Institute for Nuclear Theory, University of Washington. Seattle, WA 98195, U.S.A. and. Physics Department, Tulane University. New Orleans, LA 70118, U.S.A..
Appl. Math. Lett. Vol. 6, No. 3, pp. 55-58, 1993 Printed in Great Britain. All rights reserved
LAX
REPRESENTATION G. Institute
Copyright@
OF RIEMANN
0893-9659193 $6.00 + 0.00 1993 Pergamon Press Ltd
ELLIPSOIDS
ROSENSTEEL
for Nuclear Theory, University of Washington Seattle, WA 98195, U.S.A. and Physics Department,
Tulane University
New Orleans, LA 70118, U.S.A. (Received
and accepted November 1992)
Abstract-The fluid dynamical equations of.motion for rotating Riemann ellipsoids are presented as a Lax equation, X = [F,X]. Ftiemann ellipsoids model rotating galaxies, stars, and atomic nuclei. The elements of the Lax pair X-F are matrices in the symplectic Lie algebra sp(3, R). The matrix X is formed from the inertia and virial momentum tensors; F is composed from the potential energy, angular velocity, and pressure tensors. The Kelvin circulation is conserved by the Lax system. Riemann ellipsoidal solutions are given by symplectic isospectral deformations.
1. RIEMANN’S
MODEL
A Riemann ellipsoid is a self-gravitating fluid with an ellipsoidal boundary for which the velocity field is a linear function of the Cartesian position coordinates [l]. Riemann ellipsoids model rotating stars, galaxies, and gaseous plasmas [2-6]. If the attractive gravitational energy is replaced by the sum of the repulsive Coulomb self-energy and the attractive surface tension energy, then Riemann’s theory describes atomic nuclei in the rapidly rotating classical liquid drop regime [7-91. The purpose of this letter is to show that the Riemann equations of motion may be represented as a finite-dimensional Lax equation [lo]. Conservation of the Kelvin circulation is transparent in this elegant representation. The first objective is to establish notation and review briefly the standard formulation of the kinematics and dynamics of Riemann ellipsoids. The size, deformation, and orientation of an ellipsoid with constant density p are characterized uniquely by the inertia tensor,
Q;=/
(1)
pXiXjd3X,
where X denotes Cartesian position coordinates with respect to the ellipsoid’s inertial center of mass frame. With respect to the rotating principal axis frame, the inertia tensor is, by definition, diagonal.
where x denotes the position vector with respect to the principal axis frame, ai are the semi-axis lengths of the inertia ellipsoid, M = pv is the total mass, and v = 47raiasaa/3 is the ellipsoidal volume. In this letter, an inertial laboratory center of mass frame tensor is distinguished from the corresponding body fixed tensor by a superscript L. A time-dependent orthogonal matrix R E SO(3) that diagonalizes the inertia tensor, QL = tR . Q . R, defines the rotation from the inertial frame to the principal axis frame, x = R.X. I thank the Institute for Nuclear Theory at the University of Washington for their hospitality and the Department of Energy for partial support during this work’s completion. This work was supported in part by the National Science Foundation under Grant PI-IY-9212231.
Tywet 55
by dM’J?#
G. R~SENSTEEL
56
The angular velocity vector w of the rotating principal axis frame with respect to the inertial laboratory frame is defined by an antisymmetric matrix R E SO(~) in the rotation group Lie algebra, R = R ’ tR, f&j = EijkWke For a Riemann ellipsoid, the inertial frame velocity field U is assumed to be a linear function of the Cartesian position coordinates. A linear velocity field is characterized uniquely by the virial momentum tensor, N& = or, with respect to the principal axis frame,
J
pXiUjd3X,
(3)
where u = R.U is the projection of the velocity field onto the principal axis frame, and NL = tR . N . R. The virial momentum tensor determines the axes vibrations, Nii = (M/10) da:/dt, the body-fixed projection of the angular momentum Lk = cijkNij, and the projected Kelvin circulation Ck = eijk(Q-1'2 *N *Q1j2)ije The body-fixed kinetic tensor Tij = ~puiujd3z is the projection of the corresponding inertial frame value, TL = tR - T . R. For a linear velocity field, the kinetic tensor equals [ll] TztN.Q-l.N.
(5)
The attractive gravitational potential self-energy V = V(al, ~2, ~3) is a smooth function of the axes lengths. Define the potential tensor in the rotating frame by the diagonal matrix,
The inertia and potential tensors are simultaneously diagonalized in the principal axis frame and, therefore, commute in the inertial frame. To impose a constraint to constant volume, define the pressure tensor II = pv to be the product of the hydrostatic pressure p times the volume. The Kiemann dynamical equations are
I~=~,N]+T+w+II,
(7)
or, with respect to the inertial frame, Q” = NL + tNL, fiL=TL+WL+II.
(8)
The squared length of the Kelvin circulation vector, C2=C.C=Tr(tN.Q-1.N.Q-N2),
(9)
is conserved by the Riemann equations of motion, c2 = 0. It is convenient to put the Riemann equations in dimensionless form by choosing a length unit RJJ, an inertia unit Is = Mg/S, and an energy unit Vi = (3/S) GM2/R.+ Henceforth, an axis length ai is in units of Rc, an inertia tensor Q in units of Is, energies V, W, T, and II in units of Vo, virial momentum terms N, L, C in units of (loV~)‘/~, time in units of (&/VO)~/~, and angular velocity Sl in units of inverse time (Vo/lf~)U2. The equations of this section are unaffected by the transformation to dimensionless form except that equation (2) becomes Qij = of&j, and the diagonal virial momentum components are now Nii = (l/2) $(a:). This finishes our synopsis of the classical theory of Riemann ellipsoids; a detailed exposition of the theory is presented in [l].
Riemann
2. LAX
ellipsoids
57
REPRESENTATION
The dynamical system may be expressed in the compact and elegant Lax matrix form. Define the 6 x 6 real matrices XL and FL:
_;QNLL), FL=((WL+II;.(Qy 6>.
(10)
The Riemann ellipsoid dynamical equations (8) are equivalent to the single Lax equation,
a%’=
(11)
[FL,XL].
The matrices of the principal axis Lax pair X-F are rotations of the corresponding inertial frame quantities, X = R.X L . t R and F = R + R. FL . t R, where the rotation matrix R = diag (R, R) E SO(3) c G1(6,R), and 52 = diag(S1,R) E SO(~) C g1(6,R),
x=(; -;QN)’
F=((W+:).Q-l
;).
(12)
With respect to the body-fixed frame, the Lax equation corresponding to the Riemann dynamical system (7) is X = [F,X]. (13) The equivalence of the Lax equations (11) and (13) to the Riemann equations of motion, equai tions (8) and (7), is established by a direct computation. Thus, the matrices XL-FL and X-F constitute Lax pairs [lo, 12-141. Equilibrium Riemann ellipsoids are solutions for which the body-fixed inertia, virial momentum, and angular velocity tensors are constant in time. Hence, an equilibium solution corresponds to X = 0, or the Lax pair X and F commute, X is an equilibrium solution iff [F, X] = 0.
(14)
This is a very clean statement of the Riemann equilibrium conditions; the solutions have been studied exhaustively [11. Conservation laws for Lax equations are provided by the traces of powers of X [lo], Ip = ; n
(XL)P = ; Tr (X)“.
(15)
From the Lax equation, it is evident that every Ip is a constant of the motion,
ip=Tr
(xp-l.2 >=Tr(xp-l.[F,x])
=o.
For p = 2, this conserved quantity equals the negative of the squared length of the Kelvin circulation vector, 12 = Tr (N2 - T . Q) = -C2. The conventional proof of conservation of circulation from the dynamical equations (7) or (8) is tedious and unenlightening. The coefficients of the characteristic equation for X are combinations of the traces of powers of X itself. Hence, the trace powers determine the eigenvalues of X, which consequently are invariants themselves. Conversely, each trace of a power of X is a function of the eigenvalues of X. Since there are at most six eigenvalues, there are at most six independent invariants. In turns out that there are no additional conservation laws independent of the Kelvin circulation. The reason for this is the special form of the matrices forming the Lax pair X-F: (l)T=%Q--‘ON, and (2) X, F E sp(3,R). The special Riemann linear form for the kinetic tensor implies that the kernal of X is at least three-dimensional. To prove this, consider the column vector o E R6 given by v = (~1, Q), where ~1 E R3 is arbitrary and ‘us = Q-’ . N.q E R 3. It is easily verified that X.v = 0. Since vi is AK
6:3-E
G. ROSENSTEEL
58
arbitrary, the kernal is at least three-dimensional. Hence, the number of independent eigenvalues and invariants is at most 6 - 3 = 3. The symplectic Lie algebra sp(3,R) is defined by the 6 x 6 real matrices that preserve an antisymmetric form J: sp(3,R)
= {X E gZ(6,R) 1 tXJ
+ JX
= 0} ,
={x=(; _s)J B=tB,
ifJ=
C=tC,
(T1
i)}.
(17)
Evidently, X and F are elements of the symplectic Lie algebra. The eigenvalues of an element of the symplectic Lie algebra are pure imaginary and occur in pairs fq [15]. This eigenvalue pairing prohibits an odd dimension for the kernal of X. Therefore, the dimension of the kernal of X equals four, and the two pure imaginary eigenvalues of X are fiC. Thus, there is only one independent invariant among the I,. Suppose that X(t) is a solution to the symplectic dynamical equation, _% = [F, X], corresponding to the initial condition X = X0. If g(t) E G1(6, R) is a smooth curve satisfying the matrix differential equation 4 = F . g with the initial condition g = I, then the solution to the Lax equation is just the isospectral deformation [13], X(t) = gxog-1.
(1%
= _g-QgThis is proven using the identity 4$ ‘. It is now clear at a deeper level why each trace of a power of X(t) is a conserved quantity. The similarity transformation (19) preserves the eigenvalues of the matrix X0. It will now be shown that the matrix g must be an element of the symplectic group, EG1(6,R)Itg.J.g=J g = { (s f) Since F is an element of the symplectic Lie algebra, Sp(3,R)
0 = ‘g (tFJ
=
+ JF) g = t(Fg)Jg
But, if g equals the identity matrix is a curve in the symplectic group, symplectic isospectral deformations for the inertial frame equations (8)
+ tgJFg
= $Jg
. >
+ tgJd = g (tgJg).
(1%
PO)
at the initial time, then tg. J-g = J for all times. Hence, g(t) and the solutions to the Riemann dynamical system (7) are (18) of the initial state. A similar conclusion may be drawn and the Lax pair XL-FL. REFERENCES
1. S. Chandrasekhar, Ellipsoidal Figures of Equilibrium, Yale University Press, New Haven, CT, (1969). 2. N.R. Lebovitz, Rotating fluid masses, Ann. Rev. A&r. Ap. 5, 465 (1967). 3. N.R. Lebovitz, On the fission theory of binary stars. IV. Exact solutions in polynomial spaces, Ap. J. 284, 364 (1984). 4. N.R. Lebovitz, Bifurcation and unfolding in systems with two timescales, In Nonlinear Astrophysical Fluid Dynamics, (Edited by J.R. Buchler and S.T. Gottesman), Ann. New York Acad. Sci., New York, (1990). 5. R.H. Durison, R.A. Gingold, J.E. Tohline, and A.P. Boss, Dynamic fission instabilities in rapidly rotating n = 3/2 polytropes: A comparison of results from finite-difference and smooth particle hydrodynamics codes, Ap. J. 305, 281 (1986). 6. F.J. Dyson, Dynamics of a spinning gas cloud, J. Math. Mech. 18, 91 (1968). 7. S. Cohen, F. Plasil, and W.J. Swiatecki, Equilibrium configurations of rotating charged or gravitating liquid masses with surface tension. II, Ann. Phys. (N. Y.) 82, 557 (1974). 8. J.D. Garrett, Shape and pair correlations in rotating nuclei, In The Response of Nuclei under Extreme Conditions, (Edited by R.A. Broglia and G.F. Bertsch), Plenum, London, (1988). 9. G. Rosensteel, Rapidly rotating nuclei as Riemann ellipsoids, Ann. Phys. (N. Y.) 186, 230 (1988). 10. P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math. 21, 467 (1968). 11. R.Y. Cusson, A study of collective motion. (I). Rigid, liquid and related rotations, Nud. Phys. A114, 289 (1968). 12. H. Flaschka, The Toda lattice. II. Existence of integrals, Phys. Rev B9, 1924 (1974). 13. Ju Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16, 197 (1975). 14. A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Birkhsuser, Base& (1990). 15. R. Abraham and J.E. Marsden, Foundations of Mechanics, Benjamin, Reading, MA, (1978).
