% THIS IS THE STANDARD AMS-TEX (AMSPPT-STYLE)
% THE LIST OF REFERENCES CONTAINS REFS IN RUSSIAN, WHICH ARE CITED ON BOTH
% RUSSIAN AND ENGLISH: TO PERFORM THE COMPLETE COMPILATION ONE SHOULD HAVE
% CYRILLIC FONTS OF WASHINGTON UNIVERSITY FAMILY IN TEX-ACCESSIBLE FILES
% wncyi8.tfm and wncyr8.tfm.
\input amstex
\magnification=1200
\documentstyle{amsppt}
\NoBlackBoxes
\font\cyr=wncyr8
\font\cyi=wncyi8
\leftheadtext{Denis \ Juriev}
\rightheadtext{Nonhamiltonian \ interaction \ of \ hamiltonian \ systems}
\define\sLtwo{\operatorname{sl}(2,\Bbb C)}
\define\sUtwo{\operatorname{su}(2)}
\define\tr{\operatorname{tr}}
\define\deter{\operatorname{det}}
\define\real{\operatorname{Re}}
\document
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\eightpoint MAY \tenpoint 1994
\
\
\topmatter
\title
Topics in nonhamiltonian interaction of hamiltonian dynamic systems
\endtitle
\author Denis Juriev \endauthor
\endtopmatter
\
\
\
\head I. Introduction\endhead
The evolution of hamiltonian systems (defined by Poisson brackets and
hamiltonians) attracts a lot of attention (see f.e. [1]). Many of such systems
are associated with Lie algebras, in this case Poisson brackets (the
so--called Lie--Berezin brackets) have a linear form and are restored from the
commutator in the Lie algebra [2] (it should be marked that nonlinear Poisson
brackets are also of interest [3]). The introducing of external fields (which
interact with the system noncanonically, in general) transforms Lie algebras
into isocommutator algebras (Lie isoalgebras) [4]. If the external field
possesses a symmetry governed by the Lie algebra $\frak g$, then it is rather
convenient to describe the systems by Lie $\frak g$--bunches [5]. By use of
them one may construct various hamiltonian systems noncanonically controlled
by external fields, which dynamics maybe also hamiltonian, in particular. Such
picture is rather realistic if one supposes that a counteraction of the
system on the external fields maybe neglected. Nevertheless, it is not so in
many cases. Therefore, it is interesting to consider a dynamics of two
hamiltonian systems, which interact with each other noncanonically. A
convenient algebraic structure to describe such dynamics seems to be of the
following definition.
\definition{Definition 1} The pair $(V_1,V_2)$ of linear spaces is called {\it
an isotopic pair\/} iff there are defined two mappings
$m_1:V_2\otimes\bigwedge^2V_1\mapsto V_1$ and
$m_2:V_1\otimes\bigwedge^2V_2\mapsto V_2$ such that the mappings $(X,Y)\mapsto
[X,Y]_A=m_1(A,X,Y)$ ($X,Y\in V_1$, $A\in V_2$) and $(A,B)\mapsto
[A,B]_X=m_2(X,A,B)$ ($A,B\in V_2$, $X\in V_1$) obey the Jacobi identity for
all values of a subscript parameter (such operations will be called {\it
isocommutators\/} and the subscript parameters will be called {\it
isotopies\/}) and are compatible to each other, i.e. the identities
$$\align
[X,Y]_{[A,B]_Z}=&\tfrac12([[X,Z]_A,Y]_B+[[X,Y]_A,Z]_B+[[Z,Y]_A,X]_B-\\
&[[X,Z]_B,Y]_A-[[X,Y]_B,Z]_A-[[Z,Y]_B,X]_A)\endalign$$
and
$$\align
[A,B]_{[X,Y]_C}=&\tfrac12([[A,C]_X,B]_Y+[[A,B]_X,C]_Y+[[C,B]_X,A]_Y-\\
&[[A,C]_Y,B]_X-[[A,B]_Y,C]_X-[[C,B]_Y,A]_X)\endalign$$
($X,Y,Z\in V_1$,
$A,B,C\in V_2$) hold.
\enddefinition
This definition is a result of an axiomatization of the following
construction. Let's consider an associative algebra $\Cal A$ (f.e. any matrix
algebra) and two linear subspaces $V_1$ and $V_2$ in it such that $V_1$ is
closed under the isocommutators $(X,Y)\mapsto [X,Y]_A=XAY-YAX$ with isotopies
$A$ from $V_2$, whereas $V_2$ is closed under the isocommutators $(A,B)\mapsto
[A,B]_X=AXB-BXA$ with isotopies $X$ from $V_1$. If a family (linear space)
$V_1$ of operators forms an isocommutator algebra with a family (linear space)
$V_2$ of admissible isotopies then $(V_1,V_2)$ is an isotopic pair via the
so--called "isotopic duality" [5]\footnote{ \ The "isotopic duality" maybe
speculatively regarded as a certain "mathematical manifestation" of the Third
Law of Classical Dynamics.}.
It should be mentioned that the isocommutators define families of Poisson
brackets $\{\cdot,\cdot\}_A$ and $\{\cdot,\cdot\}_X$ ($A\in V_2$, $X\in V_1$)
in the spaces $S^{\cdot}(V_1)$ and $S^{\cdot}(V_2)$, respectively.
\definition{Definition 2} Let's consider two elements $\Cal H_1$ and $\Cal
H_2$ ("hamiltonians") in $S^{\cdot}(V_1)$ and $S^{\cdot}(V_2)$, respectively.
The equations $$\dot X_t=\{\Cal H_1,X_t\}_{A_t},\quad \dot A_t=\{\Cal
H_2,A_t\}_{X_t},$$ where $X_t\in V_1$ and $A_t\in V_2$ will be called {\it
the\/} ({\it nonlinear\/}) {\it dynamical equations associated with the
isotopic pair $(V_1,V_2)$ and "hamiltonians" $\Cal H_1$ and $\Cal H_2$\/} (it
should be marked that "hamiltonians" are not even integrals of motion in
a general situation).
\enddefinition
Let's consider several simple but crucial examples now. We shall treat the
subject purely mathematically and shall not specify its concrete physical
implications of a possible interest.
\head II. Noncanonically coupled rotators and Euler--Arnold tops \endhead
\subhead 2.1. Nonlinear integrable dynamics of noncanonically coupled rotators
\endsubhead Let's consider the Lie $\sLtwo$--bunch in $\pi_1$ (the adjoint
representation of $\sLtwo$) [5]. It is defined by the next formulas
\
\centerline{(1)$\quad$
$\aligned
[m_{-1},m_0]_{l_0}&=m_{-1}\\
[m_1,m_{-1}]_{l_0}&=0\\
[m_1,m_0]_{l_0}&=m_1
\endaligned$
$\quad$
$\aligned
[m_{-1},m_0]_{l_{-1}}&=0\\
[m_1,m_{-1}]_{l_{-1}}&=2m_{-1}\\
[m_1,m_0]_{l_{-1}}&=2m_0
\endaligned$
$\quad$
$\aligned
[m_{-1},m_0]_{l_1}&=2m_0\\
[m_1,m_{-1}]_{l_1}&=-2m_1\\
[m_1,m_0]_{l_1}&=0
\endaligned$}
\ \newline
where $l_i$ ($i=-1,0,1$) are generators of $\sLtwo$
($[l_i,l_j]=(i-j)l_{i+j}$), $m_i$ ($i=-1,0,1$) form a basis in $\pi_1$
($l_i(m_j)=(i-j)m_{i+j}$).
As it was marked in [5] the operators $l_i$ form an isocommutator Lie algebra
with respect to isotopies from $\pi_1$ via the "isotopic duality", namely
\
\centerline{(2)$\quad\qquad$
$\aligned
[l_{-1},l_0]_{m_0}&=-l_{-1}\\
[l_1,l_{-1}]_{m_0}&=0\\
[l_1,l_0]_{m_0}&=-l_1
\endaligned$
$\quad$
$\aligned
[l_{-1},l_0]_{m_{-1}}&=0\\
[l_1,l_{-1}]_{m_{-1}}&=-2l_{-1}\\
[l_1,l_0]_{m_{-1}}&=-2l_0
\endaligned$
$\quad$
$\aligned
[l_{-1},l_0]_{m_1}&=-2l_0\\
[l_1,l_{-1}]_{m_1}&=2l_1\\
[l_1,l_0]_{m_1}&=0
\endaligned$}
\
It can be easily verified that isocommutators (1) and (2) define a structure
of an isotopic pair in $\pi_1\oplus\pi_1$. Let's denote the first summand by
$\pi_1^+$ and the second summand by $\pi_1^-$. Let's also consider two fixed
elements $\Omega^{\pm}$ in $\pi_1^{\pm}$, respectively
($\Omega^+=\Omega^+_{-1}m_{-1}+\Omega^+_0m_0+\Omega^+_1m_1$,
$\Omega^-=\Omega^-_{-1}l_{-1}+\Omega^-_0l_0+\Omega^-_1l_1$), and two variables
$A$ and $B$ from $\pi_1^+$ and $\pi_1^-$ ($A=A_{-1}m_{-1}+A_0m_0+A_1m_1$ and
$B=B_{-1}l_{-1}+B_0l_0+B_1l_1$). The dynamical equations (with linear
"hamiltonians" defined by $\Omega^{\pm}$) will have the form
$$\left\{\aligned
\dot A_{-1}&=
\Omega^+_{-1}(A_0B_0-2A_1B_{-1})-\Omega^+_0A_{-1}B_0+2\Omega^+_1A_{-1}B_{-1}\\
\dot A_0&=
2\Omega^+_{-1}A_0B_1+2\Omega^+_0(A_1B_{-1}-A_{-1}B_1)+2\Omega^+_1A_0B_{-1}\\
\dot A_1&=
2\Omega^+_{-1}A_1B_1-\Omega^+_0A_1B_0+\Omega^+_1(A_0B_0-2A_{-1}B_1)
\endaligned\right.$$
$$\left\{\aligned
\dot B_{-1}&=
-\Omega^-_{-1}(B_0A_0-2B_1A_{-1})+\Omega^-_0B_{-1}A_0-2\Omega^-_1B_{-1}A_{-1}\\
\dot B_0&=
-2\Omega^-_{-1}B_0A_1-2\Omega^-_0(B_1A_{-1}-B_{-1}A_1)-2\Omega^+_1B_0A_{-1}\\
\dot B_1&=
-2\Omega^-_{-1}B_1A_1+\Omega^-_0B_1A_0-\Omega^-_1(B_0A_0-2B_{-1}A_1)
\endaligned\right.
$$
It is rather convenient to consider the compact real form of the isotopic
pair $(\pi_1^+,\pi_1^-)$. Let's denote
$$\align l_x=\tfrac{i}2(l_1-l_{-1}),\quad l_y=&\tfrac12(l_1+l_{-1}),\quad
l_z=i l_0\\
m_x=\tfrac{i}2(m_1-m_{-1}),\quad m_y=&\tfrac12(m_1+m_{-1}),\quad
m_z=i m_0.\endalign$$
The dynamical equation maybe rewritten as
$$\left\{\aligned
\dot A_x&=-\Omega^+_x(A_yB_y+A_zB_z)+\Omega^+_yA_xB_y+\Omega^+_zA_xB_z\\
\dot A_y&=-\Omega^+_y(A_xB_x+A_zB_z)+\Omega^+_xA_yB_x+\Omega^+_zA_yB_z\\
\dot A_z&=-\Omega^+_z(A_xB_x+A_yB_y)+\Omega^+_xA_zB_x+\Omega^+_yA_zB_y
\endaligned\right.$$
$$\left\{\aligned
\dot B_x&=-\Omega^-_x(B_yA_y+B_zA_z)+\Omega^-_yB_xA_y+\Omega^-_zB_xA_z\\
\dot B_y&=-\Omega^-_y(B_xA_x+B_zA_z)+\Omega^-_xB_yA_x+\Omega^-_zB_yA_z\\
\dot B_z&=-\Omega^-_z(B_xA_x+B_yA_y)+\Omega^-_xB_zA_x+\Omega^-_yB_zA_y
\endaligned\right.$$
after the change $\Omega^-\to-\Omega^-$. They maybe also written in a more
compact form
$$\left\{\aligned
\dot A&=-\left\Omega^++\left<\Omega^+,B\right>A\\
\dot B&=-\left\Omega^-+\left<\Omega^-,A\right>B
\endaligned\right. $$
here brackets $\left<\cdot,\cdot\right>$ denote $\sUtwo$--invariant inner
products in $\pi_1^{\pm}$. Such nonlinear dynamics maybe interpreted as one
for the noncanonically coupled (linear) rotators. It possesses two trivial
integrals of motion $K=\left$ and
$L=\left-\left$.
Let's denote
$\left<\Omega^+,B\right>+\frac12L=\left<\Omega^-,A\right>-\frac12L$ by $y$.
Then
$$\left\{\aligned
\dot A&=-K\Omega^++(y-\tfrac12L)A\\
\dot B&=-K\Omega^-+(y+\tfrac12L)B
\endaligned\right.
$$
whereas
$$\dot y=y^2-(\tfrac14L^2+K\beta),$$
$\beta=\left<\Omega^+,\Omega^-\right>$, so the dynamical equations are
integrable in elementary functions.
\subhead 2.2. Nonlinear integrable dynamics of noncanonically coupled
Euler--Arnold tops \endsubhead Noncanonically coupled Euler--Arnold tops are
realized by means of the same isotopic pair as noncanonically coupled rotators
but with quadratic "hamiltonians" $\Cal H_1=\left$ and $\Cal
H_2=\left$ ($T=T^*$). The corresponding dynamical equations maybe
written as
$$\left\{\aligned
\dot A&=-\leftTA+\leftA\\
\dot B&=-\leftTB+\leftB
\endaligned\right.
$$
Such system admits an integral of motion $K=\left$; the equations
maybe rewritten as
$$\left\{\aligned
\dot A&=(\rho-KT)A\\
\dot B&=(\rho-KT)B
\endaligned\right.
$$
where $\rho=\left=\left$ obeys the ordinary nonlinear
differential equation of the second degree:
$$
\ddot\rho+
(4K\alpha-6\rho)\dot\rho+4\rho^3-4K\alpha\rho^2-4K^2\beta\rho-4K^2\gamma=0,$$
$$\left.\dot\rho\right|_{t=0}=
-2K\left+\rho^2_{\left.\right|_{t=0}},$$
where $\alpha=\tr(T)$, $\beta=\frac12(\tr(T^2)-\tr^2(T))$ is a sum of main
minors of the second order of symmetric matrix $T$, $\gamma=\deter(T)$.
It is convenient to make a change of variables $y=\rho-\frac{K\alpha}3$:
$$\ddot y-6y\dot y+4y^3-\Gamma y-\Delta=0,$$
where $\Gamma=4K^2(\beta+\frac13\alpha^2)$,
$\Delta=\frac{4K^3}{27}(8\alpha^3+9\alpha\beta+27\gamma)$.
One may look for solutions of this differential equation by its splitting into
two ones $\dot y=f(y;C)$ and $ff'_y-6yf+4y^3-\Gamma y-\Delta=0$. The first
differential equation is solved in the inversed quadratures
$\int\frac{dy}{f(y;C)}=t+\tilde C$. The second one admits a trivial partial
solution $f_0(y)=2y^2+by+c$ ($b^2-2c=\Gamma$, $bc=\Delta$) and in this case
$y=y_0(t)$ is expressed in elementary functions; a general solution $f$ of the
second equation has the form $f=f_0+g$, where
$gg'_y-(2y-b)g+(2y^2+by+c)g'_y=0$.
\head III. Noncanonically coupled oscillators and their chains \endhead
\subhead 3.1. Nonlinear integrable dynamics of noncanonically coupled
oscillators \endsubhead Let's consider a standard representation of the
Heisenberg algebra with three generators $p$, $q$ and $r$ ($[p,q]=r$,
$[p,r]=[q,r]=0$) by $3\times3$ matrices:
$$p=\left(\matrix 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \endmatrix\right),\quad
q=\left(\matrix 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \endmatrix\right), \quad
r=\left(\matrix 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \endmatrix\right).$$
The linear space of admissible isotopies is generated by three elements
$a$, $b$ and $c$, which are represented by matrices
$$a=\left(\matrix 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \endmatrix\right), \quad
b=\left(\matrix 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \endmatrix\right), \quad
c=\left(\matrix 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \endmatrix\right).$$
The isocommutators have the form
\
\centerline{
$\aligned
[p,q]_a&=0\\
[p,r]_a&=r\\
[q,r]_a&=0
\endaligned$
$\quad$
$\aligned
[p,q]_b&=0\\
[p,r]_b&=0\\
[q,r]_b&=-r
\endaligned$
$\quad$
$\aligned
[p,q]_c=r\\
[p,r]_c=0\\
[q,r]_c=0
\endaligned$}
\
The elements $a$, $b$ and $c$ are closed themselves under isocommutators with
isotopies $p$, $q$ and $r$ via the "isotopic duality":
\
\centerline{
$\aligned
[a,b]_p&=0\\
[a,c]_p&=c\\
[b,c]_p&=0
\endaligned$
$\quad$
$\aligned
[a,b]_q&=0\\
[a,c]_q&=0\\
[b,c]_q&=-c
\endaligned$
$\quad$
$\aligned
[a,b]_r=c\\
[b,c]_r=0\\
[a,c]_r=0
\endaligned$}
\
The linear spaces generated by $p$, $q$, $r$ and $a$, $b$, $c$ form an
isotopic pair. One may associate noncanonically coupled oscillators with it;
namely, let's consider $p$, $q$, $r$ and $a$, $b$, $c$ as linear functionals
(we shall denote them by capitals) on dual spaces; "hamiltonians" $\Cal H_1$
and $\Cal H_2$ will be of the form $\Cal H_1=P^2+Q^2$ and $\Cal H_2=A^2+B^2$,
the dynamical equations will be written as
\
\centerline{
$\left\{\aligned
\dot Q&=2RPC\\
\dot P&=-2RQC\\
\dot R&=2PRA-2QRB
\endaligned\right.$
$\quad$
$\left\{\aligned
\dot A&=-2CBR\\
\dot B&=2CAR\\
\dot C&=2ACP-2BCQ
\endaligned\right.$}
\
It should be marked that "hamiltonians" are integrals of motion here, also
three integrals maybe written: $\Cal M=AQ-BP$, $\Cal N=BQ+AP$ and $\Cal
L=CR-BP-AQ=CR-2BP-\Cal M=CR-2AQ+\Cal M$. The presence of
five integrals essentially simplifies a picture, so an integration of the
system of two noncanonically coupled oscillators becomes an easy but
interesting exercise. Let's perform it. Put $\Cal H_1=h^2_1$, $\Cal
H^2_2=h^2_2$, $P=h_1\cos\varphi$, $Q=h_1\sin\varphi$, $A=h_2\cos\psi$,
$B=h_2\sin\psi$, then the condition $\dot{\Cal M}=\dot{\Cal N}=0$ gives
$\cos(\varphi-\psi)=-\frac{\Cal N}{h_1h_2}$, $\sin(\varphi-\psi)=\frac{\Cal
M}{h_1h_2}$, so $\vartheta:=\psi-\varphi=\arctan(\frac{\Cal M}{\Cal N})$. Also
$\dot\varphi=\dot\psi=2CR$, $\dot R=2Rh_1h_2\cos(\varphi+\psi)$, $\dot
C=2Ch_1h_2\cos(\varphi+\psi)$, hence $C=\varkappa R$ and
$\dot\varphi=2\varkappa R^2$, $\dot R=2h_1h_2R\cos(2\varphi+\vartheta)$. Let's
consider $R$ as a function of $\varphi$ then
$RR'_{\varphi}=h_1h_2\cos(2\varphi+\vartheta)$ and
$R=\sqrt{\frac{\Cal L}{\varkappa}+
\frac{h_1h_2}{\varkappa}\sin(2\varphi+\vartheta)}$ (to receive
this fact one may also use a conservation law $\dot{\Cal L}=0$). It should be
marked that the plane curve, which is defined by the equation $R=R(\varphi$)
in polar coordinates, is the Booth lemniscate. Substituting
the resulted expression for $R=R(\varphi)$ into the formula for $\dot\varphi$
one obtains that $\dot\varphi=
2\Cal L+2h_1h_2\sin(2\varphi+\vartheta)$.
\subhead 3.2. Dynamics of periodic and nonperiodic chains of noncanonically
coupled oscillators \endsubhead The dynamical equations for two noncanonically
coupled oscillators are immediately generalized on (periodic or infinite
nonperiodic) chains of them, namely
$$\left\{\aligned
\dot Q_i&=R_iP_i(R_{i-1}+R_{i+1})\\
\dot P_i&=-R_iQ_i(R_{i-1}+R_{i+1})\\
\dot R_i&=R_iP_i(P_{i-1}+P_{i+1})-R_iQ_i(Q_{i-1}+Q_{i+1})
\endaligned\right.$$
The "hamiltonians" $\Cal H_i=P^2_i+Q^2_i$ are certainly integrals of motion.
Put $\Cal H_i=h^2_i$, $P_i=h_i\cos\varphi_i$, $Q_i=h_i\sin\varphi_i$, then
$\dot\varphi_i=R_i(R_{i-1}+R_{i+1})$, $\dot R_i=R_ih_i(h_{i-1}\cos(\varphi_i+
\varphi_{i-1})+h_{i+1}\cos(\varphi_i+\varphi_{i+1}))$.
Let's introduce new "coupled" variables $\psi_i=\varphi_i+\varphi_{i+1}$ and
$S_i=R_iR_{i+1}$ as well as $H_i=h_ih_{i+1}$. The dynamical equations
are rewritten as follows
$$\left\{\aligned\dot\psi_i&=S_{i-1}+2S_i+S_{i+1}\\
\dot S_i&=S_i(H_{i-1}\cos\psi_{i-1}+2H_i\cos\psi_i+
H_{i+1}\cos\psi_{i+1})\endaligned\right.$$
Let's put $T_i=H_ie^{\sqrt{-1}\psi_i}$ now, then
$$\left\{\aligned\dot S_i&=\real (T_{i-1}+2T_i+T_{i+1})S_i\\
\dot T_i&=\sqrt{-1}(S_{i-1}+2S_i+S_{i+1})T_i\endaligned\right.$$
\subhead 3.3. A continuum limit of the noncanonically coupled oscillator chain
\endsubhead One may consider noncanonically coupled oscillator chain with
changed signs, i.e.
$$\left\{\aligned
\dot Q_i&=R_iP_i(R_{i+1}-R_{i-1})\\
\dot P_i&=-R_iQ_i(R_{i+1}-R_{i-1})\\
\dot R_i&=R_iP_i(P_{i+1}-P_{i-1})-R_iQ_i(Q_{i+1}-Q_{i-1})\endaligned\right.$$
Such chain admits a natural continuum (field) limit
$$\left\{\aligned
\dot q&=prr'_x\\
\dot p&=-qrr'_x\\
\dot r&=rpp'_x-rqq'_x
\endaligned\right.$$
$q\!=\!q(t,x)$, $p\!=\!p(t,x)$, $r\!=\!r(t,x)$. The function
$h^2\!=\!p^2\!+\!q^2$ is an integral of motion, so it is convenient to put
$p=h\cos\varphi$, $q\!=\!h\sin\varphi$. Then
$$\left\{\aligned\dot\phi&=rr'_x\\
\dot r&=-rh^2\sin{2\varphi}\varphi'_x
\endaligned\right.$$
Let's put $s\!=\!r^2$, $t\!=\!h^2e^{2\sssize\sqrt{-1}\tsize\varphi}$, then
$$\left\{\aligned\dot s&=-\ssize\real\dsize(t'_x)s\\
\dot t&=\ssize\sqrt{-1}\dsize s'_xt
\endaligned\right.$$
\head IV. Other examples \endhead
\subhead 4.1. Dynamic system connected with the isotopic pair of $3\times3$
symmetric and skew--symmetric matrices \endsubhead One of the most interesting
examples of isotopic pairs is related to $n\times n$ symmetric and
skew--symmetric matrices. Let's consider the simplest case $n=3$. It is rather
convenient to introduce the following basises:
$$l_z=\left(\matrix 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \endmatrix\right),
\quad l_x=\left(\matrix 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0
\endmatrix\right), \quad l_y=\left(\matrix 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 &
0 \endmatrix\right)$$
and
$$\align
&m_{xy}=\left(\matrix 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \endmatrix\right),
\quad m_{yz}=\left(\matrix 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0
\endmatrix\right), \quad m_{xz}=\left(\matrix 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0
& 0 \endmatrix\right),\\
&\\
&m_{xx}=\left(\matrix 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \endmatrix\right),
\quad m_{yy}=\left(\matrix 0 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0
\endmatrix\right), \quad m_{zz}=\left(\matrix 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0
& 2 \endmatrix\right).\endalign$$
Elements $l_a$ form a basis in the space of skew--symmetric matrices, whereas
$w_{ab}$ form a basis in the space of symmetric matrices. The isocommutators
have the form
$$\align
[m_{ab},m_{cd}]_{l_e}&=\epsilon_{ace}m_{bd}+\epsilon_{ade}m_{bc}+
\epsilon_{bce}m_{ad}+\epsilon_{bde}m_{ac},\\
[l_a,l_b]_{m_{cd}}&=\epsilon_{abc}l_d+\epsilon_{abd}l_c,\endalign$$
here $\epsilon_{abc}$ is a totally antisymmetric tensor.
Let's fix a skew--symmetric matrix $\Omega^-$ and a symmetric matrix
$\Omega^+$; they maybe regarded as linear "hamiltonians" for the considered
isotopic pair. If the linear functionals defined by $l_a$ and $m_{ab}$ in the
dual spaces are denoted by capitals $L_a$ and $M_{ab}$, respectively, then
the dynamical equation will have the form
$$\left\{\aligned \dot L_a&=\epsilon_{bcd}\Omega^-_bL_cM_{ad}\\
\dot M_{ab}&=\epsilon_{cde}\Omega^+_{ac}M_{bd}L_e+
\epsilon_{cde}\Omega^+_{bc}M_{ad}L_e\endaligned\right.$$
Unfortunately, I do not know whether this system is integrable.
\subhead 4.2. The hybrid coupling: "elastoplastic" spring \endsubhead It is
rather interesting to consider the case then two hamiltoinian systems are
coupled by interaction terms in a hamiltonian and simultaneously
noncanonically. Below we shall consider an "elastoplastic" string, which is a
hybrid of the ordinary "elastic" spring with noncanonically coupled
oscillators.
The isotopic pair is the same as for noncanonically coupled oscillators but
the "hamiltonian" is of the form $\Cal H=\Cal H_1+\Cal H_2-2QB-2PA$; the
dynamical equations have the form
\
\centerline{
$\left\{\aligned
\dot Q&=2RC(P-A)\\
\dot P&=-2RC(Q-B)\\
\dot R&=-2(P-A)^2R+2(Q-B)^2R
\endaligned\right.$
$\quad$
$\left\{\aligned
\dot A&=2RC(Q-B)\\
\dot B&=-2RC(P-A)\\
\dot C&=-2(P-A)^2C+2(Q-B)^2C
\endaligned\right.$}
\
Let's denote $D=P-A$, $G=Q-B$, then $\Cal J^2=D^2+G^2$ is an integral of
motion so it is convenient to put $D=\Cal J\sin\varphi$, $G=\Cal
J\cos\varphi$. Moreover, $C=\lambda R$ and $\dot R=2\Cal
J^2R^2\cos{2\varphi}$, whereas $\dot\varphi=-4\lambda R^2$. Hence,
$-2\lambda RR'_\varphi=\Cal J^2\cos{2\varphi}$ and $R=\sqrt{\frac{\Cal
L}\lambda-\frac1{2\lambda}\Cal J^2\sin{2\varphi}}$ (and the corresponding
plane curve is the Booth lemniscate again), whereas $\dot\varphi=-4\Cal
L+2\Cal J^2\sin{2\varphi}$.
\subhead 4.3. "Elastoplastic" spring with general interaction potential
\endsubhead The described picture is straightforwardly generalized on
arbitrary interaction potentials. Namely, let's consider the "hamiltonian" of
the form $\Cal H=(P-A)^2+V(Q-B)$, where $V$ is an interaction potential. The
dynamical equations are written in the form
$$\left\{\aligned
\dot Q&=2RC(P-A)\\
\dot P&=-RCV'(Q-B)\\
\dot R&=-2(P-A)^2R+(Q-B)V'(Q-B)R
\endaligned\right.$$
$$\left\{\aligned
\dot A&=RCV'(Q-B)\\
\dot B&=-2RC(P-A)\\
\dot C&=-2(P-A)^2C+(Q-B)V'(Q-B)C
\endaligned\right.$$
Let's denote $D\!=\!P\!-\!A$, $G\!=\!Q\!-\!B$; $\Cal J^2\!=\!D^2\!+\!V(G)$ is
an integral of motion so it is convenient to put $D\!=\!\Cal J\sin\varphi$,
$G\!=\!V^{-1}(\Cal J^2\cos^2\varphi)$. Moreover $C\!=\!\lambda R$ and
$\dot R\!=\!\left[-2\Cal J^2\sin^2\varphi\!+
\!V^{-1}(\Cal J^2\cos^2\varphi)V'(V^{-1}(\Cal J^2\cos^2\varphi))\right]R$,
whereas $\dot\varphi\!=\!-\frac{2\lambda R^2}{\Cal
J\cos\varphi}\times V'(V^{-1}(\Cal J^2\cos^2\varphi))$. Hence $-2\lambda
RR'_\varphi
V'(V^{-1}(\Cal J^2\cos^2\varphi))\!=\!\Cal J\cos\varphi\left[-2\Cal
J^2\sin^2\varphi\right.+\left.V^{-1}(\Cal J^2\cos^2\varphi)V'(V^{-1}(\Cal
J^2\cos^2\varphi))\right]$ and
$$R=\sqrt{\frac1{\lambda}\int\Cal J\cos\varphi
\frac{2\Cal J^2\sin^2\varphi-V^{-1}(\Cal J^2\cos^2\varphi)V'(V^{-1}(\Cal
J^2\cos^2\varphi))}{V'(V^{-1}(\Cal J^2\cos^2\varphi))}d\varphi},$$ whereas
$$\dot\varphi=\frac2{\cos\varphi}\int\cos\varphi\left[V^{-1}(\Cal
J^2\cos^2\varphi)-\frac{2\Cal J^2\sin^2\varphi}{V'(V^{-1}(\Cal
J^2\cos^2\varphi))}\right]d\varphi.$$
\Refs
\roster
\item"[1]"
{\cyr Arnolp1d V.I., {\cyi Matematicheskie metody klassicheskoe0 mehaniki},
Moskva, Nau\-ka, 1970}; English translation:\newline
Arnold V.I., {\it Mathematical methods of classical mechanics},
Springer--Verlag, 1980.
\item"[2]"
{\cyr Kirillov A.A., {\cyi E1lementy teorii predstavlenie0}, Moskva, Nauka,
1972}; English translation:\newline
Kirillov A.A., {\it Elements of the theory of representations},
Springer--Verlag, 1976.
\item"[3]"
{\cyr Karasi0v M.V., Maslov V.P., {\cyi Nelinee0nye skobki Puassona: geometriya
i kvantovanie}, Moskva, Nauka, 1991}; English translation:\newline
Karasev M.V., Maslov V.P., {\it Nonlinear Poisson brackets:
geometry and quantization}, Amer. Math. Soc., Providence, RI, 1993.
\item"[4]" Santilli R.M., {\it Foundations of theoretical mechanics. II.
Birkhoffian generalization of Ha\-miltonian mechanics}, Springer--Verlag, 1982.
\item"[5]" Juriev D., Topics in hidden symmetries; hep-th/9405050.
\endroster
\endRefs
\enddocument