
= −
∂Vplane
∂α
−
(
J2π
2
1 + J1π
2
2
)
α + (J1ξ + J2ζ) π1π2
Ξ
+
2J1J2α
(
α2 − ξζ
)
+ (J1 + J2) αJ3%
2
Ξ2
Ω (π1, π2)
−
(J1 + J2) [π3 + α (pζ − pξ)] + (J1ξ − J2ζ) pα
J1J2 (ξ + ζ)2
(pζ − pξ) , (14)
dξ
dt
= −
∂Vplane
∂ξ
− J1ξπ
2
1 + J1απ1π2
Ξ
+
J1J2ζ
(
ξζ − α2
)
+ J1ξJ3%
2
Ξ2
Ω (π1, π2)
−
J1ξp
2
α + J1 [π3 + α (pζ − pξ)] pα
J1J2 (ξ + ζ)2
+
Υ(π3 + α (pζ − pξ) , pα)
J1J2 (ξ + ζ)3
, (15)
dζ
dt
= −
∂Vplane
∂ζ
− J2ζπ2
2 + J2απ1π2
Ξ
+
J1ξJ2
(
ξζ − α2
)
+ J2ζJ3%
2
Ξ2
Ω (π1, π2)
−
J2ζp2
α − J2 [π3 + α (pζ − pξ)] pα
J1J2 (ξ + ζ)2
+
Υ(π3 + α (pζ − pξ) , pα)
J1J2 (ξ + ζ)3
, (16)
d%
dt
= −dV%
d%
− J3ρ
Ξ
(
π2
1 + π2
2
)
+
J3%
Ξ2
[
J1ξ
2 + J2ζ
2 + (J1 + J2) α2 + 2J3ρ
2
]
Ω (π1, π2) . (17)
The structure of the above expressions implies that even in the simplest case of the completely
separated potential the dynamical coupling between the parameter describing one-dimensional
oscillations orthogonal to the central plane of the body and the variables living in this central
plane is present.
4 Stationary ellipsoids as special solutions
Our equations of motion (11)–(17) are strongly nonlinear and in a general case there is hardly
a hope to solve them analytically. Nevertheless, there exists a way for imaging some features
of the phase portrait of such a dynamical system, i.e., we have to find some special solutions,
namely, the stationary ellipsoids [14, 15], which are analogous to the ellipsoidal figures of equi-
librium well known in astro- [2] and geophysics, e.g., in the theory of the Earth’s shape [3].
In the case of the two-polar (singular value) decomposition (1) we obtained the above-
mentioned special solutions just putting the deformation invariants λ, µ, % and the angular
velocities ω, ϑ equal to some constant values [6]. But now, in the case of the polar decomposi-
tion (4), we see that the Green deformation tensor G, therefore the deformation matrix S, and
the angular velocity ν of the L-top have to be constant [15], i.e.,
dG
dt
=
d
dt
(
ΦT Φ
)
=
d
dt
(
S2
)
= 0,
dν
dt
=
d
dt
(
L−1L̇
)
= 0.
This means that the L-top performs the stationary rotation, i.e., if at the initial time t = 0 we
have that the configuration of the body is L0, then at the time instant t the configuration will
On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff–Love Constraints 9
be as follows:
L0 ◦ eνt,
where ◦ is the function composition symbol. We see that the whole affinely-rigid body, which
at the initial time t = 0 has the internal configuration Φ0 = L0 ◦S, at the time instant t will be
in the following configuration:
Φ(t) = L0 ◦ eνt ◦ S = eν̂t ◦ L0 ◦ S = eν̂t ◦ Φ0, (18)
where ν̂ = L0 ◦ ν ◦ L−1
0 .
Proposition 1. While the affinely-rigid body rotates in the stationary way around the axis fixed
in the physical and material spaces, the deformation and the angular velocity of rotation are not
independent and related by some algebraic expressions.
Proof. The trajectories of type (18) are the orbits of the Euler (spatial) action of the one-
parameter orthogonal group
{
eν̂t : t ∈ R
}
⊂ SO (V, g). Nevertheless, during the motion the
configurations of the body are deformed. At the same time the Green deformation tensor does
not perform any oscillations, it is constant. This means that an equilibrium is set between the
centrifugal forces coming from the rotation of the body and the elastic forces coming from the
fact that S 6= IdU . This kind of equilibrium is possible only if between the constant values
of ν, S are set some algebraic relations that guarantee the balance of the above-described forces.
These algebraic relations between ν, S obviously come from the equations of motion (11)–(17).
Hence, we propose to divide them into the following three main branches:
(i) ν1 6= 0, ν2 = ν3 = 0, then π1, π2 6= 0 and π3 = pα = pξ = pζ = p% = 0;
(ii) ν2 6= 0, ν1 = ν3 = 0, then π1, π2 6= 0 and π3 = pα = pξ = pζ = p% = 0.
For the first two cases the relations take the same form, i.e.,
∂Vplane
∂α
= −
(
J2π
2
1 + J1π
2
2
)
α + (J1ξ + J2ζ) π1π2
Ξ
+
2J1J2α
(
α2 − ξζ
)
+ (J1 + J2) αJ3%
2
Ξ2
Ω (π1, π2) , (19)
∂Vplane
∂ξ
= −J1ξπ
2
1 + J1απ1π2
Ξ
+
J1J2ζ
(
ξζ − α2
)
+ J1ξJ3%
2
Ξ2
Ω (π1, π2) , (20)
∂Vplane
∂ζ
= −J2ζπ2
2 + J2απ1π2
Ξ
+
J1ξJ2
(
ξζ − α2
)
+ J2ζJ3%
2
Ξ2
Ω (π1, π2) , (21)
dV%
d%
= −J3ρ
Ξ
(
π2
1 + π2
2
)
+
J3%
Ξ2
[
J1ξ
2 + J2ζ
2 + (J1 + J2) α2 + 2J3ρ
2
]
Ω (π1, π2) , (22)
with the compatibility condition
(J1ξ + J2ζ) α
(
π2
1 − π2
2
)
+
[
J1
(
α2 − ξ2
)
+ J2
(
ζ2 − α2
)]
π1π2 = 0.
We see that, while our parameters ν1 or ν2 take completely arbitrary constant values, the above
equations (19)–(22) describe their interrelation with the elements of the symmetrical matrix S,
i.e., with α, ξ, ζ, %.
(iii) ν3 6= 0, ν1 = ν2 = 0, then π3, pα, pξ, pζ 6= 0 and π1 = π2 = p% = 0, whereas
pα =
(J2ζ − J1ξ) π3
J1ξ2 + J2ζ2 + (J1 + J2) α2
,
pξ =
J1απ3
J1ξ2 + J2ζ2 + (J1 + J2) α2
,
10 V. Kovalchuk
pζ = − J2απ3
J1ξ2 + J2ζ2 + (J1 + J2) α2
.
So, for the third case we obtain the following relations:
∂Vplane
∂α
=
(J1 + J2) [π3 + α (pζ − pξ)] + (J1ξ − J2ζ) pα
J1J2 (ξ + ζ)2
(pξ − pζ) , (23)
∂Vplane
∂ξ
= −
J1ξp
2
α + J1 [π3 + α (pζ − pξ)] pα
J1J2 (ξ + ζ)2
+
Υ(π3 + α (pζ − pξ) , pα)
J1J2 (ξ + ζ)3
, (24)
∂Vplane
∂ζ
= −
J2ζp2
α − J2 [π3 + α (pζ − pξ)] pα
J1J2 (ξ + ζ)2
+
Υ(π3 + α (pζ − pξ) , pα)
J1J2 (ξ + ζ)3
, (25)
dV%
d%
= 0. (26)
This time our parameter ν3 has a completely arbitrary constant value and the above equations
(23)–(26) describe the way in which α, ξ, ζ, % are related to it. �
Remark 1. It should be mentioned that the name “stationary ellipsoids” is not the most
adequate for the description of the above-obtained stationary solutions. Of course, for the
non-restricted affinely-rigid body we can visualize this kind of special solutions as follows:
• At the beginning the body stays in the equilibrium configuration.
• Then we switch on some mechanical device which deforms our body in the homogeneous
way, i.e., this deformation is the superposition of three material stretchings with the
coefficients D1, D2, D3, where Di, i = 1, 2, 3, are the diagonal elements of the defor-
mation matrix D in the two-polar decomposition Φ = RDU−1.
• After this we start to rotate our mechanical device with the constant angular velocity ϑ
around one of the main axes of the Green deformation tensor G = ΦT Φ = UD2U−1 so
that the state of material deformation follows this movement of the device with the same
angular velocity (note that our body itself does not rotate!).
• And finally, the whole system consisting of the already rotating mechanical device and the
body starts also to rotate with the constant angular velocity ω around the corresponding
main axis of the Cauchy deformation tensor C = Φ−1T Φ−1 = RD2R−1 (this time both
the state of deformation and our body rotate!).
We see that the whole system reminds the gimbals equipped with additional mechanical device
deforming the body.
If our parameters D1, D2, D3, ϑ, ω are chosen in such a way that they fulfill the algebraic
relations obtained from the equations of motion, then even when we switch off the mechanical
device which generates the state of deformation in the material of our body, nothing will change,
i.e., the stretchings will continue to rotate with the same constant angular velocity ϑ around
the same axis in the material and the body will be rotating with the same angular velocity ω
around the same axis in the space. The deformation invariants D1, D2, D3 also will be constant
during the above-described two types of rotation.
But in our case of the affinely-rigid body subject to the Kirchhoff–Love constraints we have
the homogeneous deformation only in the central plane of the body, whereas in the perpendicular
direction the body performs some nonlinear oscillations. Hence, in this situation the more
appropriate name for our special solutions is “elliptical”, but we have kept the generic name
“ellipsoidal” for the matter of convenience.
On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff–Love Constraints 11
5 Summary
It is interesting to note that the special solutions obtained for the polar decomposition case
are conceptually different from those obtained for the two-polar one [6] because here the Green
deformation tensor G = S2 has a constant value (i.e., Ġ = 2SṠ = 0) contrary to the situation
described in [6] when the Green deformation tensor G = ΦT Φ = UD2U−1, as well as the Cauchy
one C = Φ−1T Φ−1 = RD2R−1, depended on time explicitly through the time dependence of U
and R respectively, i.e.,
dG
dt
= U
(
ϑD2 −D2ϑ
)
U−1 6= 0,
dC
dt
= R
(
ωD2 −D2ω
)
R−1 6= 0,
and performed the stationary rotations around their principal axes, whereas the deformation
invariants λ, µ, % had the constant values.
So, if we additionally keep in mind that in [6] we obtained the stationary solutions only for
the isotropic model J1 = J2 = J and here the general situation J1 6= J2 is allowed, then we can
compare the four (one in [6] and three here) studied cases according to the following scheme:
• The only degrees of freedom we can manipulate are the rotational degrees of R- and U -tops,
because the deformation matrix D is constant for this type of stationary solutions.
• To achieve the constant behaviour of the Green deformation tensor G = S2 =
(
UDU−1
)2 =
UD2U−1 we have to suppose that the U -top is fixed and does not rotate at all. If U is
constant, then the principal axes of the R- and L
(
= RU−1
)
-tops (for the two-polar and
polar decompositions respectively) rotate in the same manner, i.e., at any moment ones
can be obtained from others with the help of applying some constant orthogonal transfor-
mation. This situation corresponds to the above-mentioned three cases (i)–(iii) describing
the stationary rotations of the L-top around its three principal axes.
• If U -top is not fixed, then the Green deformation is not constant and we have to con-
sider three branches of the stationary motion for R- and U -tops when they rotate not
independently but in the correlated manner, i.e., either both around their first principal
axes or both around the second ones or both around the third ones [15]. Nevertheless, for
our affinely-rigid body subject to the Kirchhoff–Love constraints only the third case is
possible and exactly this situation was studied in the previous paper [6].
Hence, we see that in the above-described sense the results obtained in this article are essentially
different from and simultaneously complementary to those obtained in [6].
Let us mention that the affine models of degrees of freedom for structured bodies have been
studied by many authors. The thorough analysis of some stationary motions for affine bodies
and their stability was presented in [5, 7, 9, 10, 13]. However, in this article we have discussed
other problems.
Acknowledgements
This paper contains results obtained within the framework of the research project 501 018
32/1992 financed from the Scientific Research Support Fund in 2007–2010. The author is
greatly indebted to the Ministry of Science and Higher Education for this financial support.
The author is also very grateful to the referees for their valuable remarks and comments
concerning this article and some propositions of the further investigation of the subject.
12 V. Kovalchuk
References
[1] Arnold V.I., Mathematical methods of classical mechanics, Springer Graduate Texts in Mathematics, Vol. 60,
Springer-Verlag, New York – Heidelberg, 1978.
[2] Bogoyavlensky O.I., Methods in the qualitative theory of dynamical systems in astrophysics and gas dy-
namics, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
[3] Chandrasekhar S., Ellipsoidal figures of equilibrium, Yale University Press, New Haven – London, 1969.
[4] Goldstein H., Classical mechanics, Addison-Wesley Press, Inc., Cambridge, Mass., 1951.
[5] Green A.E., Naghdi P.M., A thermomechanical theory of a Cosserat point with application to composite
materials, Quart. J. Mech. Appl. Math. 44 (1991), 335–355.
[6] Kovalchuk V., Rożko E.E., Classical models of affinely-rigid bodies with “thickness” in degenerate dimen-
sion, J. Geom. Symmetry Phys. 14 (2009), 51–65, arXiv:0902.3573.
[7] Lewis D., Simo J.C., Nonlinear stability of rotating pseudo-rigid bodies, Proc. Roy. Soc. London Ser. A 427
(1990), 281–319.
[8] Love A.E.H., A treatise on the mathematical theory of elasticity, Dover, New York, 1996.
[9] Nordenholz T.R., O’Reilly O.M., On steady motions of isotropic, elastic Cosserat points, IMA J. Appl.
Math. 60 (1998), 55–72.
[10] Nordenholz T.R., O’Reilly O.M., A class of motions of elastic, symmetric Cosserat points: existence, bifur-
cation, and stability, Internat. J. Non-Linear Mech. 36 (2001), 353–374.
[11] Rożko E.E., Dynamics of affinely-rigid bodies with degenerate dimension, Rep. Math. Phys. 56 (2005),
311–332.
[12] Rożko E.E., Dynamical systems on homogeneous spaces and their applications to continuum mechanics,
PhD Thesis, 2006 (in Polish).
[13] Rubin M.B., On the theory of a Cosserat point and its application to the numerical solution of continuum
problems, J. Appl. Mech. 52 (1985), 368–372.
[14] S lawianowski J.J., The mechanics of the homogeneously-deformable body. Dynamical models with high
symmetries, Z. Angew. Math. Mech. 62 (1982), 229–240.
[15] S lawianowski J.J., Analytical mechanics of deformable bodies, PWN – Polish Scientific Publishers,
Warszawa – Poznań, 1982 (in Polish).
[16] S lawianowski J.J., Kovalchuk V., Invariant geodetic problems on the affine group and related Hamiltonian
systems, Rep. Math. Phys. 51 (2003), 371–379.
[17] S lawianowski J.J., Kovalchuk V., S lawianowska A., Go lubowska B., Martens A., Rożko E.E., Zawistows-
ki Z.J., Affine symmetry in mechanics of collective and internal modes. I. Classical models, Rep. Math.
Phys. 54 (2004), 373–427, arXiv:0802.3027.
[18] S lawianowski J.J., Kovalchuk V., S lawianowska A., Go lubowska B., Martens A., Rożko E.E., Zawistows-
ki Z.J., Affine symmetry in mechanics of collective and internal modes. II. Quantum models, Rep. Math.
Phys. 55 (2005), 1–46, arXiv:0802.3028.
http://dx.doi.org/10.1093/qjmam/44.3.335
http://arxiv.org/abs/0902.3573
http://dx.doi.org/10.1098/rspa.1990.0014
http://dx.doi.org/10.1093/imamat/60.1.55
http://dx.doi.org/10.1093/imamat/60.1.55
http://dx.doi.org/10.1016/S0020-7462(00)00021-4
http://dx.doi.org/10.1016/S0034-4877(05)80087-4
http://dx.doi.org/10.1115/1.3169055
http://dx.doi.org/10.1002/zamm.19820620604
http://dx.doi.org/10.1016/S0034-4877(03)80029-0
http://dx.doi.org/10.1016/S0034-4877(04)80026-0
http://dx.doi.org/10.1016/S0034-4877(04)80026-0
http://arxiv.org/abs/0802.3027
http://dx.doi.org/10.1016/S0034-4877(05)80002-3
http://dx.doi.org/10.1016/S0034-4877(05)80002-3
http://arxiv.org/abs/0802.3028
1 Introduction
2 Two-polar decomposition
3 Polar decomposition
4 Stationary ellipsoids as special solutions
5 Summary
References
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