/ • < • _ . . .
IC/81/193
INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS
CLASSICAL MECHANICS OF A BREATHING TOP
INTERNATIONALATOMIC ENERGY
AGENCY
UNITED NATIONSEDUCATIONAL.
SCIENTIFICAND CULTURALORGANIZATION
Heinz Peter Berg
1981 MIRAMARE-TRIESTE
IC/81/193
International Atomic Energy Agency
and
United Nations Educational Scientific and Cultural Organization
INTERNATIONAL CEHTRE FOE THEORETICAL PHYSICS
CLASSICAL MECHANICS 0*' A BREATHING TOP •
Heinz Peter Berg
International Centre for Theoretical Physics, Trieste, Italy,and
I n s t i t u t fur Theoretiscne Faysik, Technische Univers i ta t C laus tha l ,D-3392 Claus tha l -Ze l l e r f e ld , Federal Republic of Germany.
ABSTRACT
We develop the classical mechanics of a breathing
top and consider the left- and right-invariant actions on the
configurations space SL(n,R). This model is compared with the
affinely-rigid body, based on GL(n,R). Applications of the
breathing top can be found in the theory of elasticity and
hydromechanics.
MIRAMARE - TRIESTE
September 1981
* To appear in fieports in Mathematical Physics.
1. INTRODUCTION
Most of the dynamical systems in quantum theory consist of a large
number of particles,but there exists no mathematical procedure
solving such many body problems; on the other hand,in the clas-
sical description of complex systems, collective generalized
coordinates are often introduced.
The concept of collective motion is to isolate a subsystem of a
gj.ven system describing the motion of the system, when only the
collective degrees of freedom are excited, i.e. the other degrees
of freedom are "frozen".
CEUt modfel of a breathing top is based on the Lie group SL(n,R).
SL(n,R) as configuration space supplies the so-called affine
degrees of freedom, i.e. the rotational and vibrational degrees
of freedom of the body as a whole.
A possible application of SL(n,R) in classical mechanics is recently
shown by Leach [2^ : the complete dynamical symmetry group of the
onedimensional time-dependent harmonic oscillator is SL(n,R),
where the two linear and the three quadratic constants of motion
form a five parameter subgroup.
In section 2 we develop the analytical mechanics of the breathing
top. The idea of a so-called affinely-rigid body has been introduced
by SJawianowski \5~\ , [ej , \7~] ; for example a homogeneously
deformable medium is an affinely-rigid body. The configuration space
of the affinely-rigid body with frozen translational degrees of free-
dom is GL{n,R), the group of rotations and deformations.
The configuration space of our model is SL(n,R), the group of
rotations and volume preserving deformations.
-2-
We mainly discuss the physically important cases n=2 and n=3.
The application of the breathing top in the theory of elasticity
and hydrodynamics will be handled in section 3 and we show how
our model differs from the affinely-rigid body.
2. KINEMATICS OF THE BREATHING TOP
In this section we want to describe the properties of the
breathing top in classical kiner.iati.es and the connection with the
model of an affinely-rigid body.
L«t (K,V,>) be an affine space, where M is a manifold, V a linear
space of translations in M and •* the mapping M x M -» V.
Linear deformations result from displacing all the material points
of the medium, according to some fixed affine transformations
of M.
We handle the special case in which the body is fixed at one point,
i.e. we exclude translations.
The configuration of a body which undergoes linear deformations
and rotations without translations is uniquely described by a
linear isomorphism ip :U •• V, where V is the physical space and
U the material space.
In the following L(U,V) is the space of all linear mappings of
U into V, LI(O,V) the set of all linear isomorphisms of 0 into V.
In the case of the affinely-rigid body we denote LI{O,V) by GL(U,V),
where GL(U,V) is an open submanifold of L(U,V), in the case of
the breathing top we denote LI(U,V) by SL(u,V), which is a closed
submanifold of L(U,V) , and SL(U,V) is the set of volume preser-
ving linear isomorphisms.
-3-
The configuration space of the considered systems is den^oted by
M = LI(U,V), dim M = n.
The restriction to volume preserving mappings is equivalent to
reducing the dimension of M to n - 1.
In the linear spaces U and V the automorphism groups GL(U) and GL(V)
act in a natural way. These actions give rise to the natural actions
of GL(U) und GL(V) on the configuration space GL(U,V). The correspon-
ding transformations are defined by
Ad = A ' if
(f B = <f « B
(2.1a)
(2.2a)
A t G L ( V ) , B SGL(U) , <Ji. GL(U,V).
GL(V) acts on GL(U,V) on the left and GL(U) on the right.
We choose now the cartesian coordinate system and hence can identify
0 and V with Rn.
Consider the set of all linear invertible operators from H n into Rn.
The set of these operators, written as matrices with respect to
a given basis, is the full linear group GL(n,R). The restriction
to the subgroup with det = 1 is called the special linear (or
unimodular) group SL(n,R).
Hence the configuration space LI(U,V) can be identified by
LI(Rn,R") = LI(n,R).
The configuration space of the breathing top M = SL(n,R) possesses
two natural actions on SL(n,R), namely
A J
if B
A o if
if • B
(2.1b)
(2.2b)
for arbitrary A« SL(n,B)1, BtSL(n,R)r and <ft SL(n,R),
-k-
The transformations of. the group SL{n,R) have a direct physical
interpretation as deformations and rotations of the system referred
to a fixed space system.
On the other hand SL(n,R)r describes possible symmetries of the
body itself, for example isotropy.
The kinematic of a system is fully determined by the configuration
and the generalized velocity or equivalently by the configuration
and the momentum.
For the special case of GL(n,R) it is sufficient to use the genera-
lized velocities y '(t) e L(U,V) t<] > but in general one has to
combine components of y •(t) with coefficients which depend on
-the configuration <f t LI(U,V). They are conveniently called non-
holonomic velocities. Analogously to non-holonomic velocities Jl
we define nibn-holonomic momenta Y, .
The velocities and momenta are usually introduced in the general
theory of the tangent and cotangent bundles. However, the configuration
space in our cases is a Lie group and therefore
( 2 . 7 )
TG ? G x IiG
T*G = G x LG*
where G is a Lie group and LG a Lie algebra.
For the affinely-rigid body we get
TG = GL(n,R) X L(n,R)
T"G ^ GL(n,R) x L(n,R)"
(2.3)
(2.4)
(2.5)
(2.6)
where L(n,R) is a Lie algebra and L(n,R)* is canonically isoiuorphic
to L(n,R) what easily can be seen from
- 5 -
tj is a variable and )t acts like a linear function.
For the breathing top we yield
TG : SL(n,R) x sl(n,R)
T*G ~ SL(n,R) x s l ( n , R ) '
(2.8)
(2.9)
The Lie algebra sl(n,R) is canonically isomorphic to sl(,R)*.
Differences between the affinely-rigid body and the breathing
top appear in the definitions of the non-holonomic velocities
and momenta.
In the first case the left velocity Jl is defined as a mapping
il:GL(n,R) x L(n,R) * L(n,R) such that
Jl(tfrV) s II o (9 . (2.10)
These mappings identify the state space with GL(n,R) x L(n,R):
(>f.<f) •*• (ifrif "f"1) • (2.11)
Such a cross product doesn't exist in the case of. a breathing
top. We consider a mapping Jl :TSL(n,R) -» sl(n,R) with
- 1J l = vf • if
We have to require that for fixed
TSL(n,R) =U^
(2.12a)
Tr( <y <f~1) = O, i .
where
T SL(n,R) if =>fA» Tr(Jl). C L(n,R)
In analogy we define J l :
Jl = ^ " 1 » ^ (2.12b)
as the right velocity.
If we want to consider Jl, Jl as matrices Jl fc, Jl R , we need metrics.
Then we can define
-6-
JLlj(t) g k j
and
ki KJwhere g J , ̂ are the contravariant components of a fixed
metric tensor g, -̂ in V and U respectively.
The independent elements of the antsymmetric part Jl •'it) are the
components of the angular velocity at time t with respect to the
laboratory system and A*- J (t) the angular velocity referred to the
body fixed frame. -
The antisymmetric parts describe the rotational behaviour of the
body and the symmetric ones the deformative behaviour.
The momentum map I :T*G •> LG for the affinely-rigid body is defined
as I :GL(n,R> x L(n,R) -> L(n,R) such that
2. (>f,t) = y o TT (2.13a)
with tf c GL(n,R) and n £ L(n,r).
In the case of the breathing top ¥ is a linear mapping for a given
(f on T\,SL(n,R) = N. The adjoint space of N consists of the
equivalence classes [W] <- L(n,R) defined by
where Trdf^if) =Tr(B2,(j) with (f = Jl«f , Cf« SL(n,R), Jl t si (n,R).
The momentum map is then Z :T*SL(n,R) •* sl(n,R)
) =
In analogy
with Tr(T,» ) - O . (2.13b)
(2.14)
We can write T? in (2.1 3) , (2.14) in terms of if (within the Lagrange
description). This is also reasonable for the model of the brea-
-T-
thing top, if we consider y as a mapping <| : R -> SL(n,R). Then if are
automatically tangent vectors to SL(n,R).
The transformation rules for the non-holonomic velocities are
given by
JL -^AJIA"1 , AsSL(n,R) , Jl •* Jl (2.15a)
for left regular actions and
SL(n,R) (2.15b)
for right regular actions.
In analogy, the transformation rules for the non-holonomic momenta
are given by
and
Z •* hi A-1 Z -» Z
Z *B'1
(2.16a)
(2.16b)
for left and right regular actions respectively, A,BcSL(n,r).
The Z- mappings can be identified with non-holonomic canonical
momenta conjugate to Jl-s. Similarly, T - mappings are conjugate
to Jl - mappings:
TrUJL) - Tr(£jl) (2.17)
I , I generate extended point transformations corresponding to
the left and right regular translation groups.
3. ELASTICITY THEORY AND HYDRODYNAMICS
A possible application of the breathing top can be found in the
classical theory of elasticity.
We introduce a coordinate system which is connected with the in-
finitesintal representations of the t ie group SL(n.rR).
Every element »f*SL(n>.HJ can be : represented" tmiqualy In. the f tein
if = exp{« ia i)exp(J3b j) , Tr(bj) = 0 , (3.1)
where <* , S" ** are constants and a^, b. are symmetric and antisym-
metric matrices- respectively, which form a basis for the Lie algebra.
i % iWe use these coordinates<K ,J , because they enable us to separate
rotational and pure deformative parts.
Unfortunately even the threedimensional breathing top is difficult
to handle. The main reason is the different structure of the two-
and threedimensional orthogonal groups: S0(3,R) is a simple three-
dimensional group, S0(2,R) is the one-dimensional group of plane
rotations and therefore commutativ.
The simplicity of S0(3,H) implies that in all formulas for impor-
tant physical quantities - for example the kinetic energy - and
in the equations of motion the invariants <j (cf. 3.4} and X , which
describes dilatations, are mixed in a_ nonseparable way. Hence it
is impossible to solve the classical' Hamilton-Jacobi equations
with a realistic,isotropic potential by the method of the separa-
tion of variables.
Thereforet we discuss the two-dimensional case.
As a basis of sl(2,R) we can use
Every if E SL(2,R) can be written as
or explicitely
-9-
(3.3)
+ sinb^ sin (:|-f) cosh^sinl + sin
-cosh^siti? + -•)")• cosh^eosv"-
(3.4)
with ̂ e (0,2"D) ,if t (0,2t) and % e{0,») .
The parameters fully describe the configuration of the breathing
top; they possess a physical meaning: ? is a rotation and ^,^
are polar coordinates which describe a pure shear.
If we want to state the kinetic energy T, we have to notice that
the invariance group for T is only the special orthogonal group.
In linear coordinates the kinetic energy T is given by
T = 1 Tr(ip fc(f 9) (3.5)
with if € SL(2,R) and if f. SL(2,R) .
The inertial tensor e = e"111 in (3.5) is constant and is comoving.
Moreover we have to pay attention to the requirements of section 2.
Let now 9 =f I (i.e. spherical symmetry),
£ *-<f) . (3.6)
Using the new parameters we can express the energy in the form
x=H~"^ - (3.7)T = h ( ̂ cosh2«; + v cosh ? + y sinh ̂
The dynamical properties of an elastic body are characterized by
the potential energy V. V only depends on the configuration and
on 3 . Moreover, we restrict ourselves to the case of small de-
formations around ^ = 0. Therefore let the potential be
V = V<<;) = k$2 . (3. )
Such a potential i s physically reasonable (cf. Hooke's law). The
.correspondtngr 'I«pniltonian . equations can be estimated by means
-10-
Q£ the Hamilton:-Jacob! theory Dl • t*l ( because the Hamiltan-Jacobi
equation is separable and the formulas for the general shape of
integrable potentials follow from the classical StSckel theorem
M . • •
This example shows that the breathing top is completely solvable
in the theory of elasticity, but only in the two-dimensional case
and with SO(2,R) as the symmetry group for the kinetic energy.
We want to use the SL(3,R) group not only for the description of
Jtineroatical but also of the dynamical structure of a system. Hence
the application of the breathing top as a model in hydromechanics
is reasonable.
Noll [3} characterizes a system as a fluid, if its isotropy group
(i.e. the group of all material isomorphisms) is the unimodular
group.
This means fluids are not invariant against changes of the volume,
but invariant against changes of the shape.
Hence only itochor deformations are allowed. Therefore the
breathing top is a model for fluids, in contrast to the affinely-
rigid body, which admits volume changes.
If SL{3,H) acts on the configuration space as a symmetry group,
we can discuss two possible models:
1. This model is connected with the model that we had used in the* -1
theory of elasticity (3-5); T is a quadratic form of A = if • ̂
with constant coefficients
T = j Tr(JL tAB)
We assume additional right orthogonal invariance; 6
* 4-
T - I Tr(Jl il ) ,
-11-
(3.9)
(3.1O)
the invariance group becomes then: SL{3rR).^ x SO(3,R)r.
The most, general quadratic form which is invariant under
the group above is
T = £ +• |
The last summand in (3.11) violates the positive definiteness
of the kinetic energy. Hence the positive- definite energy
is given by
* . (3,12)
In.the case of the affinely-rigid body, metric tensors on GL(3,R)
are invariant simultaneously under left and right regular trans-
lations, i.e. under groups GL(3,R>, x GL(3,R) .
The corresponding kinetic energies would have the following form:
T = -£ Tr(Jl2) + £(TrJl)2 = - 5 Tr(iL2) + ̂ (TrA) (3.13)
Such metrics must not be positive-definite because of the non-
compactness of GL(3,R). Hence they are useless as physical models
of the kinetic energy. This concerns in particular the Killing
metric on GL(3,R):
~ Tr[(Jl - ̂ (3.14)
Such a metric is degenerate - the one-parameter dilatation subgroup
is isotropic - and orthogonal to the whole of GL(3,R).
When restricted to SL(3,R) it becomes non-singular and non-defi-
nite. Contributions corresponding to rotations (skew-symmetric
part of-fl.) and shears (symmetric part ofJl) have opposite signs.
-12-
If we want to consider the Lagrangian L = T - V (i.e. dynamical
systems) where T is one of the above forms and V a potential,
then the Legendre transformation only depends on the structure
of T. Let us describe it for the models discussed above.
A * , A
1. Using (3.9) the variation Jl -=» Si + 4JL results in
IT = Trfflji 4ji ) = Tr(IfA) . (3.15)
Hence Legendre transformation implies in the following dependenceA A
of the momentum mapping I of .ft :
z-eJL*
The canonical form of T is given by
T = j Tr(© I ) = -j ) = j Tr(2t ©
or in terms of coordinates
2 i j k
(3.16)
(3.17)
(3.18)
we finally get
(Tr (3.22)
ACKNOWLEDGEMENT
The author is indepted to Prof. Dr. H.D. Doebner and Doc. Dr.
J.J. Sjawianowski for many suggestions. He would like to thank
Prof. Abdus Salam, the International Atomic Agency and UNESCO
for hospitality at the International Centre for Theoretical
Physics, Trieste, during the main stage of this work.
2. Using (3.-12) the variation results in
where Z =j*Xt +XTrJL I
With the inversed Legendre transformation
i -} Tr Z I
-13-
(3.19)
(3.20)
(3.21)
- l l i -
REFERENCES
1. A.E. Greenf J . E . Adkins, Large e l a s t i c deformations.
Clarendon P r e s s , Oxford 1960
2. P.G. Leach, a. Math. Phys. 2J.' 3 0 ° (1980)
3 . W. N o l l , R.A. Toupin, C.C. Wang, Continuum theory of
inhomogeneit ies in simple b o d i e s , Springer-Verlag,
New York 1968
4. M. Seredynska, Bul l . Acad. Po l . S c i . 2^, 445 (1975)
5. J . J . S^awianowski, Arch. Mechanics 2Q, 569 (1974)
6. J . J . SJawianowski, I n t . J . Theor. Phys. V2, 271 (1975)
7. J . J . SZawianowski, Rep. Math. Phys. K>» z 1 9 (1976)
8. J . J . SJawianowski, IFTR Reports , 1 s t Pol ish-Swedish
Conference on Micropolar E l a s t i c i t i e s , Jablonna 1977,
Warsaw 1979
IC/61/29
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IC/81/100INT.REP.*
IC/81/101INT.REF.*
:c/8i / io2IHT.REP.*
ic/ei/103MT.REP."
ic/ei/iouINT.REP.•
ic/ai/io5
IC/81/106
IC/81/10T
IC/81/103INT.REP.•
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