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A few remarks about integrability of the equations of motion of a rigid body in ideal fluid
Volume
80A, number
2,3
PHYSICS
A FEW REMARKS ABOUT INTEGRABILITY
24 November
LETTERS
1980
OF THE EQUATIONS
OF MOTION OF A RIGID BODY IN IDEAL FLUID A.M. PERELOMOV Institute of Theoretical and Experimental Physics, 1 I7259 Moscow, USSR Received
13 August
1980
We demonstrate that the problem of motion of a rigid body in an ideal fluid for a nondegenerate case is completely integrable by means of the quadratic constants of motion for the Clebsch and Steklov cases only. We construct a Lax pair
for a high-dimensional generalization of the Clebsch case.
The classical problem of motion of a rigid body in an ideal fluid (MRBIF) was the subject of intensive study during the last and the beginning of the present century (see e.g. [l-lo]). Recently, S.P. Novikov has noticed that within the framework of Lie groups this problem could be so recast as to have the meaning of finding geodesics for a right-invariant Riemannian metric on the six parameter Lie group E(3) of motions of the three-dimensional euclidean space. Starting from this observation he indicated the possible links of the problem with the SO(4)generalization of the Euler top, and considered the contraction of the group SO(4) onto the group E(3) for which the Euler equations of the SO(4) top are transformed into equations for the so-called Clebsch case of MRBIF. It is worth noticing that the Manakov L-M pair of the S0(4)-top, as was demonstrated by S.P. Novikov, diverges in the limit of the deformation, though the constants of motion for the S0(4)-top are transformed into the constants for the Clebsch case. In this paper, we construct an L-M pair for a n-dimensional generalization of the Clebsch case which is closely related to the Moser [ 121 L-M pair for geodesic flow on (n - 1) dimensional ellipsoid. It is worthwhile to notice that our L-M pair is associated with the Lie algebra of the group SL(n, R) (see below). We demonstrate also that the MRBIF problem for nondegenerate case is completely integrable by means of the quadratic integrals of motion only for the Clebsch [6] and Steklov [8] cases. 156
(1) Let us give the group-theoretical description of the equations MRBIF. Let G = E(3) be the group of motions of the three-dimensional euclidean space R3. Notice that E(3) is generated by rotations and translations of R3. Let 9 be its Lie algebra with the standard basis xi,yk,j, k = 1,2,3: txj, +]
= EjkmXm,
bj?j,&l
[xj,Ykl
= fjkmym 3
=0 .
(1)
Here Ed%is the standard completely skew-symmetric tensor. Let $, pk be coordinates of a point of g- the space of linear fupctionals on Q and 7 be the space of functions on g , belonging to the class C”. In the space 7 the commutation relations (1) generate the Poisson structure that is defined by the equations {$J lk> = Ejkm lm 9 ($9 Pk) = fj/cmPm > (pi, Pk} = 0 . (2) The function tem
HE 7 determines
I$ = apjiat = &‘, Pj) ,
the hamiltonian
ik= {H,lk}.
sys-
(3)
It is easy to see that functions 12 = PjPj ?
13 = $pi
(4)
are integrals of motion for any hamiltonian H. Considering them as some constants we,@tain the orbit R of the co-adjoint representation Ad(G) of the group G = E(3). This orbit a is the phase space of the considered system.
PHYSICS LETTERS
Volume 80A, number 2,3
Hence the dynamic equations
where
can be written as
i=Z(N*L.
(5)
+ 2bjk$pk t clkpjpk = 2(Hu + HI t Hz) , (6)
where the quadratic form (6) must be positive. In the present note we consider only the nondegenerate case, when all eigenvalues of the symmetric matrix aij are positive and differ from each other. Without any loss of generality we may still assume that the matrix aij is diagonal: aij = aidij and the matrix bij is symmetric which we can always achieve using the transformations belonging to the group of motion of euclidean space. Proposition 1. If all eigenvalues of the matrix au differ from each other, the fourth quadratic constant exists only for the Clebsch [6] and Steklov [8] cases. Let us outline the proof. Notice that the equation {H, fi = 0 where (7)
amounts to the following set of equations IHoJo)=O,
WoJll
+ V&Z01
=O,
IH1,~1}+CHo,z2}+{H2,10}=0, Vf1J21
+ W2,1,1=0
9
(8)
which generate a set of equations for the coefficients AJh B1%’ CJk After some algebra which aJ7V bJk, ‘Jk, we shall omit due to lack of space, one can convince oneself that solutions to the equations exist only for the Clebsch and Steklov cases. These cases are the following: (I) The Clebsch case [6]. The energy is of the form 2H = C (ajf t CjPf t 2bj$pj) 3 i
- cl) t aGjl(cl - c2) = 0 .(lO)
of motion is of the form
(11)
(2) Now, as was noticed by S.P. Novikov, the equations MRBIF have the form (3) with the hamiltonian quadratic in Pi and Zk:
21= Aijli’i t 2Bjk$pk t qkpjpk = 2(Io t I, t 12)
ar1(c2 - c3) t a;‘(c3 The constant
HereLEg,M=VHEQ,V:T-+‘Sisthestandard gradient map. In the present situation dim Sz = 4, so that we need only one more global integral of motion to indicate the equations of motion.
2H = a&k
24 November 1980
It is worth noticing that the equations of motion in the Clebsch case can be integrated with the help of the B-functions of two variables [7]. (II) The Steklov case [8] which was further developed by Liapunov [9] and Kolosof [lo]. Here H is again given by eq. (9); the coefficients bj, Cj and the constant of motion I are of the form bj =p(ala2a3)aJT1 t v,
~1 =p2al(a2 - a3)2 + v’ , ... , (12)
21~~(~-2/_l(a~tV)$p~)t~2((a2-a3)2tV”)p~t.... (13) Let us notice that in the Clebsch case the quantities
lyk)-lz$,
$~=pi’tc’(cYjk
@l-&2=-
Cl -C2 a3
I,, = 13, *..
‘...’
(14)
are the integrals of motion and are in involution. Note also that 2H=CCjIj
(15)
2
and that after the substitution $k + (qjpk - qkpj) the integrals Ii are transformed into Uhlenbeck’s integrals [ 11 ,121 for the problem of geodesic flow on two-dimensional ellipsoid. Hence, these two problems are closely related. The essential point is that the Clebsch case can be naturally generalized to a high-dimensional situation. To this end, let us take the group G of motion of ndimensional euclidean space, the standard cozdinates l,% = -I,+ p,(j, k, m = 1, .... n) in the space $? and the standard Poisson brackets. In this case we have
C
2H= j
.
al%and c,
(16) satisfy the constraints
(9) 157
Volume
80A, number
UJ$ (Cj -
2,3
PHYSICS
CJJ +a;&-cm)ta,f(c,,,
-c$=O
.(17)
The quantities
zj=pTt;
(CY-U.&l%, crj-(Yk=ajlcl(cj-Ck) (18)
are constants of motion in involution. This problem, like the previous one, is related to the problem of the geodesics on an (n - l)-dimensional ellipsoid, studied in detail by Moser [ 11,121. The similar quantities for the Steklov case are of the form
9 1 = (a1 - a2)-11f2 + P2](Ql ~ a&4
+ (a 1 - Q-l + (a1 -
1:3 + 2~~223
@;I
-
(19)
They are constants of motion in involution. It is worth noticing that the quantum analogues of Ik and ,Cm commute with each other. (3) It should be noticed that the Lie algebra E(3) is not semi-simple, and its invariant metric is degenerate; consequently, the equations of motion (5) are of no Lax form. Nontheless, for the n-dimensional generalization of the Clebsch case we can indicate a L-Mpair associated, however, with the Lie algebra GL(n, R). The reason for this is that in the Clebsch case the momenta pi appear only in combinations Pjpk which together with the $k generate an n2dimensional Lie algebra of a Lie group (non-semisimple) that is obtained by contraction of GL(n, R). Proposition 2. The equations
of motion for the ndimensional Clebsch case are equivalent to the Lax equation with the spectral parameter X:
i = [L,M]
.
(20)
Here L=LOtXA-h-lP,
Pjk = I_Ipipk )
M=-(MOtXC),
(21)
A = diag(al , .... a,) ) c = diag(c1, .*., c,). (22)
It should be noticed that 2H=
tr(-$L”lMo
t PC,) +
const;
aik =~-~-~~. J
158
(23)
LETTERS
24 November
1980
The constants of motion are obtained by expanding the quantities I,,(X) = tr(Ln (A)) in powers of h and the algebraic curve related to this problem is given by equation det@f - L(h)) = 0 .
(24)
It should be noticed that by the substitution
the system indicated above is transformed into a system that was studied in the problem of geodesics on a (IZ - I)-dimensional ellipsoid by Moser [ 11 ,121. The above substitution transforms the L-M pair (2 1) into the L-M pair of ref. [ 121 in which, however, the spectral parameter h was not introduced. Not also that at /J = 0 we obtain the known Manakov’s L-M pair [ 131 for pz-dimensional rigid body. 1 would like to thank S.P. Novikov and V.L. Go10 who directed my attention to the problem of motion of a rigid body in ideal fluid and indicated the papers on the subject. The discussions of this problem at S.P. Novikov’s seminar at Moscow University were illuminating and helpful. References
111 G. Kirchhoff,
Vorlesungen iiber mathematische Physik, 1, Mechanik (Teubner, Leipzig, 1876). 121 L. Dirichlet, Monatsber. Berliner Acad. 30 (1852) 10. [31 A. Clebsch, J. Reine, Angew. Math. 52 (1856) 103. [41 W. Thomson and P. Tait, Treatise on natural philosophy, vol. 1 (Oxford, 1867). [51 G. Kirchhoff and J. Reine, Angew. Math. 71 (1870) 237. (61 A. Clebsch, Math. Ann. 3 (1871) 238. [71 H. Weber, Math. Ann. 14, 2 (1878) 173. [81 V. Steklov, Math. Ann. 42 (1893) 273. Reports Kharkov Math. Sot. ser. 2, 4, [91 A. Liapunoff, l-2 (1893) 81 (in Russian); Gesammelte Werke, vol. 1 (Moscow, 1954) p. 320-324 (in Russian). [lOI G.V. Kolosoff, Compt. Rend. Acad. Sci. 169 (1919) 685. [Ill J. Moser, Various aspects of integrable Hamiltonian systems, to appear in: Proc. CIME Conference, held in Bressanone, Italy, June 1978. I121 J. Moser, Geometry of quadric and spectral theory, Courant Institute preprint, 1979. [I31 S.V. Manakov, Funct. Anal. Appl. 10,4 (1976) 93.
80A, number
2,3
PHYSICS
A FEW REMARKS ABOUT INTEGRABILITY
24 November
LETTERS
1980
OF THE EQUATIONS
OF MOTION OF A RIGID BODY IN IDEAL FLUID A.M. PERELOMOV Institute of Theoretical and Experimental Physics, 1 I7259 Moscow, USSR Received
13 August
1980
We demonstrate that the problem of motion of a rigid body in an ideal fluid for a nondegenerate case is completely integrable by means of the quadratic constants of motion for the Clebsch and Steklov cases only. We construct a Lax pair
for a high-dimensional generalization of the Clebsch case.
The classical problem of motion of a rigid body in an ideal fluid (MRBIF) was the subject of intensive study during the last and the beginning of the present century (see e.g. [l-lo]). Recently, S.P. Novikov has noticed that within the framework of Lie groups this problem could be so recast as to have the meaning of finding geodesics for a right-invariant Riemannian metric on the six parameter Lie group E(3) of motions of the three-dimensional euclidean space. Starting from this observation he indicated the possible links of the problem with the SO(4)generalization of the Euler top, and considered the contraction of the group SO(4) onto the group E(3) for which the Euler equations of the SO(4) top are transformed into equations for the so-called Clebsch case of MRBIF. It is worth noticing that the Manakov L-M pair of the S0(4)-top, as was demonstrated by S.P. Novikov, diverges in the limit of the deformation, though the constants of motion for the S0(4)-top are transformed into the constants for the Clebsch case. In this paper, we construct an L-M pair for a n-dimensional generalization of the Clebsch case which is closely related to the Moser [ 121 L-M pair for geodesic flow on (n - 1) dimensional ellipsoid. It is worthwhile to notice that our L-M pair is associated with the Lie algebra of the group SL(n, R) (see below). We demonstrate also that the MRBIF problem for nondegenerate case is completely integrable by means of the quadratic integrals of motion only for the Clebsch [6] and Steklov [8] cases. 156
(1) Let us give the group-theoretical description of the equations MRBIF. Let G = E(3) be the group of motions of the three-dimensional euclidean space R3. Notice that E(3) is generated by rotations and translations of R3. Let 9 be its Lie algebra with the standard basis xi,yk,j, k = 1,2,3: txj, +]
= EjkmXm,
bj?j,&l
[xj,Ykl
= fjkmym 3
=0 .
(1)
Here Ed%is the standard completely skew-symmetric tensor. Let $, pk be coordinates of a point of g- the space of linear fupctionals on Q and 7 be the space of functions on g , belonging to the class C”. In the space 7 the commutation relations (1) generate the Poisson structure that is defined by the equations {$J lk> = Ejkm lm 9 ($9 Pk) = fj/cmPm > (pi, Pk} = 0 . (2) The function tem
HE 7 determines
I$ = apjiat = &‘, Pj) ,
the hamiltonian
ik= {H,lk}.
sys-
(3)
It is easy to see that functions 12 = PjPj ?
13 = $pi
(4)
are integrals of motion for any hamiltonian H. Considering them as some constants we,@tain the orbit R of the co-adjoint representation Ad(G) of the group G = E(3). This orbit a is the phase space of the considered system.
PHYSICS LETTERS
Volume 80A, number 2,3
Hence the dynamic equations
where
can be written as
i=Z(N*L.
(5)
+ 2bjk$pk t clkpjpk = 2(Hu + HI t Hz) , (6)
where the quadratic form (6) must be positive. In the present note we consider only the nondegenerate case, when all eigenvalues of the symmetric matrix aij are positive and differ from each other. Without any loss of generality we may still assume that the matrix aij is diagonal: aij = aidij and the matrix bij is symmetric which we can always achieve using the transformations belonging to the group of motion of euclidean space. Proposition 1. If all eigenvalues of the matrix au differ from each other, the fourth quadratic constant exists only for the Clebsch [6] and Steklov [8] cases. Let us outline the proof. Notice that the equation {H, fi = 0 where (7)
amounts to the following set of equations IHoJo)=O,
WoJll
+ V&Z01
=O,
IH1,~1}+CHo,z2}+{H2,10}=0, Vf1J21
+ W2,1,1=0
9
(8)
which generate a set of equations for the coefficients AJh B1%’ CJk After some algebra which aJ7V bJk, ‘Jk, we shall omit due to lack of space, one can convince oneself that solutions to the equations exist only for the Clebsch and Steklov cases. These cases are the following: (I) The Clebsch case [6]. The energy is of the form 2H = C (ajf t CjPf t 2bj$pj) 3 i
- cl) t aGjl(cl - c2) = 0 .(lO)
of motion is of the form
(11)
(2) Now, as was noticed by S.P. Novikov, the equations MRBIF have the form (3) with the hamiltonian quadratic in Pi and Zk:
21= Aijli’i t 2Bjk$pk t qkpjpk = 2(Io t I, t 12)
ar1(c2 - c3) t a;‘(c3 The constant
HereLEg,M=VHEQ,V:T-+‘Sisthestandard gradient map. In the present situation dim Sz = 4, so that we need only one more global integral of motion to indicate the equations of motion.
2H = a&k
24 November 1980
It is worth noticing that the equations of motion in the Clebsch case can be integrated with the help of the B-functions of two variables [7]. (II) The Steklov case [8] which was further developed by Liapunov [9] and Kolosof [lo]. Here H is again given by eq. (9); the coefficients bj, Cj and the constant of motion I are of the form bj =p(ala2a3)aJT1 t v,
~1 =p2al(a2 - a3)2 + v’ , ... , (12)
21~~(~-2/_l(a~tV)$p~)t~2((a2-a3)2tV”)p~t.... (13) Let us notice that in the Clebsch case the quantities
lyk)-lz$,
$~=pi’tc’(cYjk
@l-&2=-
Cl -C2 a3
I,, = 13, *..
‘...’
(14)
are the integrals of motion and are in involution. Note also that 2H=CCjIj
(15)
2
and that after the substitution $k + (qjpk - qkpj) the integrals Ii are transformed into Uhlenbeck’s integrals [ 11 ,121 for the problem of geodesic flow on two-dimensional ellipsoid. Hence, these two problems are closely related. The essential point is that the Clebsch case can be naturally generalized to a high-dimensional situation. To this end, let us take the group G of motion of ndimensional euclidean space, the standard cozdinates l,% = -I,+ p,(j, k, m = 1, .... n) in the space $? and the standard Poisson brackets. In this case we have
C
2H= j
.
al%and c,
(16) satisfy the constraints
(9) 157
Volume
80A, number
UJ$ (Cj -
2,3
PHYSICS
CJJ +a;&-cm)ta,f(c,,,
-c$=O
.(17)
The quantities
zj=pTt;
(CY-U.&l%, crj-(Yk=ajlcl(cj-Ck) (18)
are constants of motion in involution. This problem, like the previous one, is related to the problem of the geodesics on an (n - l)-dimensional ellipsoid, studied in detail by Moser [ 11,121. The similar quantities for the Steklov case are of the form
9 1 = (a1 - a2)-11f2 + P2](Ql ~ a&4
+ (a 1 - Q-l + (a1 -
1:3 + 2~~223
@;I
-
(19)
They are constants of motion in involution. It is worth noticing that the quantum analogues of Ik and ,Cm commute with each other. (3) It should be noticed that the Lie algebra E(3) is not semi-simple, and its invariant metric is degenerate; consequently, the equations of motion (5) are of no Lax form. Nontheless, for the n-dimensional generalization of the Clebsch case we can indicate a L-Mpair associated, however, with the Lie algebra GL(n, R). The reason for this is that in the Clebsch case the momenta pi appear only in combinations Pjpk which together with the $k generate an n2dimensional Lie algebra of a Lie group (non-semisimple) that is obtained by contraction of GL(n, R). Proposition 2. The equations
of motion for the ndimensional Clebsch case are equivalent to the Lax equation with the spectral parameter X:
i = [L,M]
.
(20)
Here L=LOtXA-h-lP,
Pjk = I_Ipipk )
M=-(MOtXC),
(21)
A = diag(al , .... a,) ) c = diag(c1, .*., c,). (22)
It should be noticed that 2H=
tr(-$L”lMo
t PC,) +
const;
aik =~-~-~~. J
158
(23)
LETTERS
24 November
1980
The constants of motion are obtained by expanding the quantities I,,(X) = tr(Ln (A)) in powers of h and the algebraic curve related to this problem is given by equation det@f - L(h)) = 0 .
(24)
It should be noticed that by the substitution
the system indicated above is transformed into a system that was studied in the problem of geodesics on a (IZ - I)-dimensional ellipsoid by Moser [ 11 ,121. The above substitution transforms the L-M pair (2 1) into the L-M pair of ref. [ 121 in which, however, the spectral parameter h was not introduced. Not also that at /J = 0 we obtain the known Manakov’s L-M pair [ 131 for pz-dimensional rigid body. 1 would like to thank S.P. Novikov and V.L. Go10 who directed my attention to the problem of motion of a rigid body in ideal fluid and indicated the papers on the subject. The discussions of this problem at S.P. Novikov’s seminar at Moscow University were illuminating and helpful. References
111 G. Kirchhoff,
Vorlesungen iiber mathematische Physik, 1, Mechanik (Teubner, Leipzig, 1876). 121 L. Dirichlet, Monatsber. Berliner Acad. 30 (1852) 10. [31 A. Clebsch, J. Reine, Angew. Math. 52 (1856) 103. [41 W. Thomson and P. Tait, Treatise on natural philosophy, vol. 1 (Oxford, 1867). [51 G. Kirchhoff and J. Reine, Angew. Math. 71 (1870) 237. (61 A. Clebsch, Math. Ann. 3 (1871) 238. [71 H. Weber, Math. Ann. 14, 2 (1878) 173. [81 V. Steklov, Math. Ann. 42 (1893) 273. Reports Kharkov Math. Sot. ser. 2, 4, [91 A. Liapunoff, l-2 (1893) 81 (in Russian); Gesammelte Werke, vol. 1 (Moscow, 1954) p. 320-324 (in Russian). [lOI G.V. Kolosoff, Compt. Rend. Acad. Sci. 169 (1919) 685. [Ill J. Moser, Various aspects of integrable Hamiltonian systems, to appear in: Proc. CIME Conference, held in Bressanone, Italy, June 1978. I121 J. Moser, Geometry of quadric and spectral theory, Courant Institute preprint, 1979. [I31 S.V. Manakov, Funct. Anal. Appl. 10,4 (1976) 93.