AbstractThe principle of form-invariance upon relativisation of the Hamiltonian structures of fluids and plasmas is an empirical observation that the (special) relativistic version of a given non-relativistic Hamiltonian system of classical Physica (a) is also a Hamiltonian system, and moreover, (b) has the same noncanonical Hamiltonian structure as its nonrelativistic counterpart. It is shown that one-dimensional gas dynamics of a polytropic gas violates this principle

Kupershmidt, B.A.

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Form-invariance upon relativisation of the Hamiltonian structures of fluids is not universal 

Computers Math. Applie. Vol. 21, No. 5, pp. 95-99, 1991 0097-4943/91 $3.00 + 0.00 
Printed in Great Britain. All rights reserved Copyright(~ 1991 Pergamon Press plc 
FORM- INVARIANCE UPON RELAT IV ISAT ION 
OF  THE HAMILTONIAN STRUCTURES OF  FLU IDS 
IS  NOT UNIVERSAL  
B. A. KUPER.SHMIDT 
The University of Tennessee Space Institute, TuUahoma, TN 37388 USA 
Abst ract - -The  principle of form-invariance upon relativisation of the Hamiltonian structures of 
fluids and plasmas is an empirical observation that the (special) relativistic version of a given non- 
relativistic Hamiltonian system of classical Physics (a) is also a Hamiltonian system, and moreover, 
(b) has the same noncanonical Hamiltonian structure as its nonrelativistic counterpart. It is shown 
that one-dimensional gas dynamics of a polytropic gas violates this principle. 
INTRODUCTION 
Continuous dynamics of nonrelativistic conservative systems is often governed by the noncanon- 
ieal Hamiltonian formalism (see, e.g., bibliography in [1,2]). Experimental observations show 
that the (special) relativisation of such systems preserves their Hamiltonian character and even 
retains the nonrelativistic form of their Hamiltonian structures. This fact has been by now estab- 
lished for all the basic systems: adiabatic fluid dynamics in flat [3-5] and in curved [6, 1] space, 
nonabelian hydrodynamics [7], Yang-Mills-Vlasov Plasma [7], adiabatic Magnetohydrodynamics 
(MHD) [8], and anisotropic MHD [9]. This Hamiltonian rigidity with respect o special relativi- 
sation has been found to apply not only to individual dynamical systems but also to nontrivial 
morphisms between these systems, such as the reduction with respect o the direct sum oforthog- 
onal groups [10]. It appears that the principle of Hamiltonian form-invariance under relativisation 
is universally applicable, but this is not to be. Below I show that relativisation of one-dimensional 
dynamics of arbitrary nonrelativistic polytropic gas destroys its second Hamiltonian structure. 
NONRELATIVISTIC ONE-DIMENSIONAL GAS DYNAMICS 
The motion equations of nonrelativistic one-dimensional barotropic fluid (or gas) dynamics are 
-Mr  = (Mv + P)x (1.1) 
-p ,  = (pv)  (1.2) 
v = M/p  (2) 
where M is the momentum density, p is the fluid density, v is the velocity, subscripts (except he 
subscript zero later on) denote partial derivatives, and P = P(p)  is the pressure, 
P = p2ep (3) 
where e = e(p) is the specific internal energy. With tile Ilamiltonian being the energy density 
M 2 
H (4) 
I thank Y. Nutku for making his preprint available to me. This work was partially supported by the National 
Science Foundation. 
Typeset by M A/~-TF_~ 
95 
96 B.A. KUPERSHMIDT 
the system (1), (2) has the standard Hamiltonian structure in the (M,p)-variables (with 0 = 
0/0x): 
B 1 (MO+OMPo0) 
= op (5) 
which means two things: first, that the system (1), (2) can be written in the form 
M ['6H/6M (6) 
where 8H/8(.) is the variational (in our case, partial) derivative of H (4) with respect o the 
variable (-); and second, that the matrix (5) is Hamiltonian (see, e.g., [12]). 
The I-Iamiltonian representation (6) is available for arbitrary pressure function P(p) or, what is 
the same, arbitrary specific internal energy function e(p). In the case when the gas is polytropic, 
p = ~pl/~, /~ and 0 ¢ a are constants. (7) 
Nutku observed [11] that there exists also another Hamiltonian structure of the dynamical system 
(1), (2): 
B2 ( (Mv + P)O + O(Mv + P) pro + aOM'~ 
= cgpv + c~MO ot(pO + Op) ] (S) 
with the Hamiltonian function 
H = H1 = M. (9) 
It is this structure that will be shown to be impossible to deform with the parameter c -2, c being 
the speed of light. Before proceeding further, a few remarks are in order. 
REMARK 1. Nutku shows that for the polytropic gas there exists not only the second tIamiltonian 
structure B 2 but also a third Hamiltonian B 3, with the Hamiltonian 
H=Ho=p (10) 
and that the third Hamiltonian structure services, in addition to polytropic gases (7), also the 
case  
P = ~gnp (I1) 
REMARK 2. For the case c~ = 1/2, the gas dynamical equations (1), (2) are isomorphic to the 
classical one-dimensional long-wave quations for which the three Hamiltonian structures were 
found earlier in [13] in the general context of deformations of Hamiltonian structures. 
REMARK 3. For the case of a free fluid, i.e., when the pressure P vanishes [fl = 0 in (7)], the 
multi-Hamiltonian representations were given in [14,15] (in more general relativistic and super 
cases). 
REMARK 4. To check the Hamiltonian property of a given matrix B one has to check the following 
identity for arbitrary column-vectors X and Y (equation (2.17) in [15]): 
B($[YtB(X)]) = D[B(Y)][B(X)] - D[B(X)][B(Y)] (12) 
where 6(-) is the column-vector f functional derivatives of (.) and D(.) is the Fr~chet derivative 
of (.). For the case 
(aO+Oa fO.-t-Og) (13) 
B= \Of+gO bO+Ob 
Form-invariance upon relativisation 97 
where a, b, f ,  g are arbitrary functions of u and h, the equation (12) reduces to the following 
system (equations VI (1.15) in [16]): 
2bgh + (f + g)gu = 2abu + (f  + g)bh (14.1) 
2afu + ( f  + g)fh = 2bah + ( f  + g)au (14.2) 
gh(au - - fh )=ah(gu  -- bh) (14.3) 
bu(au -- fh) = fu(gu -- bh) (14.4) 
ahb,, = ahYu (14.5) 
Applying formulae (14) to the matrix (8) we find that B 2 is Hamiltonian if and only if the gas 
is polytropic: 
apPp = P. (15) 
Similarly one shows that a third Hamiltonian structure B a exists if and only if P is given either 
by (7) or (11). 
REMARK 5. The function g and b (13) are taken in (8) to be 
g=aM,  b=ap (16) 
under the assumption that the conserved ensity p/a  of the dynamical equations (1), (2) is the 
momentum of the Hamiltonian matrix B 2, i.e., it produces the uniform velocity-one shift in the 
z direction. A slightly more general choice would be to take 
b = ap + p, p a constant, (17) 
but one may check that no advantage is gained by allowing nonzero p in (17). 
REMARK 6. For the case (16), the system (14) collapses to the single equation (14.2), which 
takes the form 
2afM + ( f  + aM)( fp  - aM) = 2apap (18) 
Allowing v and P in (8) to be arbitrary functions of M and p, equation (18) becomes 
Mv M v -a - -+ P +pvp v -a  =PM v+a +2aPp,  
P 
(19) 
which reduces to equation (15) when v = M/p and PM = O. 
REMARK 7. The matrices B 1 (5) and B 2 (8), (2), (7) are not only separately Hamiltonian but 
form a Hamiltonian pair, i.e., their arbitrary linear combination (with constant coefficients) is 
Hamiltonian as well. Once again allowing v and P in (8) to be arbitrary functions of M and p 
(anticipating relativistic complications later on), we find that the Hamiltonian matrices B 1 (5) 
and B ~ (8) form a Hamiltonian pair iff the equation (19) is satisfied for v + A, for every constant 
A, which in turn results in the equations 
MVM + PVv = PM (20) 
(MPM + pPp) = pPv~ 
Again, when v = M/p,  one at once recovers equation (15). 
(21) 
98 B.A.  KUPERSHMIDT 
RELAT IV IST IC  ONE-DIMENSIONAL GAS DYNAMICS 
Relativistic gas dynamical equations have the same form (I) but with relations (2) and (3) 
being replaced by [4] 
M = Ov (22) 
0 = p?w (23) 
= (1 - (24) 
w = 1 + (eo + P/po)c -2 (25) 
p = (26) 
P= P(Po)= p~ep o, e=e(po) .  (27) 
As shown in [4,5], the relativistic system (1), (22)-(27) possesses the same Hamiltonian form (5), 
with the Hamiltonian 
H = H2 = c2(0 - p) - P (28) 
To show that second Hazniltonian form (8) does not survive relativisation we need to show that 
the defining relation (19) is not satisfied whatever P(Po) is provided P ¢ 0. (When P -- 0, one 
gets w = 1 and since, in general, 
0 = (p~w 2+ M21c2) z/2, (29) 
for the case w = 1 one obtains 0 = p(1 + M2/p2c2) 1/2. Thus, 
v = - -  1 + (ao) p p2C2 J
is of homogeneous degree zero in M and p. Therefore, since P = 0, both equations (20) and (21) 
are satisfied.) Now suppose P ¢ 0. Since 
M = .T2vpow, P = "YPo, (31) 
we can re-express the partial derivatives operators OM and Op entering into the equation (19) in 
terms of O~ and Opo, as follows: 
OM = A (RlOpo "-I- R2Ov), Of, = A (QlOpo + O2Ov) (32) 
= [    vp0(p0W),o - ¢ o, (33) 
Rl=%p0,  R2=-7 ,  Ql=- (T2v)~pow,  Q2=72v(pow)po. (34) 
Since equation (19) is linear in the (M, p)-partial derivatives, dividing out by A we can rewrite 
equation (19) in the form 
MR2 (v - o~ M (v  - M 
Since MR" + pQ2 = 7avp~wpo, equation (35) becomes 
7av2P~Wpo( 1 - aTw) - 27P = Ppo [%vp0(1 + aTw) - 23(72v)~pow] . (36) 
Form-invariance upon relativisation 99 
In the l imit  c -2  ---* O, one has 7 ~ 1, % --+ O, w --~ 1, wpo ~ O, and equat ion (36) yields 
P = PpoaP0, which is exact ly  equat ion (15). Using the formulae 
wpo = PpoPolC -2 (37) 
and 
P = PpoaPO (38) 
in equat ion (36), we can reduce equat ion (36), by div iding it out by PpoPo (PpoPO = a - lP  ~ 0), 
to the equat ion 
73v2c-2(1 - c~Tw) - 27a = %v( l  + aVw ) - 2aw (72v)v. (39) 
Since 
equat ion (39) becomes 
% = 73vc -2, (40) 
7 = w (72 +74v2c-2) ,  (41) 
or 
7 = w72 (1 + 72v2c -2)  = w74 (42) 
which is obviously false. This  contradict ion shows that  the equat ions (36) and (19) are not 
satisfied. Therefore,  the relat iv ist ic  matr ix  B 2 (8) is not Hami l ton ian except when P = 0. 
REFERENCES 
1. D.D. Holm, Hamiltonian Formalism for General-Relativistic Adiabatic Fluids, Physica 17D, 1 (1985). 
2. D.D. Holm, J.E. Marsden, T. Ratiu, and A. Weinstein, Nonlinear Stability of Fluid and Plasma Equilibria, 
Phys. Rcp. 123,  1 (1985). 
3. I. Bialynicki-Birada and Z. Iwinski, Canonical Formulation of Relativistic Hydrodynamics, Rcp. Math. Phys. 
4, 139 (19"73). 
4. D.D. Holm and B.A. Kupershmidt, Relativistic Fluid Dynamics as a Handltouian System, Phys. Left. 101A, 
23 (1984). 
5. D.D. Holm and B.A. Kupershmidt, Lyapunov Stability of Relativistic Fluids and Plasmas, Phys. Fluids 29, 
49 (1986). 
6. D. Bao, J.E. Marsden, and R. Walton, The Hamiltonian Structure of General Relativistic Perfect Fluids, 
Comm. Math. Phys. 99,319 (1985). 
7. D.D. Holm and B.A. Kupershmidt, Relativistic Chromohydrodynaruics and Yang-Mills Vlasov Plasma, 
Phys. Left. 105A, 225 (1984). 
8. D.D. Holm and B.A. Kupershmidt, Relativistic Magnetohydrodynamics as a Hamiltonian System, Compt. 
Rend. (Paris) 300 (Ser. I), 153 (1985). 
9. D.D. Holm and B.A. Kupershmidt, Hamiltonian Theory of Relativistic Magnetohydrodynamics with Aniso- 
tropic Pressure, Phys. Fluids 29, 3889 (1986). 
10. B.A. Kupershmidt, Cylindrically invariant flows in relativistic fluid dynamics, Comp. Math. AppL 13, 881 
(1987). 
11. Y. Nutku, On a New Class o] Completely Integrable Nonlinear Wave Equations II. Multi-Hamiltonian 
Structure, Bosphorus University preprint, (1986). 
12. D.D. Holm and B.A. Kupershrnidt, Poisson Brackets and Clebsch Representations for Magnetohydrody- 
narnics, Multifluid Plasma, and Elasticity, Physica 6D, 347 (1983). 
13. B.A. Kupershmidt, DeJormations oJ Hamiltonian Structures, M.I.T. preprint, (1978). 
14. B.A. Kupershmidt, Supersymmetric Fluid in a Free One-Dimensional Motion, Left. Nuovo Cim. 40, 476 
(1984). 
15. B.A. Kupershmidt, Odd and Even Poisson Brackets in Dynamical Systems, Left. Math. Phys. 9,323 (1985). 
16. B.A. Kupershmidt, Discrete Laz Equations and Differential-Di.~erence Calculus, Asterisque, Paris, (1985). 


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Form-invariance upon relativisation of the Hamiltonian structures of fluids is not universal

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