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\document\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$\boxed{\boxed{\aligned
&\text{\eightpoint"Thalassa Aitheria" Reports}\\
&\text{\eightpoint RCMPI-96/01}\endaligned}}$\newline
\ \newline
\ \newline
\ \newline
\topmatter
\title
Topics in nonHamiltonian (magnetic--type) interaction of classical
Hamiltonian dynamical systems. II. Generalized Euler--Amp\`ere equations and
Biot--Savart operators.
\endtitle
\author
Denis V. Juriev
\endauthor
\affil
"Thalassa Aitheria"\linebreak
Research Center for Mathematical Physics and
Informatics,\linebreak
ul.Miklukho--Maklaya 20-180, Moscow 117437 Russia\linebreak
E--mail: denis\@juriev.msk.ru
\endaffil
\abstract
The paper is devoted to the algebraico--analytic presentation of the
Euler--Amp\`ere dynamical equations for two magnetically interacting massive
systems (the charged Euler tops) and their generalizations. The interaction
dynamics is based on the algebraic structure of isotopic pairs, which underlies
the noncanonical version of Poisson brackets depending explicitely on the
states of the acting system.
\endabstract
\endtopmatter
\
\head I. Introduction\endhead
The Hamiltonian approach lies undoubtly in a core of analytic mechanics
elaborated by J.L.Lagrange, A.M.Legendre, S.Poisson, W.R.Hamilton,
C.G.J.Jacobi, J.Liouville, their contemporaries and successors. Many
important classical Hamiltonian dynamical systems (e.g. the Euler tops
[Ar,Fo,LL1]) are somehow related to the Lie algebras [Ar,DNF,Fo,TF] or their
nonlinear generalisations [KM]. Really, just the algebraic approach to
classical mechanics, which goes back to S.Lie, \'E.Cartan and H.Poincar\'e,
supplies analytic dynamics by many elegant and physically important examples.
The interaction of Hamiltonian systems may be either Hamiltonian, i.e. defined
by a subsidiary interaction term $\Cal H_{\tni}$ of the Hamiltonian and,
perphaps, by a deformation of initial ("free") Poisson brackets, or
nonHamiltonian. The least case, corresponding to certain nonpotential
interaction forces, is of an interest from both theoretical and practical
points of view.
First, indeed, the Hamiltonian approach to the interaction of systems
is no more than a formal abstraction of the prerelativistic mechanics, which
ignores a fundamental principle of a short-range action underlying the
field--theoretic picture. Zero approximation of the classical mechanics is
correct only for small distances between systems and slow velocities of
their motion. Though the idea of the short-range action is, in general,
incompatible with mechanics of material points and instantly spreading forces,
the first most precise field approximation admits a formulation
on the language of the nonHamiltonian classical mechanics in terms of
nonpotential magnetic--type forces, which depend on velocities. In
electrodynamics such forces obey the Amp\`ere-Lorentz law [A,M,L], which
is timer than the Hamiltonian Darwin approximation [LL2]. One can formulate
more subtle field effects such as induction on the language of the
nonHamiltonian classical mechanics, though it claims to introduce the
forces explicitely depending on the second time derivatives (the Weber law [W]).
Note that a general law of interaction of two charged particles [Fe] may be
formulated on the language of the integrodifferential mechanics (the retarded
potentials), which goes back to C.Gauss (a letter to W.Weber, 1845) and to
B.Riemann [R].
Some analogs of the Amp\`ere-Lorentz law may be written also for particles,
interacting via a nonlinear (e.g. nonabelian gauge or gravitational) field.
For the gravitational interaction an analog of the Darwin approximation is
one of Einstein, Infeld, Hoffman; Eddington, Clark and Fichtenholz [F,LL2].
Certainly, in the nonlinear case the interaction forces are not pairwise (and,
moreover, can not be splitted via reduction to the Weber form), though the
nonHamiltonian terms themselves are represented as a sum of two-particle ones.
Note that one of the most known effects of the general relativity, the shift of the
Mercury perihelion (as well as the secular shift of a particle orbit in the
gravitational field of the rotating body, the relativistic precession of a
spherical top, etc.), is described very precisely by the nonHamiltonian
magnetic-type corrections to the Newton force of gravity.
Second, the nonconservative interaction forces appear also in the classical
mechanics itself (general nonholonomic constraints [H,Ap,Ch,LCA,NF,P,Wh],
gyroscopic systems [TT,Gl,I,LCA,Bo,AKN], Rayleigh--Lienart type interactions
[AVK, AKN,P]), in the servomechanics [Ap], the electrotechnics and in the
general theory of (natural or artificial) systems with a feedback
[B,Bu,St,La,RSC].
Note that the nonHamiltonian corrections, which are caused by a finite speed of
interactions (cf. the cited letter of C.Gauss to W.Weber), are not obligatory
relativistic. They also appear in the continuum media mechanics, e.g. in the
simple model of the interaction of $n$ vortices "freezed" into the ideal
fluid [AKN]. In general, the nonHamiltonian terms are the first order
corrections related to the retarded potentials and so they are peculiar to
a wast class of linear differential equations.
Really, the nonHamiltonian forces of both kinds (magnetic--type and
Rayleigh--Lienart type) underlie also the soliton--soliton interactions
governed by the $n$--soliton solutions of nonlinear equations of mathematical
physics or their algebrogeometric perturbations. The presence of a "hidden
potential action at distance" in the 1--dimensional dynamics of poles of
solutions of some nonlinear equations of mathematical physics [Pe] is an
exception, which only confirms the general rule (1--dimensional case does not
admit magnetic--type forces). The dynamics of zeroes of the polynomial
solutions of linear equations is almost always nonHamiltonian [Pe] (with
pairwise forces of the Rayleigh--Lienart type).
So the theoretical and practical significance of exploration of nonHamiltonian
systems is out of any doubts.
An important problem of mathematical physics is to describe various algebraic
structures, governing the nonHamiltonian interaction. The reasonableness of
such setting has no need in any proofs. Among many arguments the following ones
are mostly convincing: (1) the algebraic approach is a new vision of the
known differential equations, solutions of which in their turn may encode
an important algebraic or geometric information (e.g. instantons or monopoles),
(2) often algebraically constructed differential equations possess {\it a
priori\/} or {\it a posteriori\/} important and interesting properties (e.g.
some form of integrability), (3) if experimental data are not complete and do
not allow to restore the law of an interaction uniquely, the use of the
algebraically constructed differential equations gives a correct answer as
a rule (cf. with the Amp\`ere-Lorentz law of the elementary current
interaction [A,G,Gr,M,L]), (4) the solutions of the Wigner problem (the
dynamical inverse problem of the representation theory) [Wi,Ju6] of a
derivation of the quantum commutation relations from classical equations of
motion are constructed by use of the algebraic structures, which underlie such
equations. Some aspects of these topics for the nonHamiltonian interactions
were discussed in [Ju7].
The aim of this paper is to imbed the nonHamiltonian magnetic--type
interactions into the algebraico--analytic scheme analogous to the well--known
one for the Hamiltonian systems.
The author apologizes to the English language reader for the numerous
references to the literature in Russian. Several Russian books or papers from
the reference list are really translated into English but they are not
available to the author himself; however, the main part of the rest Russian
publications is essential and the material is not reproduced in the English
language literature adequately.
\head II. Interaction of classical Hamiltonian dynamical systems:
general classes\endhead
Let us describe the general classes of interactions of Hamiltonian systems.
\definition{Definition 1} Let $\dot X=\{\CH_1(X),X\}_1$ and
$\dot A=\{\CH_2(A),A\}_2$ be two classical Hamiltonian dynamical systems
defined by Poisson brackets $\{\cdot,\cdot\}_1$ and $\{\cdot,\cdot\}_2$ and
Hamiltonians $\CH_1$ and $\CH_2$. Their interaction is called
{\it Hamiltonian\/} iff it has the Hamiltonian form
$$\left\{\aligned
\dot X=&\{\CH(X,A),X\}\\
\dot A=&\{\CH(X,A),A\}
\endaligned\right.$$
where $\{\cdot,\cdot\}$ are the new Poisson brackets on both variables $A$ and
$X$, and $\CH$ is a new Hamiltonian: $\{\cdot,\cdot\}=\{\cdot,\cdot\}_1+
\{\cdot,\cdot\}_2+\varepsilon\Phi(\cdot,\cdot)$ and $\CH(X,A)=\CH_1(X)+\CH_2(A)+
\varepsilon\CH_{int}(X,A)$. Here $\varepsilon$ is a coupling constant and
$\CH_{int}$ is an interaction term.
\enddefinition
Often, $\Phi(\cdot,\cdot)=0$ and $\CH_{int}$ is defined by the interaction
potential.
The interaction is called {\it nonHamiltonian\/} if it can not be defined by
the construction above.
It is reasonable to consider certain classes and types of nonHamiltonian
interactions.
{\it Classes of nonHamiltonian interaction.}
\definition{Definition 2} Let $\dot X=\{\CH_1(X),X\}_1$ and
$\dot A=\{\CH_2(A),A\}_2$ be two classical Hamiltonian dynamical systems defined
by Poisson brackets $\{\cdot,\cdot\}_1$ and $\{\cdot,\cdot\}_2$ and Hamiltonians
$\CH_1$ and $\CH_2$. Their interaction
$$\left\{\aligned
\dot X=&\{\CH_1(X),X\}+\varepsilon v_1(X,A)\\
\dot A=&\{\CH_2(A),A\}+\varepsilon v_2(X,A)
\endaligned\right.$$
is called {\it conservative\/} iff $\dot\CH_1+\dot\CH_2=\CL_{v_1}\CH_1+
\CL_{v_2}\CH_2=0$ and {\it ultraconservative\/} (or {\it gyroscopic\/}
[Bo,N,AKN]) iff $\dot\CH_1=\CL_{v_1}\CH_1=0$, $\dot\CH_2=\CL_{v_2}\CH_2=0$.
\enddefinition
{\it Types of nonHamiltonian interaction.}
\definition{Definition 3} Let $\dot X=\{\CH_1(X),X\}_1$ and
$\dot A=\{\CH_2(A),A\}_2$ be two classical Hamiltonian dynamical systems defined
by Poisson brackets $\{\cdot,\cdot\}_1$ and $\{\cdot,\cdot\}_2$ and Hamiltonians
$\CH_1$ and $\CH_2$. Their interaction is called {\it magnetic--type
interaction\/} iff it has the form
$$\left\{\aligned
\dot X=&\{\CH_1(X),X\}_A\\
\dot A=&\{\CH_2(A),A\}_X
\endaligned\right.$$
with new Poisson brackets explicitely depending on the states of the acting
systems. The interaction is called {\it electromagnetic--type (hybrid)
interaction\/} iff it has the similar form but with the potential interaction
term included into the Hamiltonian.
\enddefinition
\proclaim{Lemma 1} Magnetic--type interaction is ultraconservative.
\endproclaim
\demo{Proof} Really, $\dot\Cal H_1(X)=\{\Cal H_1(X),\Cal H_1(X)\}_A=0$ and
$\dot\Cal H_2(A)=\{\Cal H_2(A),\Cal H_2(A)\}_X=0$. \qed
\enddemo
\remark{Remark 1} If the interaction is of a magnetic type and $\Cal I(X,A)$
is a magnitude such that $\{\Cal H_i,\Cal I\}_{(\cdot)}=0$ ($i=1,2$) for
all values of the subscript parameters then $\Cal I$ is an integral of motion.
However, this construction does not provide us with {\it all\/} integrals of
motion contrary to the Hamiltonian dynamics.
\endremark
The simplest case of magnetic-type interaction is one of linear dependence on
the states of the opposite system. To describe such dynamics one needs in
purely algebraic requisites, which are discussed in the next paragraph.
\head III. The algebra of isotopic pairs\endhead
There exist, at least, two different approaches to the problem of
a description of algebraic structures underlying the nonHamiltonian
interaction. The first one relates such interaction to certain deformations
of the initial algebraic structures (Lie algebras) such as isotopic pairs
[Ju1,Ju4,Ju5] or general nonlinear I-pairs [Ju2]. The second one relates
the interaction to certain subsidiary algebraic structures (designs) on Lie
algebras [Ju3].
The least approach is effective for nonultraconservative interactions
of Lienart type, whereas the first one is convenient for nonHamiltonian
magnetic-type interactions.
This paragraph is devoted to a systematic introduction to the algebra of
the isotopic pairs.
\definition{Definition 4 {\rm [Ju1,Ju4,Ju5,Ju6]}} The pair $(V_1,V_2)$ of
linear spaces is called {\it an (even) 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 isotopic
elements\/}) 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
Let's discuss this definition.
First, it may be considered as a result of an axiomatization of the following
trivial construction: let $\Cal A$ be an associative algebra (f.e. any matrix
one) and $V_1$, $V_2$ be two linear subspaces in it such that $V_1$ is closed
under the isocommutators $(X,Y)\mapsto [X,Y]_A=XAY-YAX$ with isotopic elements
$A$ from $V_2$, whereas $V_2$ is closed under the isocommutators $(A,B)\mapsto
[A,B]_X=AXB-BXA$ with isotopic elements $X$ from $V_1$.
\remark{Remark 2 {\rm [Ju5,Ju6]}} Let $H$ be a (finite dimensional) linear
space. If $\Cal A$ is a subspace of $\End(H)$ let's put $\Cal
A^{{\vz}}=\{X\in\End(H),\forall A\in\Cal A, \forall B\in\Cal A,
AXB-BXA\in\Cal A\}$. Then $\Cal A\subseteq\Cal A^{{\vz}{\vz}}$ and
$(\Cal A,\Cal A^{{\vz}})$ is an isotopic pair.
\endremark
\remark{Remark 3 {\rm [Ju5,Ju6]}} Let $\Cal A\subseteq\End(H)$, $\dim\Cal A=n$,
$\Lie_n$ be the space of all Lie algebras of dimension $n$. Then there exists
a natural mapping $\Cal L:\Cal A^{{\vz}}\mapsto\Lie_n$. It should be mentioned
that $\card\Cal L(\Cal A^{{\vz}})$ may be not equal to $1$ so $\Cal L(X)$ and
$\Cal L(Y)$ are not the same in general for different $X$ and $Y$. It means
that the isocommutators $[\cdot,\cdot]_X$ and $[\cdot,\cdot]_Y$ determine
structures of nonisomorphic Lie algebras on the space $\Cal A$ in general
(though they may be isomorphic in particular).
\endremark
\remark{Remark 4 {\rm [Ju6]}} The construction of the remark 1 may be
immediately generalized. Namely, let $(V_1,V_2)$ be an isotopic pair,
$\Cal A\subseteq V_1$, $\Cal A^{\vz}:=\{X\in V_2:(\forall A,B\in\Cal A)
[A,B]_X=0\}$. Then $(\Cal A,\Cal A^{\vz})$ is an isotopic pair.
\endremark
\remark{Remark 5 {\rm [Ju4]}} One may also consider the following construction.
Let $(V_1,V_2)$ be an isotopic pair, $A$ and $X$ be elements of $V_1$ and $V_2$,
respectively. Put $U_1=\{B\in V_1: [B,A]_X=0\}$ and $U_2=\{Y\in V_2:
[Y,X]_A=0\}$, then $(U_1,U_2)$ is an isotopic pair.
\endremark
Numerous examples of isotopic pairs were considered in [Ju1,Ju4,Ju6]. Very
intriguing related topics (isotopic pairs and Lie superalgebras, isotopic
pairs and Lie $\frak g$-bunches; isotopic pairs, Lie hybrids and
L.V.Sabinin's nonlinear geometric algebra, etc.) were discussed in [Ju5,Ju6];
the knowledge of these topics is not necessary for the understanding of
the following material.
Let us concentrate an attention on some new aspects of the algebraic theory.
Let $(V_1,V_2)$ be an isotopic pair and $e_1$, $e_2$ be certain fixed elements
of $V_1$, $V_2$, respectively. It is convenient to define four operators
$\ad^{(1)}(x):V_1\mapsto V_1$, $\ad^{(2)}(u):V_2\mapsto V_2$,
$\coad^{(1)}(x):V_2\mapsto V_2$, $\coad^{(2)}(u):V_1\mapsto V_1$:
$$\ad^{(1)}(x)=[x,\cdot]_{e_2},\ \ad^{(2)}(u)=[u,\cdot]_{e_1},\quad
\coad^{(1)}(x)=[e_2,\cdot]_x,\ \coad^{(2)}(u)=[e_1,\cdot]_u,$$
where $x\in V_1$, $u\in V_2$.
\proclaim{Lemma 2} Operators $\ad^{(i)}$ supply $V_1$ and $V_2$ by the
structures of Lie algebras $\frak g_1$ and $\frak g_2$:
$$\aligned
[\ad^{(1)}(x),\ad^{(1)}(y)]=& \ad^{(1)}([x,y]_{e_2}),\\
[\ad^{(2)}(u),\ad^{(2)}(v)]=& \ad^{(1)}([u,v]_{e_1}).
\endaligned$$
Operators $\coad^{(i)}$ supply $V_1$ and $V_2$ by the structures of
$\frak g_2$-- and $\frak g_1$--modules, respectively:
$$\aligned
[\coad^{(1)}(x),\coad^{(1)}(y)]=& \coad^{(1)}([x,y]_{e_2}),\\
[\coad^{(2)}(u),\coad^{(2)}(v)]=& \coad^{(2)}([u,v]_{e_1}),
\endaligned$$
where $x,y\in V_1$, $u,v\in V_2$.
\endproclaim
\demo{Proof} The statement of the Lemma is a sequence of the Jacobi and
Faulkner--Ferrar identities
$$[X,Y]_{[A,B]_Z}=[[X,Z]_A,Y]_B+[[Z,Y]_A,X]_B-[[X,Y]_B,Z]_A,$$
which hold in any isotopic pair.
\enddemo
\definition{Definition 5 {\rm [Ju6]}} An isotopic pair $(V_1,V_2)$ is called
{\it contragredient}, if $V_1=V_2^*$, $V_2=V_1^*$ and for all $A,B\in V_2$,
$X,Y\in V_1$ the following sequence holds:
$$\left<[X,Y]_B,A\right>=-\left<[X,Y]_A,B\right>=
\left<[A,B]_X,Y\right>=-\left<[A,B]_Y,X\right>,$$
where $\left<\cdot,\cdot\right>$ is a natural pairing of $V_1$ and $V_2$,
$X,Y\in V_1$, $A,B\in V_2$.
\enddefinition
\remark{Remark 6} In the contragredient isotopic pair $(V_1,V_2)$ the operators
$\coad^{(i)}$ are conjugate to the operators $\ad^{(i)}$.
\endremark
\definition{Definition 6} The pair of mappings $L_i:V_i\mapsto V_i$ is
called {\it a derivative\/} of the isotopic pair
$(V_1,V_2)$ iff
$$\aligned
L_1([x,y]_u)=& [L_1x,y]_u+[x,L_1y]_u+[x,y]_{L_2u},\\
L_2([u,v]_x)=& [L_2u,v]_x+[u,L_2v]_x+[u,v]_{L_1x},
\endaligned$$
where $x,y\in V_1$, $u,v\in V_2$.
\enddefinition
\remark{Remark 7} The derivatives of an isotopic pair $(V_1,V_2)$ form
a Lie algebra, which will be denoted by $\Der(V_1,V_2)$.
\endremark
\proclaim{Lemma 3} The pairs of mappings $(\ad^{(i)},\coad^{(i)})$ realize
homomorphisms $\frak g_i\mapsto\Der(V_1,V_2)$ of Lie algebras $\frak g_i$
into the Lie algebra $\Der(V_1,V_2)$.
\endproclaim
\demo{Proof} One should use the Faulkner--Ferrar identities.
\enddemo
\definition{Definition 6} Two elements $e_1$ and $e_2$ of the isotopic pair
$(V_1,V_2)$ ($e_i\in V_i$) are called {\it polar\/} if
$$[e_1,\cdot]_{e_2}=0,\qquad [e_2,\cdot]_{e_1}=0.$$
\enddefinition
\remark{Remark 8} Let $(V_1,V_2)$ be an isotopic pair, $e_1$ and $e_2$ be
two arbitrary elements of $V_1$ and $V_2$, respectively, $(U_1,U_2)$ be a
subpair of $(V_1,V_2)$ of the remark 4. Then $e_1$, $e_2$ are polar elements
in $(U_1,U_2)$.
\endremark
Note that two elements $A$ and $B$ of the isotopic pair $(\End(H),\End(H))$
are polar iff $AB=BA=C$, where $C$ is the scalar matrix.
\proclaim{Lemma 4} The operators $\ad^{(i)}$ and $\coad^{(i)}$ realize
the Lie algebras $\frak g_1$ and $\frak g_2$ as subalgebras of
$\Der(\frak g_2)$ and $\Der(\frak g_1)$, respectively, iff
$e_1$ and $e_2$ are polar elements.
\endproclaim
\demo{Proof} The statement of the Lemma is a sequence of the compatibility
identities between isocommutators in the isotopic pairs and the definition
of the polar elements.
\enddemo
So the most important case is one of $\frak g_1\simeq\frak g_2$.
\definition{Definition 7 {\rm [Ju4]}} A Lie algebra $\frak g$ is called
{\it a magnetic Lie algebra\/} iff the pair of spaces $(\frak g,\frak g)$ is
supplied by the $\frak g$--equivariant structure of isotopic pair.
\enddefinition
Note that any isotopic pair $(V_1,V_2)$ such that $\dim V_1=\dim V_2=n$ is a
magnetic Lie algebra with the abelian Lie algebra $\frak g=\Bbb R^n$. Various
notrivial examples of the magnetic Lie algebras were considered in [Ju4].
\proclaim{Lemma 5} Let $\frak g$ be a magnetic Lie algebra, then $(V_1,V_2)$
($V_1=\frak g\oplus\Bbb R\cdot e_1$, $V_2=\frak g\oplus\Bbb R\cdot e_2$)
admits a natural structure of the isotopic pair with polar elements $e_1$ and
$e_2$. Operators $\ad^{(i)}$ and $\coad^{(i)}$ are essentially the operators
of the adjoint representation of $\frak g$.
\endproclaim
\demo\nofrills{}
The proof of the Lemma is straightforward.
\enddemo
Thus, the necessary algebraic requisites are collected so it is reasonable now
to enter upon a construction of the generalized Euler--Amp\`ere dynamics.
\head IV. Biot-Savart operators and the generalized Euler--Amp\`ere equations
\endhead
In the paper [Ju1] there were proposed certain versions of nonHamiltonian
dynamical equations associated with isotopic pairs. However, the considered
toy equations are initially more typical for problems of classical mechanics
(gyroscopic systems) and electromechanics (electromotors) than for general
ones of Amp\`ere-Lorentz electrodynamics and its analogues (the first
nonHamiltonian approximations of nonlinear field interactions), though
they may be useful for other interesting physical problems, which were
discussed in [Ju1,Ju5].
Here the generalizations of the dynamical equations of [Ju1] will be
constructed and certain their properties will be explored. The crucial role
will be played by the (generalized) Biot-Savart operators in the construction.
First, note that the isocommutators in an isotopic pair $(V_1,V_2)$ define
families of compatible Poisson brackets $\{\cdot,\cdot\}_A$ and
$\{\cdot,\cdot\}_X$ ($A\in V_2$, $X\in V_1$) in the spaces
$S^{\cdot}(V_1)\subset C^{\infty}(V_1^*)$ and
$S^{\cdot}(V_2)\subset C^{\infty}(V_2^*)$, respectively. Here $S^{\cdot}(H)$ is
the sum of all symmetric degrees of the space $H$ and $C^{\infty}(H)$ is the
space of all smooth functions on $H$. The compatibility means that a linear
combination of any two Poisson brackets is also a Poisson bracket.
If $\frak g$ is a magnetic Lie algebra then one may define composite Poisson
brackets in $S^{\cdot}(\frak g)\subset C^{\infty}(\frak g^*)$ as a sum of the
Poisson brackets related to the Lie algebra $\frak g$ itself (Lie--Berezin
brackets) and to the isotopic pair structure in $(\frak g,\frak g)$. Such
composite brackets will be also denoted by $\{\cdot,\cdot\}_A$ and
$\{\cdot,\cdot\}_X$, where $X$ belongs to the first copy and $A$ belongs to
the second copy of $\frak g$.
Let us define the generalized Euler--Amp\`ere dynamics now.
\definition{Definition 8} Let $\frak g$ be a magnetic Lie algebra. Let us
consider two elements $\Cal H_1$ and $\Cal H_2$ ({\it Hamiltonians\/}) in
$S^{\cdot}(V_1)$ and $S^{\cdot}(V_2)$, respectively. Let us also consider two
symmetric mutually anticonjugate operators
$R_1\in\Hom(V_2^*,V_2)=V_2\otimes V_2$ and
$R_2\in\Hom(V_1^*,V_1)=V_1\otimes V_2$ (the symmetricity means that
$R_i\in S^2(V_i)$), $R_1+R_2^*=0$ ({\it the generalized Biot-Savart
operators\/}). The equations
$$\aligned
(\forall X\in V_1\subset C^{\infty}(V_1^*)) &\quad
\dot X(P_t)=\{\Cal H_1,X\}_{R_2(Q_t)}(P_t),\\
(\forall A\in V_2\subset C^{\infty}(V_2^*)) &\quad
\dot A(Q_t)=\{\Cal H_2,A\}_{R_1(P_t)}(P_t),
\endaligned
$$
where $P_t\in V_1^*$ and $Q_t\in V_2^*$ are called {\it the generalized
Euler--Amp\`ere equations associated with the magnetic Lie algebra $\frak g$,
Biot-Savart operators $R_i$ and Hamiltonians $\Cal H_1$ and $\Cal H_2$}.
\enddefinition
Certainly, the conditions $X\in V_1\subset C^{\infty}(V_1^*)$,
$A\in V_2\subset C^{\infty}(V_2^*)$ may be changed to the general ones
$X\in C^{\infty}(V_1^*)$, $A\in C^{\infty}(V_2^*)$.
\remark{Remark 9} If $V_i$ are somehow identified with $V_i^*$ than one may
consider Biot-Savart operators $R_i$ as elements of $\End(\frak g_i)$. For
example, such situation is realized if $\frak g$ is a compact magnetic Lie
algebra (i.e. a compact Lie algebra, which is a magnetic one). Note that all
semisimple Lie algebras $\frak g$ admit a $\frak g$--equivariant structures of
the contragredient isotopic pairs [Ju4]. Some dynamical equations of the papers
[Ju1,Ju5] are particular case of the generalized Euler--Amp\`ere equations with
$R_i$ equal to identical operators.
\endremark
\remark{Remark 10} If $\frak g$ is a magnetic Lie algebra $\soa(3)$ [Ju4],
$\Cal H_i$ are defined by the quadratic Casimir functions, operators $R_i$
are traceless and have two equal eigenvalues, than one receives the equations
of motion of two spherical charged particles with fixed centers (the Euler
tops), interacting via Amp\`ere-Biot-Savart forces. The quantities $X$ and $A$
are identified with angular velocities and the absolute coordinate system is
used.
The situation is the same after an account of the first order corrections
produced by the Bernett and Einstein-de Haas gyromagnetic effects [V]. However,
the dynamical self--induction processes are neglected (see the remarks at the
Introduction).
The same data but with $\tr R_i\ne 0$ describes the nonHamiltonian
approximations for the gravitational and nuclear [Da] interactions.
\endremark
\remark{Remark 11} Certainly, generalized Euler--Amp\`ere dynamics realizes a
nonHamiltonian (mag\-ne\-tic--type) interaction and, therefore, it is
ultraconservative (gyroscopic).
\endremark
\remark{Remark 12 {\rm (hystorical notes)}} Note that a geometric formalism for
systems under {\it external\/} magnetic or gyroscopic forces is really
well-known (see e.g. [N,DKN,AKN, Bo]). There exists an equivalent canonical
realization in this case, so the nonHamiltonian approach is excessive. The
crucial point of the algebraic description of the {\it interaction\/} forces of
those types is played by the compatibility conditions in the definition of the
isotopic pair. Such conditions were written in a slightly weaker form in the
paper by Faulkner J.R. and Ferrar J.C. [FF]. The present form was introduced in
[Ju1,Ju4], where its supergeneralization was also formulated. The classical
essentially nonHamiltonian dynamics, associated with isotopic pairs, was
introduced in [Ju1]. Its quantum counterpart was considered in [Ju5], where the
Wigner problem (the dynamical inverse problem of representation theory) was
solved for the discussed model. Note that the generalized Euler--Amp\`ere
dynamics is also essentially nonHamiltonian.
\endremark
\remark{Remark 13} It seems that some kind of dynamical equations, associated
with isotopic pairs, may help to construct their {\it globalizations}. Such
globalizations and their constructions are undoubtly important for the
nonlinear geometric algebra [S1-S3]. In its turn methods of the nonlinear
geometric algebra may be useful for the global analysis of the long time
behaviour of solutions of the dynamical equations. The simple examples of
isotopic pairs indicate a relation of this topic with the theory of classical
$r$--matrices [STS,KM] and its quantum counterpart [D,RTF].
\endremark
Note that the methods of the nonlinear geometric algebra (pseudo(quasi)groups
of transformations) are effectively used in the classical theory of nonlinear
Poisson brackets and their asymptotic quantization [KM]. So their penetration
into the nonHamiltonian mechanics would be very natural. The unified
geometroalgebraic approach to the nonlinear Poisson brackets and nonHamiltonian
systems is really welcome.
Let us discuss some properties of the generalized Euler--Amp\`ere equations now.
Let us supply a compact Lie algebra $\frak g$ by a canonical structure of
the magnetic Lie algebra, defined via the Killing form [Ju2]. The related
isotopic pair is just the Okubo pair [Ju2].
\proclaim{Theorem 1} If Hamiltonians $\Cal H_1$ and $\Cal H_2$ are defined by
quadratic Casimir functions then the Euler--Amp\`ere dynamics on the compact
magnetic Lie algebra $\frak g$ has the solution
$$\aligned
X_t=&\ \frac{\Cal E_1}{\| Y_t\|}Y_t,\\
A_t=&\ \frac{\Cal E_2}{\| B_t\|}B_t,
\endaligned$$
where $\Cal E_1^2=\Cal H_1(X_0)$, $\Cal E_2^2=\Cal H_2(A_0)$, and
$$\left(\matrix Y_t \\ B_t\endmatrix\right)=\exp\left\{\left(\matrix 0 & R \\
-R^* & 0\endmatrix\right)t\right\}\left(\matrix Y_0 \\ B_0\endmatrix\right),$$
where $R$ is the Biot--Savart operator.
\endproclaim
\demo{Proof} First, note that $\| Y_t\|=\| B_t\|$ for all $t$. The
Euler--Amp\`ere equations
$$
\dot X_t=(R_1(A_t),X_t)X_t-\|X_t\|^2R_1(A_t),\quad
\dot A_t=-((R_2(X_t),A_t)A_t-\|A_t\|^2R_2(X_t))$$
follows from this fact immediately (here $(\cdot,\cdot)$ is the Killing form,
$\|Z\|\!=\!\sqrt{(Z,Z)}$)\qed
\enddemo
\proclaim{Corollary} A motion of a pair of the magnetically interacting
generalized Euler--Amp\`ere tops is almost periodic with almost periods
$\tfrac{2\pi}{\omega_i}$, where $\omega_i$ are the eigenvalues of the
Biot--Savart operator.
\endproclaim
\remark{Remark 14} If the Biot--Savart operator is traceless and
$\SO(n-1)$--invariant ($n=\dim\frak g$), then the generalized Euler--Amp\`ere
dynamics is periodic. In particular, the dynamics of a pair of two magnetically
interacting spherical tops is periodic.
\endremark
\head V. The generalized Euler--Amp\`ere dynamics on the magnetic Lie groups
\endhead
Let us consider a certain generalization of dynamics discussed earlier.
\definition{Definition 9} {\it A magnetic Lie group\/} is a Lie group $G$,
which Lie algebra $\frak g$ is a magnetic Lie algebra.
\enddefinition
Let us consider two copies $G_1$ and $G_2$ of a magnetic Lie group $G$ and
the cotangent fiber bundles $T^*G_1$, $T^*G_2$ over them. These bundles are
supplied by canonical symplectic structures, which define Poisson brackets
in the spaces $C^{\infty}(T^*(G_i))$ of smooth functions over $T^*G_i$.
On the other hand, the structure of isotopic pair in $\frak g\oplus\frak g$
perturbes these Poisson brackets, so that the new brackets in
$C^{\infty}(T^*_1)$ and $C^{\infty}(T^*2)$ become to depend on the points of
$T^*G_2$ and $T^*G_1$, respectively.
\definition{Definition 10} Let $G$ be a magnetic Lie group, $\Cal H_1$,
$\Cal H_2$ be two smooth $G_i$--invariant functions on the cotangent bundles
$T^*G_i$ over $G_i$, $R_1$ and $R_2$ be two mutually anticonjugate $G_1\times
G_2$--invariant sections of the bundles $\Hom_{G_1\times G_2}(T^*G_1,TG_2)$ and
$\Hom_{G_2\times G_1}(T^*G_2,TG_1)$, respectively ({\it the generalized
Biot--Savart operators\/}). The equations
$$\aligned
(\forall X\in C^{\infty}(T^*G_1)) &\quad
\dot X(P_t)=\{\Cal H_1,X\}_{R_2(P_t,Q_t)}(P_t),\\
(\forall A\in C^{\infty}(T^*G_2)) &\quad
\dot A(Q_t)=\{\Cal H_2,A\}_{R_1(P_t,Q_t)}(P_t),
\endaligned
$$
where $P_t\in T^*G_1$ and $Q_t\in T^*G_2$ are called {\it the generalized
Euler--Amp\`ere equations associated with the magnetic Lie group $G$,
Biot-Savart operators $R_i$ and Hamiltonians $\Cal H_1$ and $\Cal H_2$}.
\enddefinition
\remark{Remark 15} Note that now the Biot--Savart operators depend on both
$P_t$ and $Q_t$. However, the $P_t$ in the first equation and $Q_t$ in the
second equation appears in the Biot--Savart operators only via its projection
on the base $G_1$ ($G_2$) of the cotangent bundle $T^*G_1$ ($T^*G_2$).
\endremark
\remark{Remark 16} Iff $G=\SO(3)$ and $\Cal H_i$ are the Euler hamiltonians,
then the generalized Euler--Amp\`ere equations describe a motion of two
asymmetric magnetically interacting charged particles with fixed centers.
\endremark
\remark{Remark 17} The generalized Euler--Amp\`ere dynamics of Def.10 is
ultraconservative (gyroscopic).
\endremark
The main topic of this paragraph is a nonHamiltonian reduction ([AKN])
in the generalized Euler--Amp\`ere equations associated with magnetic Lie
groups.
To perform the nonHamiltonian reduction by $G_1\times G_2$ means to consider
the motion of Euler tops interacting via Amp\`ere--Biot--Savart forces in their
proper coordinates. The result is rather straightforward.
\proclaim{Theorem 2} After the nonHamiltonian reduction the generalized
Euler--Amp\`ere equations will have the form
$$\aligned
(\forall X\in C^{\infty}(T^*G_1)) \quad
\dot X(P_t)=&\ \{\Cal H_1,X\}_{R_2(Q_t)}(P_t),\\
(\forall A\in C^{\infty}(T^*G_2)) \quad
\dot A(Q_t)=&\ \{\Cal H_2,A\}_{R_1(P_t)}(Q_t),\\
\dot R_1=&\ \ad(Q_t)\circ R_1+R_1\circ\ad^*(P_t),\\
\dot R_2=&\ \ad(P_t)\circ R_2+R_2\circ\ad^*(Q_t),
\endaligned
$$
where $R_i$ are {\it dynamical\/} variables from $\Hom(V_2^*,V_1)$ and
$\Hom(V_1^*,V_2)$, respectively. Operators $\ad$ and $\ad^*$ are the adjoint
and coadjoint operators in the Lie algebra $\frak g$.
\endproclaim
The variables $R_i$ will be called {\it the dynamical Biot--Savart operators}.
Note that $R_i$ are mutually conjugate. So the reduced Euler--Amp\`ere
equations on the magnetic Lie groups may be considered as certain analogs of
the Euler--Amp\`ere equations on the magnetic Lie algebras with dynamical
Biot--Savart operators. So it is reasonable to explicate the algebraic
structure of the dynamical Biot--Savart operators.
Let $x\in V_1$, $u\in V_2$, then one may define an element $R_{x,u}$ of
$\Der(V_1,V_2)$ in the following way:
$$R_{x,u}(y)=[x,y]_u\quad(u\in V_1),\qquad R_{x,u}(v)=[u,v]_x\quad(v\in V_2).$$
Such derivations will be called {\it internal}.
\proclaim{Lemma 6} The internal derivations of an isotopic pair $(V_1,V_2)$
form the Lie algebra $\Inn(V_1,V_2)$, which is an ideal of $\Der(V_1,V_2)$.
\endproclaim
\demo{Proof} One should use the Faulkner--Ferrar identities once more to prove
that $\Inn(V_1,V_2)$ is the Lie algebra and the definition of the derivation
to prove the second part of the Lemma.
\enddemo
If $(V_1,V_2)$ is the contragredient isotopic pair then the elements of
$\Inn(V_1,V_2)$ define pairs of mappings from $\Hom(V_2^*,V_1)=\End(V_1)$ and
$\Hom(V_1^*,V_2)=\End(V_2)$. So it is rather natural to suppose that the
dynamical Biot--Savart operators are corresponded to certain elements of
$\Inn(\frak g_1,\frak g_2)$ ($\frak g_1\simeq\frak g_2\simeq\frak g$) or
more generally to elements of $\Der(\frak g_1,\frak g_2)$. Such setting
explicates an algebraic structure of the reduced Euler--Amp\`ere dynamics
of Theorem 2.
\remark{Remark 18} Note that the dependence of the Poisson brackets of
any system on the states of the same system is hidden into the dynamical
Biot--Savart operators so that the reduced Euler--Amp\`ere equations
receive "visually" a form claimed by Def.3.
\endremark
\head VI. Perspectives: Quantum Euler--Amp\`ere dynamics
\endhead
The performed algebraisation of the nonHamiltonian magnetic--type interactions
is sufficient for the quantization of the Euler--Amp\`ere dynamics, i.e.
a certain solution of the dynamical inverse problem of the representation
theory in sense of [Ju6].
\remark{Remark 19} Quantum analogs of the magnetically interacting Euler
tops may be used as models for the magnetic interactions of atomic nuclei,
nucleons and nucleonic pairs, Cooper pairs, charged solitons, quantum
vortices, etc. They may be also adopted for a description of the first
quantum gauge field (wionic) approximations of the electroweak interactions
of particles (the analog of the quantized Maxwell electromagnetic theory is
the gauge Weinberg--Salam electroweak theory, and the analog of the Hamiltonian
Darwin approximation is the Fermi theory or the V-A current theory
of Gell--Mann, Feinmann, Marshak and Sudarshan [O1,MR,O2]). It seems that the
quantum nonHamiltonian constructions may be important for many concrete
applications, e.g. the quantum gyroscopes [PS,PRS:Ch.XI], the high $T_c$
superconductive technologies, the spin generators and amplifiers [PRS:Ch.VI],
quantum processors and quantum ROM/RAM media, etc.
\endremark
The quantum mechanical version of the Euler--Amp\`ere dynamics is proposed to
be discussed in the forthcoming paper, where the general questions of the
quantum nonHamiltonian interactions will be considered.
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\enddocument