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This article is cited in 1 scientific paper (total in 1 paper) On tensor invariants for integrable cases of Euler, Lagrange and Kovalevskaya rigid body motion A. V. Tsiganov Saint Petersburg State University
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    § 1. IntroductionConsider an autonomous differential vector field $X$ on a smooth phase space with coordinates $x=(x^1,\dots,x^n)$. This vector field is involved in the definition of a system of ordinary differential equations or a dynamical system and in the definition of the invariance equation where $\mathcal L_X$ is the Lie derivative of the tensor field $T$ along the vector field $X$. The solutions of equation (1.2) are constant, on the solution of equations (1.1). For a modern account of relations between equations (1.1) and (1.2), and the corresponding references, see Kozlov [1]–[4] and the recent paper [5].The Lie derivative $L_X T$ determines the rate of change of the tensor field $T$ under the phase space deformation defined by the flow of system (1.1). In local coordinates, the Lie derivative of the tensor field $T$ of type $(p, q)$ is given by
$$
\begin{equation}
\begin{aligned} \, \!\!\! (\mathcal{L}_XT)^{i_1\dots i_p}_{j_1\dots j_q}&=\sum_{k=1}^n X^{k}(\partial _{k}T^{i_1\dots i_p}_{j_1\dots j_q}) -\sum_{\ell=1}^n (\partial _{\ell}X^{i_1})T^{\ell i_2\dots i_p}_{j_1\dots j_s}-\dots - \sum_{\ell=1}^n (\partial _{\ell}X^{i_p})T^{i_1\dots i_{p-1}\ell}_{j_1\dots j_s} \nonumber \\ &\quad+\sum_{m=1}^n(\partial _{j_1}X^m)T^{i_1\dots i_p}_{m j_2\dots j_q}+\dots +\sum_{m=1}^n(\partial _{j_q}X^m)T^{i_1\dots i_p}_{j_1\dots j_{q-1}m}=0, \end{aligned}
\end{equation}
\tag{1.3}
$$
where $\partial_s= {\partial}/{\partial x^s}$ is the partial derivative in the $x^s$ coordinate (see [6]). The Lie derivative commutes with the exterior differentiation operation and satisfies the Leibniz rule. These facts allows us to construct tensor invariants from a set of basic invariant tensor fields which either have a simpler functional dependence on the variables $x$, or have some special properties or a physical interpretation. As a standard example, we take the Hamiltonian vector fields on the symplectic manifold $M=\mathbb R^{2n}$ where $P$ is an invariant Poisson bivector satisfying the Poisson identity and independent of the coordinates and momenta. The second factor $dH$ in the tensor product (1.4) is an invariant 1-form constructed by differentiating the Hamilton function $H$, that is, the scalar invariant of equation (1.2), which usually coincides with the mechanical energy of the dynamical system (1.1) (see[7]–[9]).Following Jacobi [7], we usually impose an additional constraint on the invariant tensor field $P$ in expression (1.4) for the vector field $X$ (the Jacobi identity) This identity ensures that the Poisson bracket of integrals of motion $f_1$ and $f_2$ is again an integral of motion
$$
\begin{equation*}
f_3=P\, df_1\wedge df_2\quad \Longrightarrow\quad \mathcal L_X f_3=0.
\end{equation*}
\notag
$$
Here, [[X,Y]] is the Schouten–Nijenhuis bracket of the tensor fields $X$ and $Y$ (see [6]). In the Poincaré–Cartan theory of invariants [10], [11], the Jacobi condition (1.5) is redundant, because so is the tensor product of invariants
$$
\begin{equation*}
\mathcal L_X(P\, df_1\wedge df_2)=(\mathcal L_X P)\, df_1\wedge df_2+P\, d(\mathcal L_X f_1)\wedge df_2+ P\, df_1\wedge d(\mathcal L_X f_2)=0,
\end{equation*}
\notag
$$
irrespective of other properties of tensor invariants. So, the properties of the Lie derivative do not depend on the properties of the tensor fields to which it is applied, that is, they do not depend on symmetry or antisymmetry at index permutation or on the equality to zero of the Schouten–Nijenhuis bracket, which is equivalent to fulfillment of the Jacobi identity for the Poisson brackets corresponding to an invariant bivector of type $(2,0)$ and of its analogues for fields of other types [12]–[15]. For the outer product of vector fields, the Schouten–Nijenhuis bracket is defined using the Lie bracket $[\,{\cdot}\,,{\cdot}\,]$ as follows:
$$
\begin{equation}
\begin{aligned} \, &[\![X_1\wedge\dots \wedge X_k,\, Y_1\wedge \dots \wedge Y_\ell]\!] \nonumber \\ &\qquad=\sum_{i,j}(-1)^{i+j} [X_i,Y_j] \wedge X_1 \wedge \dots \wedge \widehat{X}_i\wedge \dots \wedge X_k\wedge Y_1\wedge \dots \wedge \widehat{Y}_j\wedge \dots \wedge Y_\ell. \end{aligned}
\end{equation}
\tag{1.6}
$$
Here, $\widehat{X}_i$ and $\widehat{Y}_j$ mean that the corresponding vector fields ${X}_i$ and ${Y}_j$ are missing in the outer product of the vector fields [16]. If the divergence of the vector field $X$ is zero
$$
\begin{equation}
\operatorname{div} X=\partial_1 X^1+\partial_2 X^2+\dots+\partial_n X^n=0,
\end{equation}
\tag{1.7}
$$
then, according to (1.3), the Lie derivative of completely antisymmetric unit tensor fields $\Omega$ of type $(0,n)$ and $\mathcal E$ of type $(n,0)$ is zero:
$$
\begin{equation}
\Omega =dx^1\wedge dx^2\wedge\dots\wedge x^n, \qquad \mathcal L_X\Omega =0,
\end{equation}
\tag{1.8}
$$
$$
\begin{equation}
\mathcal E = \partial_1\wedge\partial_2\wedge\dots\wedge\partial_n, \qquad \mathcal L_X \mathcal E =0.
\end{equation}
\tag{1.9}
$$
In the general case, the invariant tensor $\mathcal E$ does not satisfy the Jacobi identity (see the discussion in Takhtajan [15]), but it does not play any role in the theory of invariants (unlike, for example, the Poisson geometry). The study of invariance of the differential form $\Omega$, which began in 1838 with the Liouville theorem [17] on conservation of the volume form for the divergence-free vector fields is continued in the Poincaré–Cartan theory of absolute, relative, and conditional invariants [10], [11]. The invariance of the multivector field $\mathcal E$ is a key element of the Jacobi theory of functional determinants [18]–[20], [7], which was developed in 1841–1845s. We will consider this theory below.The invariant tensor fields $\Omega$ and $\mathcal E$ give rise to a tower of solutions of the invariance relation (1.2) if we know the scalar integrals $f_1,\ldots,f_k$ or integral invariants. For instance, we can construct the invariant multivector fields
$$
\begin{equation}
T=\mathcal E,\quad T_i=\mathcal E\, df_i,\quad T_{ij}=\mathcal E \, df_i \, df_j,\quad T_{ijk}=\mathcal E\, df_1\, df_j\, df_k,\quad \dots
\end{equation}
\tag{1.11}
$$
and the corresponding invariant brackets of functions on the phase space. We emphasise here that we are talking about global tensor invariants uniquely defined on the whole phase space, since locally, for example, near the Liouville tori, for Kolmogorov non-degenerate systems, all solutions of the invariance equation (1.2) in the spaces of differential 2-forms and Poisson bivectors can be constructed using $\Omega$, $\mathcal E$, and the first integrals according to the Bogoyavlenskij classification [21]. A natural question here is about existence of invariance solutions which cannot be obtained via tensor operations from some family of base tensor invariants (for example, of the form (1.11)). According to Poincaré, we then should be able to acquire information on solutions of ODE (1.1) from all existing tensor invariants satisfying the invariance equation (1.2). In this paper, we will discuss solutions of the invariance equation (1.2) for the equations of motion describing the rotation of a rigid body for three well-known integrable cases: Euler, Lagrange, and Kovalevskaya, a description of which can be found in the book [22]. In all these cases the divergence of the vector field is zero and to find the tensor invariants we will solve the invariance equation (1.2) using polynomial substitution on the variables $x_1,\dots,x_n$ for the components of the tensor field $T$ of the form
$$
\begin{equation}
T^{i_1\dots i_p}_{j_1\dots j_q}=\sum_{k,\ell,m=1}^n (a^{i_1\dots i_p}_{j_1\dots j_q})^{k\ell m}x_k x_\ell x_m +\sum_{k,\ell}^n (b^{i_1\dots i_p}_{j_1\dots j_q})^{k\ell}x_k x_\ell +\sum_{k=1}^n (c^{i_1\dots i_p}_{j_1\dots j_q})^{k}x_k+d^{i_1\dots i_p}_{j_1\dots j_q},
\end{equation}
\tag{1.12}
$$
where $a$, $b$, $c$ and $d$ with the corresponding indices are the unknown real parameters, which are determined in the process of solving the system of algebraic equations resulting from the substitution of (1.12) into the invariance equation (1.2). We use various computer algebra systems to find, classify, and verify solutions. According to [23], computer computation is capable of finding new solutions and then drawing attention to necessary analytical study of properties of computer-generated solutions. The main outcome of this mathematical experiment is the observation that, for Lagrange and Kovalevskaya systems, all solutions to the invariance equation (1.2) in the space of differential forms can be constructed from $\Omega$ and first integrals by means of differentiation, tensor multiplication, etc. Conversely, in the space of multivector fields, there exist solutions that cannot be obtained from the basic invariant $\Omega$, $\mathcal E$ and first integrals by means of the same operations. 1.1. The Jacobi theory of functional determinantsIn modern literature, the term “Jacobian” refers to both the Jacobian matrix and its determinant. The Jacobian matrix collects all first-order partial derivatives of a multivariate function, which can be used, for example, for the backpropagation algorithm in training a neural network (see, for example, the studies of Nobel laureate in physics for 2024). The Jacobian determinant (or the functional determinant) is useful in changing between variables, where it acts as a scaling factor between coordinate spaces. Of course, Jacobi was not the first to study the functional determinant, which now bears his name — the first study dates back to a 1815 paper by Cauchy, see the historical remarks in [24]. The name “Jacobian” for functional determinants was proposed in 1853 by Sylvester (see p. 476 in [25]), with a reference to Jacobi’s paper [18] of 1841. In this paper, Jacobi proved, among many other things, that if $n$ functions of $n$ variables are functionally related, then the Jacobian is identically zero, while if the functions are independent the Jacobian cannot be identically zero. A few years later, in 1844–45s, Jacobi published several memoirs [19], [20] on the method of multipliers, which extends, to systems of ordinary differential equations, the Euler multiplier method [26]. In his lectures on dynamics delivered in 1842–43s (published posthumously in 1866, a reprinted (slightly revised) appeared in 1884 as a supplement to his Gesammelte Werke [7] and translated into English in [7]), Jacobi developed in Lecture 13 the theory of functional determinants, essentially following [18], and, in Chap. 14, the theory of multipliers. On this occasion, in [7], pp. 104-106, and in [7], pp. 112–114, he again proved the main properties of Jacobian without isolating they as separate statements. The Jacobi theory rapidly became a classical tool, for instance, see textbooks on analytical mechanics by Appel [8] and Whittaker [9], the course of lectures for engineers by de la Vallée–Poussin [27], the course of mathematical analysis by Goursat [28], and by Bianchi’s course on differential geometry [29]. The first editions of these textbooks were published at almost the same time at the beginning of the 19th century. Instead of the original Jacobi notation for functional determinants
$$
\begin{equation}
Q=\sum \pm \frac{\partial g_1}{\partial x_1}\, \frac{\partial g_2}{\partial x_2}\cdots\frac{\partial g_n}{\partial x_n},
\end{equation}
\tag{1.13}
$$
these textbooks used the notation
$$
\begin{equation*}
Q=\frac{\partial (g_1,g_2,\dots, g_n)}{\partial (x_1,x_2,\dots,x_n)},
\end{equation*}
\notag
$$
which we will use below. Following Jacobi [20] and the textbooks [10], [11], consider the system of differential equations
$$
\begin{equation}
\frac{dx_1}{X_1}=\frac{dx_2}{X_2}=\dots=\frac{dx_n}{X_n},
\end{equation}
\tag{1.14}
$$
where $X_i$ are functions on phase space with coordinates $x_1,\dots,x_n$. In § 1.1, for components of vectors and vector fields, we use the lower indexes, in lieu of the upper indexes adopted in modern studies [4], [5]. Any function $f(x_1,\dots,x_n)$ that is constant for $x_1,\dots,x_n$ and satisfies (1.14) is called the first integral; Jacobi wrote this condition in the form
$$
\begin{equation}
X(f)=\frac{\partial f}{\partial x_1}X_1+\frac{\partial f}{\partial x_2}X_2+\dots+\frac{\partial f}{\partial x_n}X_n=0
\end{equation}
\tag{1.15}
$$
(for the “sake of brevity”, as he puts). With the help of the Lie derivative, this equation can be written as System (1.14) can only have $n-1$ independent first integrals $f_1,\dots,f_{n-1}$, and any other integral $f$ is a function of $f_1,\dots,f_{n-1}$ because the functional determinant is zero
$$
\begin{equation*}
\frac{\partial(f,f_1,\dots,f_{n-1})}{\partial(x_1,\dots,x_n)}=0.
\end{equation*}
\notag
$$
Expanding this determinant along the top row, this gives
$$
\begin{equation}
\frac{\partial f}{\partial x_1}\Delta_1+\frac{\partial f}{\partial x_2}\Delta_2+\dots+\frac{\partial f}{\partial x_n}\Delta_n=0,
\end{equation}
\tag{1.16}
$$
where $\Delta_i$ is the algebraic complement of the $i$th element on the first row of the Jacobian. For $X(x)\neq 0$, the coefficients on the derivatives in equations (1.15) and (1.16) are proportional, that is, there exists a function $M(x)$ such that The function $M(x)$ is called the multiplier of the system of equations (1.14); this multiplier satisfies the linear partial differential equation
$$
\begin{equation}
\operatorname{div}(MX)=\partial_1(MX_1)+\partial_2 (MX_2)+\dots+\partial_n(MX_n)=0,
\end{equation}
\tag{1.18}
$$
where the first integrals $f_1,\dots,f_{n-1}$ are missing. The importance of the Jacobi multiplier $M$ comes from the fact that if $M$ and $M'$ are solutions of (1.18), then their quotient $M/M'$ is a first integral of the dynamical system (1.1). Relations between these first integrals and symmetries are discussed in the Bianchi textbook [29], see also [30]. From the definition of multiplier (1.17), one immediately gets a representation of the functions $X_i$ involved in the definition of equations (1.14) in terms of the functional determinant and the multiplier
$$
\begin{equation*}
MX=\frac{\partial(x,f_1,\dots,f_{n-1})}{\partial(x_1,\dots,x_n)},
\end{equation*}
\notag
$$
see p. 253 in [27] and the discussion by Kozlov [4], [31]. Thus, Jacobi obtained a representation of the vector field $X$ via the functional determinant and the multiplier
$$
\begin{equation}
X=\frac{1}{M(x)}\,\frac{\partial(x,f_1,\dots,f_{n-1})}{\partial(x_1,\dots,x_n)},
\end{equation}
\tag{1.19}
$$
that is,
$$
\begin{equation*}
X_i=\frac{1}{M(x)}\,\Delta_i=\frac{1}{M(x)}\, \frac{\partial(x_i,f_1,\dots,f_{n-1})}{\partial(x_1,\dots,x_n)}.
\end{equation*}
\notag
$$
Jacobi also proved that if the divergence of the field $X$ is zero (1.7), then the multiplier $M(x)$ is a function of the first integrals, which can always be chosen so that $M(x)=1$ and
$$
\begin{equation*}
X=\frac{\partial(x,f_1,\dots,f_{n-1})}{\partial(x_1,\dots,x_n)}.
\end{equation*}
\notag
$$
The other way around, if a dynamical system is representable via the Jacobians of the first integrals, then it is divergence-free, see the corresponding theorems in § 229 of the de la Vallée–Poussin lectures [27]. Definition (1.19) includes the completely antisymmetric unit tensor $\mathcal E$, and hence the vector field $X$ can be rewritten as
$$
\begin{equation}
X=\frac{1}{M(x)}\,\mathcal E\, df_1\cdots df_{n-1}=T_n\omega_{n-1},
\end{equation}
\tag{1.20}
$$
where
$$
\begin{equation*}
T_n=\frac{1}{M(x)}\,\mathcal E \quad\text{and}\quad \omega_{n-1}=df_1\cdots df_{n-1}
\end{equation*}
\notag
$$
are two tensor invariants of $X$, that is $\mathcal L_X T_n=0$ and $\mathcal L_X \omega_{n-1}=0$. To prove these results, Jacobi uses the fundamental lemma given on p. 203 of the first part of [20], written in Latin, according to which the algebraic complements $\Delta_{ij}$ in the Jacobian of the mapping $g\colon\mathbb R^n\to \mathbb R^n$ satisfy
$$
\begin{equation}
\sum_{j=1}^n \frac{\partial}{\partial x_j}\,\Delta_{ij}=0,
\end{equation}
\tag{1.21}
$$
so that
$$
\begin{equation}
\frac{\partial(g_1,g_2,\dots,g_{n})}{\partial(x_1,\dots,x_n)}=\sum_{j}^n \biggl( \frac{\partial g_i}{\partial x_j}\biggr)\Delta_{ij} =\sum_{j}^n \frac{\partial }{\partial x_j}(g_i\Delta_{ij}).
\end{equation}
\tag{1.22}
$$
The Jacobi complete proof of this lemma is reproduced in the Appel textbook [8], pp. 460–461, which, according to his contemporary D. K. Bobylev, was the best course on theoretical mechanics of its time. The Jacobi theory of functional determinants is applied not only in the theory of ordinary differential equations and analytical mechanics. For example, in 1910 Hadamard, in an appendix to Tannery’s textbook on real functions [32], introduced the notion of Kronecker index for continuous non-vanishing mappings on smooth boundary using a generalization of equations (1.21) and (1.22). The same formulas are used in the definition of the Brouwer degree in [33], in the Brouwer fixed point theorem [34], in the theory of quasi-convex functions [35], in the theory of Jacobian in the sense of distributions [36], and in the theory of null Lagrangians [37], etc. 1.2. Systems with $n-1$ first integralsIn [7], Jacobi treats the theory of functional determinants as an introduction to the construction and study of Hamiltonian systems, see the last paragraph of his Lecture 18. Indeed, for divergence-free vector fields one can set $M(x)=1$ and represent the functional determinant (1.19), (1.20) in various ways as a combination of elements of the Jacobian and their algebraic complements. The tower of brackets corresponding to these combinations for arbitrary functions $g_1,\dots,g_m$, $2\leqslant m\leqslant n$, on the phase space depends on the base tensor invariants $\mathcal E$ and $f_1,\dots, f_{n-1}$,
$$
\begin{equation}
\begin{aligned} \, \{g_1,g_2,\dots,g_n\} &=\frac{\partial(g_1,g_2,\dots,g_{n})}{\partial(x^1,x^2,\dots,x^n)}=\mathcal E\, dg_1\cdots dg_{n}, \\ \{g_1,\dots,g_{n-1}\}_{\alpha_1} &= \frac{\partial(g_1,\dots,g_{n-1}, f_{\alpha_1})}{\partial(x^1,x^2,\dots,x^n)}=T_{\alpha_1}\, dg_1\cdots dg_{n-1}, \\ \{g_1,\dots,g_{n-2}\}_{\alpha_1\alpha_2} &= \frac{\partial(g_1,\dots,g_{n-2},f_{\alpha_1},f_{\alpha_2})}{\partial(x^1,x^2,\dots,x^n)} =T_{\alpha_1,\alpha_2}\, dg_1 \cdots dg_{n-2}, \\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \\ \{g_1,g_2\}_{\alpha_1\dots\alpha_{n-2}} &=\frac{\partial(g_1,g_2,f_{\alpha_1},\dots,f_{\alpha_{n-2}} )}{\partial(x^1,x^2,\dots,x^n)}=T_{\alpha_1,\dots,\alpha_{n-2}} \, dg_1 \, dg_2. \end{aligned}
\end{equation}
\tag{1.23}
$$
If $[\![\mathcal E,\mathcal E]\!]=0$, both these brackets and their linear combinations satisfy the Jacobi identity — this is a consequence of the fundamental Jacobi lemma. Invariant Poisson brackets for $m$ functions $g_1,\dots,g_m$ on the phase space can be used for description of the original invariant vector field $X$. For example, for $m=2$, we have standard Poisson brackets for two functions, and so we can assert that the vector field $X$ is a Hamiltonian vector field (1.4). Proposition 1. The divergence-free vector field (1.7) with $n-1$ independent first integrals is Hamiltonian with respect to each of its first integrals The corresponding Poisson invariant bivectors have the form To prove the Jacobi identity for these brackets, we can use the fundamental Jacobi lemma and the expression for the components of the bivector $P_k$ in terms of the algebraic complements $\Delta^{ij}_k$ of the Jacobian elements of dimension $(n-2)\times(n-2)$
$$
\begin{equation*}
P_k^{ij}=(-1)^{i+j}\Delta^{ij}_k,\qquad \Delta^{ij}_k=\frac{\partial(\widehat{x},f_1,\dots,\widehat{f}_k,\dots, f_{n-1})}{\partial(x^1,\dots,\widehat{x}^i,\dots,\widehat{x}^j,\dots,x^n)},
\end{equation*}
\notag
$$
which are obtained by subtracting the first and $k$th rows and the $i$th and $j$th columns from the Jacobian (1.19). On the other hand, since
$$
\begin{equation*}
\{g_1,g_2\}_k=\{g_1,g_2,f_1,\dots,\widehat{f}_k,\dots,f_{n-1},\},
\end{equation*}
\notag
$$
we can say that the Jacobi identity follows from the equality to zero of the Schouten–Nijenhuis bracket for the basic tensor invariant $\mathcal E$. The same Poisson brackets can be obtained using the second basic invariant $\Omega$ (see [38]). In classical mechanics, examples of the use of the Jacobi theory of functional determinants include the Euler equations describing the rotation of a rigid body [39], the problem of the attraction of a point to a fixed centre in a resistive medium and in empty space [7], the Jacobi system of three bodies attracted by a force proportional to the cube of the inverse of the distance between them (a particular case of the Calogero–Moser system) [40], and other dynamical systems considered by Jacobi himself [30]. We should also mention the series of Volterra papers of 1895 (see [41]) on the Chandler oscillation of the Earth poles, where a representation of the vector field in terms of the functional determinant (1.19) is used for the study of the Euler equations of a rigid body with gyrostat, and Jacobi’s lectures [7], where, as an exercise in the multiplier theory, additional first integrals are found in the Kepler problem — these integrals are known as components of the Runge–Lenz vector in physics. For homogeneous dynamical systems with quadratic integrals, a representation of the vector field via Jacobian of the first integrals was obtained in [42]; see also [43] and [31]. The representation of a vector field as a functional determinant
$$
\begin{equation*}
X=\frac{\partial(x,f_1,\dots,f_{n-1})}{\partial(x^1,\dots,x^n)}
\end{equation*}
\notag
$$
for a Euler system without gyrostat was rediscovered by Nambu [44], which, after the work of Takhtajian [15], led to the creation of generalized Nambu mechanics — clearly, this is a particular case of the Jacobi multiplier theory. The Jacobi results for the Kepler problem were rediscovered in [45], see also the paper [46] concerned with dynamical systems with three degrees of freedom. A discussion of the Poisson–Nambu geometry and related references can be found, for example, in [47]–[49]. Similarly to the Kepler problem and the Jacobi three-body problem (a particular case of the Calogero–Moser system), the Jacobi representation of the vector field (1.19), (1.20) exists for all maximally degenerate or superintegrable Hamiltonian systems on a symplectic manifold of dimension $n=2m$, since the divergence of the Hamiltonian field is zero and maximally degenerate systems have $n-1$ independent first integrals — this number of integrals is necessary for this representation. There are many examples of such degenerate systems, see examples and references in [50]–[53]. A generalization of representation (1.19) to the more general case, where the system of differential equations (1.1) admits $n -1$ absolute integral invariants, is discussed by Kozlov [4], where the equations of motion for the Goryachev–Chaplygin top are considered as one of the examples. 1.3. Systems with $n-2$ first integralsThe equations of motion (1.1) for the vector field $X$ (see (1.19)) with known $n-1$ independent first integrals are integrable in quadratures; this is also so for the equations of motion for a volume preserving vector field $X$
$$
\begin{equation*}
\Omega=M(x) \, dx^1\wedge dx^2\wedge\dots\wedge dx^n,
\end{equation*}
\notag
$$
and $n-2$ independent first integrals $f_1,\dots, f_{n-2}$ (see the modern account of the Euler–Jacobi theorem on integrability and its generalizations in [2]–[4]). Here, $M(x)$ is an invariant measure. In this case, $M(x)$ still satisfies equation (1.18), but is not a first integral of the system of equations (1.14) (see [19] and [20]). The proof of this theorem can be reduced to the previous case with $n-1$ first integrals by constructing the $(n-1)$st independent first integral where $y^{1,2}$ and $Y^{1,2}$ are the coordinates and components of the vector field near the invariant manifold given by $f_1=c_1$, $\dots$, $f_{n-2}= c_{n-2}$.It is quite natural to ask about the representation of the vector field $X$ satisfying the conditions of the Euler–Jacobi theorem in a form analogous to (1.20): so that
$$
\begin{equation*}
L_X X=(L_XT_m)\omega_{m-1}+T_m(\mathcal L_X\omega_{m-1})=0.
\end{equation*}
\notag
$$
Here, $T_m$ is a multivector field of type $(m,0)$ with components that do not depend on the coordinates $x$, and $\omega_{m-1}$ is an invariant differential form of type $(0,m-1)$. In what follows, we will prove that, in both the Lagrange and Kovalevskaya cases, the invariance condition $\mathcal L_X T=0$ (see (1.2)) can actually be replaced by the weaker one
$$
\begin{equation}
(\mathcal L_X T_m)\omega_{m-1}=0,\qquad \mathcal L_X\omega_{m-1}=0.
\end{equation}
\tag{1.26}
$$
Following the terminology of the Poincaré–Cartan theory of invariants [10], [11]. solutions $T$ of the invariance equation (1.2) will be called absolute tensor invariants, and solutions $T_m$ of equation (1.26) will be referred to as conditional tensor invariants depending on the absolute invariant $\omega_{m-1}$. For a vector field of the form (1.25) , one can similarly construct a counterpart for the tower of Poisson brackets and Poisson bivectors, and so the topic of tensor invariants under consideration is closely related to the problem of Hamiltonization of vector fields possessing an invariant measure. Recall that the Poisson brackets appearing with Hamiltonization of concrete dynamical systems (integrable by the Euler–Jacobi theorem) usually depend on the phase space variables in a rather complicated way [54]–[59]. We can assume that the corresponding Poisson bivectors can be represented as a tensor product of numerical tensor fields $T_m$, more simple vector fields $T$, and $1$-forms of the first integrals $df_i$. § 2. The Euler caseWe start with the Euler equations, which describe the motion of a rigid body
$$
\begin{equation*}
\dot{M}_1=(a_2 - a_3)M_2M_3,\qquad \dot{M}_2=(a_3 - a_1)M_3M_1,\qquad \dot{M}_3=(a_1 - a_2)M_1M_2,
\end{equation*}
\notag
$$
where $M_1$, $M_2$ and $M_3$ are the coordinates on the phase space of this autonomous system of differential equations (1.1), $a_1$, $a_2$ and $a_3$ are arbitrary real parameters [22]. A decomposition of the corresponding vector field $X$ in the basis of the vector fields $\partial_i=\partial/\partial M_i$, $i=1,2,3$, has the form
$$
\begin{equation}
X=b_1M_2M_3\,\partial_1+b_2M_3M_1\,\partial_2+b_3M_1M_2\,\partial_3,
\end{equation}
\tag{2.1}
$$
where Following [22], we choose polynomials of second degree in the variables $M_k$ as independent scalar invariants
$$
\begin{equation*}
f_1=\frac12(a_1M_1^2+a_2M_2^2+a_3M_3^2),\qquad f_2=M_1^2+M_2^2+M_3^2.
\end{equation*}
\notag
$$
Any function of these first integrals $g(f_1,f_2)$ is also a solution of the invariance equation (1.2), as well as the differential forms of type $(0,1)$ and $(0,2)$ generated by the first integrals
$$
\begin{equation}
\omega_1=dg(f_1,f_2),\qquad \omega_2=dg_1(f_1,f_2)\, dg_2(f_1,f_2).
\end{equation}
\tag{2.2}
$$
Here, $g_{1,2}(f_1,f_2)$ are arbitrary functions of the first integrals, and the tensor product can be symmetrized or alternated, see the discussion in [4]. The degenerate differential forms thus obtained can be used analogously to their application in vortex theory [10], [11], [60]. The divergence of the vector field $X$ is zero and, therefore, in the space of differential forms of type $(0,3)$, the basic solution of the invariance equation (1.2) is a volume form and the basic solution in the space of multivector fields of type (3,0) has the form
$$
\begin{equation}
\mathcal E =\partial_1\wedge \partial_2 \wedge \partial_3.
\end{equation}
\tag{2.3}
$$
The Schouten–Nijenhuis bracket (1.6) is zero $[\![\mathcal E,\mathcal E]\!]=0$ and, therefore, this trivector defines the Poisson bracket
$$
\begin{equation*}
\{g_1,g_2,g_3\}=\sum_{i,j,k=1}^3 \mathcal E^{ijk}\, \frac{\partial g_1}{\partial M_i} \, \frac{\partial g_2}{\partial M_j}\, \frac{\partial g_3}{\partial M_k}
\end{equation*}
\notag
$$
on the phase space, such that The products of the invariant trivector $\mathcal E$ and the invariant 1-forms $df_1$ and $df_2$ are compatible Poisson bivectors on the Lie–Poisson algebra $\mathrm{so}^*(3)$
$$
\begin{equation*}
\begin{aligned} \, P_1&=\mathcal E \, df_1=\begin{pmatrix} 0 & a_3M_3 & -a_2M_2 \\ -a_3M_3 & 0 & a_1M_1 \\ a_2M_2 & -a_1M_1 & 0 \end{pmatrix}, \\ P_2 &=\mathcal E \, df_2= \begin{pmatrix} 0 & M_3 & -M_2 \\ -M_3 & 0 & M_1 \\ M_2 & -M_1 & 0 \end{pmatrix}. \end{aligned}
\end{equation*}
\notag
$$
These bivectors satisfy
$$
\begin{equation*}
\begin{alignedat}{2} X &=P_1\, df_2=\mathcal E \, df_1\wedge df_2,&\qquad P_1\, df_1&=\mathcal E \, df_1\wedge df_1=0, \\ X&=-P_2 \, df_1=-\mathcal E \, df_2\wedge df_1,&\qquad P_2\, df_2&=\mathcal E \, df_2\wedge df_2=0. \end{alignedat}
\end{equation*}
\notag
$$
Recall that two Poisson bivectors $P_1$ and $P_2$ are compatible if their linear combination $P_1 +\lambda P_2$ is also a Poisson bivector for any $\lambda\in\mathbb R$. The Poisson tensor $T\, dg(f_1,f_2)$ constructed from the function $g$ of $f_{1,2}$ is also compatible with the Poisson tensors $P_1$ and $P_2$. Instead of using a basis of bivectors $P_{1,2}$ in the space of $(2,0)$ type vector fields, we can also use a frame of three invariant bivectors
$$
\begin{equation}
P'_i=\partial_i\wedge \mathcal L_X \, \partial_i,\qquad \partial_i=\frac{\partial}{\partial M_i},\quad i=1,2,3,
\end{equation}
\tag{2.4}
$$
related by The Schouten–Nijenhuis bracket (1.6) between these bivectors is zero and so we have three Poisson bivectors compatible to each other. The vector field $X$ (see (2.1)) is Hamiltonian with respect to any of these invariant Poisson tensors
$$
\begin{equation*}
X=\frac{1}{a_i}P'_i\, df_1,\quad X=\frac{1}{2}P'_i\, df_2,\qquad i=1,2,3.
\end{equation*}
\notag
$$
Let us now describe the results of our mathematical experiment. If we substitute (1.12) into (1.2) and solve the resulting equations, we get all the above tensor invariants and their combinations. In addition, in the space of tensor fields of type $(3,0)$, we found another invariant tensor field $T$ whose components are homogeneous cubic polynomials (1.12) of $M_i$. This solution depends on four parameters, cannot be represented as a sum of absolutely antisymmetric and absolutely symmetric parts, and is uniquely determined by the fact that the multiplication of $T$ by the invariant 1-forms $df_1$ and $df_2$ gives a single invariant symmetric tensor of type $(2,0)$ up to the factor
$$
\begin{equation*}
\begin{aligned} \, P_s^{ij} &=c_{11}\sum_{k=1}^n T^{ijk}\, \frac{\partial f_1}{\partial M_k}= c_{12} \sum_{k=1}^n T^{ikj}\, \frac{\partial f_1}{\partial M_k}=c_{13}\sum_{k=1}^n T^{kij}\, \frac{\partial f_1}{\partial M_k}, \\ P_s^{ij} &=c_{21}\sum_{k=1}^n T^{ijk}\, \frac{\partial f_2}{\partial M_k}= c_{22} \sum_{k=1}^n T^{ikj}\, \frac{\partial f_2}{\partial M_k}=c_{23}\sum_{k=1}^n T^{kij}\, \frac{\partial f_2}{\partial M_k}, \end{aligned}
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
c_{11}+c_{12}+c_{13}=c_{21}+c_{22}+c_{23}=0,\qquad c_{ij}\in\mathbb R.
\end{equation*}
\notag
$$
We also have the relation $T(df_1\wedge df_2)=0$. For brevity, we will not write all the components of this invariant tensor explicitly — the main thing for us is that this additional solution exists. In the space of tensor fields of type (1,1), the only solution we obtained in the experiment has the form The invariant mixed tensors of type $(2,1)$ and $(1,2)$ are combinations of the invariant tensor fields considered above.§ 3. The Lagrange caseThe equations of motion have the form
$$
\begin{equation*}
\begin{gathered} \, \dot{\gamma}_1=2(\gamma_3M_2 -\gamma_2M_3),\qquad \dot{\gamma}_2=2(\gamma_1M_3-\gamma_3M_1),\qquad \dot{\gamma}_3=2(\gamma_2M_1 - \gamma_1M_2), \\ \dot{M}_1=-a\gamma_2,\qquad \dot{M}_2=a\gamma_1,\qquad \dot{M}_3=0, \end{gathered}
\end{equation*}
\notag
$$
where $\gamma_1$, $\gamma_2$, $\gamma_3$ and $M_1$, $M_2$, $M_3$ are the coordinates on the phase space of this autonomous system of differential equations, $a$ is an arbitrary real parameter. For a discussion of the equations, and the physical meaning of the coordinates and of the parameter $a$, see, for example, the textbook [22]. The divergence of the corresponding vector field $X$ is zero, and so the phase flow preserves the volume form $\Omega$ (see (1.8)) and the fully antisymmetric unit tensor $\mathcal E$ (see (1.9)). We choose the following polynomials as independent scalar invariants
$$
\begin{equation*}
\begin{alignedat}{2} f_1&=\gamma_1^2+\gamma_2^2+\gamma_3^2,&\qquad f_2&=\gamma_1M_1+\gamma_2M_2+\gamma_3M_3, \\ f_3&=M_1^2+M_2^2+M_3^2+a\gamma_3,&\qquad f_4&=M_3; \end{alignedat}
\end{equation*}
\notag
$$
they give rise to independent invariant differential 1-forms (2.2). Proposition 2. In the Lagrange case, all solutions of the invariance equation in the space of differential forms of type $(0,2)$ and $(0,3)$ with components (1.12) are tensor products of the invariant 1-forms The proof depends on an explicit solution of the invariance equation via substitution (1.12) and analysis of the solutions thus obtained. The experimental fact that there are no other solutions in the space of differential forms for this and other integrable systems seems to be one of the reasons for the limited applicability of the mathematical theory of Poincaré–Cartan integral invariants in classical mechanics. In contrast to the space of differential forms, in the space of multivector fields of type $(n, 0)$, the set of solutions of the invariance equation is not exhausted by tensor operations on some small set of basic invariants. Indeed, in the space of $(1,0)$ tensor fields; the general solution of the invariance equation is as follows: In the space of vector fields of type $(1,0)$, a general solution of the invariance equation has the form where
$$
\begin{equation}
Y=-\gamma_2\, \partial_1+\gamma_1\, \partial_2-M_2\, \partial_4+M_1\, \partial_5, \qquad \partial_i=\frac{\partial}{\partial \gamma_i},\quad \partial_{i+3}=\frac{\partial}{\partial M_i},\qquad i=1,2,3.
\end{equation}
\tag{3.1}
$$
In addition, in the space of vector fields, there exists a conditional invariant $\partial_6$ associated with the linear first integral $f_4=M_3$ and the corresponding Noether symmetry
$$
\begin{equation*}
L_X e_6\neq 0,\quad\text{but}\quad (L_X e_6)\, df_4=0.
\end{equation*}
\notag
$$
In what follows, we will use this symmetry-related conditional invariant for construction of other conditional invariants. 3.1. Invariant tensor fields of type $(2,0)$Substituting a tensor field $P$ of type $(2,0)$ with components of the form (1.12) into the equation $\mathcal L_XP=0$ and equating to zero the coefficients with different degrees of the variables $\gamma$ and $M$, we obtain a system of 6860 algebraic equations for 3024 parameters. A solution of this equations with standard commands on a standard PC takes about several minutes. The use of polynomial substitutions of higher order increases substantially both the number of equations and the number of possible solutions, and the code usually hangs-up (if standard commands are used). In this case, for obtaining partial solutions, it is customary to use special strategies to limit the search space for solutions. So, solving algebraic equations via computer algebra systems we obtain absolute tensor invariants, which break out into the symmetric and antisymmetric tensor fields, among the latter there are fifty-one Poisson bivectors, which in turn involve Poisson bivectors compatible with the canonical Poisson bivector $P_{\mathrm{c}}$ on the Lie–Poisson algebra $e^*(3)$. Thus, we have the following invariant Poisson bivectors with $(\gamma,M)$-linear components:
$$
\begin{equation}
\begin{aligned} \, P_1 &=\begin{pmatrix} 0& -M_3& M_2& 0& -\dfrac{a}2& 0 \\ M_3& 0& -M_1& \dfrac{a}2& 0& 0 \\ -M_2& M_1& 0& 0& 0& 0 \\ 0& -\dfrac{a}2& 0& 0& 0& 0 \\ \dfrac{a}2& 0& 0& 0& 0& 0 \\ 0& 0& 0& 0& 0& 0 \end{pmatrix}, \nonumber \\ P_2 &=\begin{pmatrix} 0& 2\gamma_3& -2\gamma_2& 0& 0& 0 \\ -2\gamma_3& 0& 2\gamma_1& 0& 0& 0 \\ 2\gamma_2& -2\gamma_1& 0& 0& 0& 0 \\ 0& 0& 0& 0& -a& 0 \\ 0& 0& 0& a& 0& 0 \\ 0& 0& 0& 0& 0& 0 \end{pmatrix}, \nonumber \\ P_3 &=P_{\mathrm{c}}= \begin{pmatrix} 0 &0 & 0 & 0 &\gamma_3 &-\gamma_2 \\ 0 & 0 & 0 & -\gamma_3 & 0 & \gamma_1 \\ 0 & 0 &0 & \gamma_2 & -\gamma_1 & 0 \\ 0 & \gamma_3 & -\gamma_2 & 0 & M_3 & -M_2 \\ -\gamma_3 & 0 & \gamma_1 & -M_3 & 0 & M_1 \\ \gamma_2 & -\gamma_1 & 0 & M_2 & -M_1 & 0 \end{pmatrix}, \end{aligned}
\end{equation}
\tag{3.2}
$$
where $P_3=P_{\mathrm{c}}$ is the canonical Poisson bivector which defines the standard Lie–Poisson bracket on the algebra $e^*(3)$
$$
\begin{equation*}
\{\gamma_i,\gamma_j\}=0,\qquad \{\gamma_i,M_j\}=\varepsilon_{ijk}\gamma_k,\qquad \{M_i,M_j\}=\varepsilon_{ijk}M_k,
\end{equation*}
\notag
$$
and $\varepsilon_{ijk}$ is is the Levi-Civita tensor [22]. The vector field $X$ is written in multi-Hamiltonian form because these bivectors are compatible with each other $[\![P_i,P_j]\!]=0$, $i,j=1,2,3$.That the Schouten–Nijenhuis brackets vanish can be checked using the representation
$$
\begin{equation*}
\begin{aligned} \, P_1&=\frac{1}2(\partial_1\wedge\mathcal L_{Z}\, \partial_1+\partial_2\wedge \mathcal L_{Z}\, \partial_2), \qquad P_2=\frac{1}{2}\sum_{i=1}^3 \partial_i\wedge\mathcal L_X\, \partial_{i+3}+a\, \partial_{4}\wedge \partial_{5}, \\ P_3&=\frac{1}{2}(\partial_4\wedge L_{Z}\, \partial_4+\partial_5\wedge \mathcal L_{Z}\, \partial_5) +\partial_6\wedge\mathcal L_{f_4Y} \, \partial_6,\qquad Z=X-f_4Y. \end{aligned}
\end{equation*}
\notag
$$
These solutions of the invariance equation were obtained and studied in [61]–[63]. We now proceed with the description of the previously unknown solutions. Proposition 3. In the space of Poisson bivectors whose components are inhomogeneous polynomials of second order in the variables $(\gamma,M)$, there are two independent solutions of the invariance equation In the space of symmetric tensor fields with components cubic in the variables $(\gamma,M)$, in addition to the tensor product of vector fields $X$ and $Y$, there are three more solutions of the invariance equation of rank 1, 2, and 4. For example, the explicit form of the solution of rank four is as follows:
$$
\begin{equation*}
P_6=X\vee\biggl( \sum_{i=1}^3 M_i\, \partial_i+a\, \partial_6 \biggr)-aY\vee Z,
\end{equation*}
\notag
$$
where $\vee$ is a symmetrized tensor product, $Y$ is the vector field (3.1) invariant with respect to the action of the initial vector field $X$, and the third vector field
$$
\begin{equation*}
Z=\sum_{i=1}^3(2 \gamma_i\, \partial_i+M_i\, \partial_{i+3})
\end{equation*}
\notag
$$
included in this definition satisfies that is, it is a conformal invariant. The products of the symmetric tensor invariant $P_6$ with the invariant differential 1-forms $df_i$, $i=1,\dots,4$, are linear combinations of the invariant vector fields $X$ and $Y$
$$
\begin{equation*}
\begin{alignedat}{2} P_6 \, df_1 &= f_2 X- 2af_1Y, &\qquad P_6\, df_2 &= \frac{f_3}{2}\,X -\frac{3af_2}{2}\,Y, \\ P_6 \, df_3 &= \frac{3af_4}{2}\,X - af_3Y, &\qquad P_6 \, df_4 &= \frac{a}{2}\, X - \frac{af_4}{2}\,Y. \end{alignedat}
\end{equation*}
\notag
$$
According to Poincaré [10], we can use symmetric tensor fields to study dynamical systems in parallel with antisymmetric tensor invariants. The divergence of the vector field $X$ (see (1.7)) is zero, and, therefore, the completely antisymmetric unit tensor $\mathcal E$ (see (1.9)) is a solution of the invariance equation (1.2). Its product with the outer product of the invariant 1-forms $df_i$
$$
\begin{equation*}
P_7=\mathcal E \, df_1\wedge df_2\wedge df_3\wedge df_4,
\end{equation*}
\notag
$$
is an invariant Poisson bivector of rank two such that $P_7\, df_i=0$, $i=1,\dots,4$. 3.2. Invariant tensor fields of type $(3,0)$The tensor product of absolute invariants of type $(2,0)$ and $(1,0)$ is an absolute tensor invariant of type $(3,0)$. The question naturally arises of whether there are other absolute invariants whose components are also inhomogeneous cubic polynomials in the variables $(\gamma,M)$. Proposition 4. The invariance equation $\mathcal L_XT=0$ has a solution of the form The proof is a direct verification of the above equations. In the space of tensor fields of type $(3,0)$ with numerical components, the invariance equation $\mathcal L_X T=0$ has no solutions. Let us therefore proceed with equation (1.26),
$$
\begin{equation}
(\mathcal L_X T)\, dH=0,\qquad H=\sum_{k=1}^4e_k f_k,\quad T\, dH\neq 0,\qquad e_k\in\mathbb R,
\end{equation}
\tag{3.5}
$$
which describes the absolute invariance of the non-trivial tensor field $P=T\, dH$ of type $(2,0)$ with respect to the action of the vector field $X$. Here, $H$ is the linear combination of the first integrals. The real parameters $e_k$ are found by solving this equation. Proposition 5. In the Lagrange case, in the space of tensor fields of type $(3,0)$ with constant components, there are only three solutions of equation (3.5) For a proof, one should solve the system of algebraic equations using computer algebra systems. The question of construction of conditional numerical invariants $T_i$ of type $(3,0)$ from an absolute numerical invariant $\mathcal E$ of type $(6,0)$ is open. The conditional invariants $T_i$ (see (3.6)) are solutions of equation (3.5) and are constructed without participation of the second vector field $Y$ under which they are absolute invariants $\mathcal L_YT_i=0$, $i=1,2,3$. The tensor product of an absolute invariant of type $(2,0)$ and a conditional invariant of type $(1,0)$ is a conditional tensor invariant of type $(3,0)$ according to the Leibniz rule of differentiation. As an example, consider the tensor product
$$
\begin{equation*}
T=P_i \, \partial_6,\qquad \partial_6=\frac{\partial}{\partial M_3},
\end{equation*}
\notag
$$
where $P_i$ is one of the absolute invariants of type $(2,0)$ found earlier, whose components are inhomogeneous polynomials in the variables $(\gamma,M)$. The conditional invariant $T$ of type $(3,0)$ obtained in this way satisfies the relations
$$
\begin{equation*}
P_i=T\, df_4,\qquad (L_X T)\neq 0,\qquad (L_X T)\, df_4=0.
\end{equation*}
\notag
$$
As another example, consider a tensor field of type $(4,0)$ represented as
$$
\begin{equation*}
T=(\partial_1\wedge\partial_2\wedge \partial_3)\,\partial _6+\widehat{T},
\end{equation*}
\notag
$$
where $\mathcal E=\partial_1\wedge\partial_2\wedge \partial_3$ is a Levi-Civita tensor of type $(3,0)$, and in the tensor field $\widehat T$, there are only two non-zero components $\widehat{T}^{4233}= 1$ and $\widehat{T}^{5133} = -1$. In this case,
$$
\begin{equation*}
X=\frac{1}{2f_4}\,T\, df_1\, df_2\, df_3,\qquad \mathcal L_XT\neq 0,\qquad (\mathcal L_X T)\, df_4=0.
\end{equation*}
\notag
$$
However, the tensor products $T$ with $df_i$ or $\omega_{i,j}=df_i\wedge df_j$ will not satisfy any of the invariance conditions under consideration. This example shows that, to find conditional invariants in spaces of tensor fields of higher valence, the number of imposed conditional invariance equations should be increased accordingly. Thus, in the space of tensor fields of type $(3,0)$ with $(\gamma,M)$ components cubic in variables (1.12), there exists a non-empty set of absolute conditional tensor invariants. The question of the properties of these invariants, their classification, and construction via arbitrary basic invariants remains open. § 4. The Kovalevskaya caseIn the Kovalevskaya case, the six equations of motion of a rigid body
$$
\begin{equation*}
\begin{gathered} \, \dot{\gamma}_1=2\gamma_3M_2 - 4\gamma_2M_3,\qquad \dot{\gamma}_2=-2\gamma_3M_1 + 4\gamma_1M_3,\qquad\dot{\gamma}_3=2\gamma_2M_1 - 2\gamma_1M_2, \\ \dot{M}_1=-2M_2M_3,\qquad \dot{M}_2=2M_1M_3 - a\gamma_3,\qquad \dot{M}_3=a\gamma_2 \end{gathered}
\end{equation*}
\notag
$$
possess three first integrals
$$
\begin{equation*}
f_1=\gamma_1^2+\gamma_2^2+\gamma_3^2, \qquad f_2=\gamma_1M_2+\gamma_2M_2+\gamma_3M_3,\qquad f_3=M_1^2 + M_2^2 + 2M_3^2 + a\gamma_1,
\end{equation*}
\notag
$$
which are polynomials of degree two, and have the first integral
$$
\begin{equation*}
f_4=(M_1^2 - M_2^2 - a\gamma_1)^2 + (2M_1M_2 - a\gamma_2)^2,
\end{equation*}
\notag
$$
with is a polynomial of degree four. The divergence of the corresponding vector field $X$ is zero, and hence the phase flow preserves the volume form $\Omega$ (see (1.8)), and, therefore, these equations are integrable in quadratures according to the Euler– Jacobi theorem [22]. Proposition 6. In the Kovalevskaya case, all solutions of the invariance equation in the space of differential forms of type $(0,2)$ and $(0.3)$ with components of the form (1.12) are tensor products of the invariant 1-forms The proof depends on an explicit solution of the invariance equation via substitution (1.12) and analysis of the solutions thus obtained. The corresponding vector field $X$ is a Hamiltonian vector field where $P_{\mathrm{c}}=P_3$ (3.2) is the canonical Poisson bivector on the Lie–Poisson algebra $e^*(3)$ (see [22]). The components of the second invariant vector field are inhomogeneous polynomials of degree four in the variables $(\gamma,M)$
$$
\begin{equation*}
\begin{aligned} \, Y&=4\gamma_3\bigl((M_1^2 + M_2^2)M_2+a(\gamma_1M_2-\gamma_2M_1)\bigr)\, \partial_1 \\ &\!\quad-4\gamma_3\bigl((M_1^2 + M_2^2)M_1-a(\gamma_1M_1+\gamma_2M_2) \bigr) \, \partial_2 \\ &\!\quad+4\bigl( (M_1^2 + M_2^2)(\gamma_2M_1 -\gamma_1 M_2)-a(\gamma_1^2 + \gamma_2^2)M_2\bigr) \, \partial_3 \\ &\!\quad+ 2\bigl(2(M_1^2 + M_2^2)M_2M_3 -2a \bigl((\gamma_2M_1 - \gamma_1M_2)M_3 +2\gamma_3M_1M_2\bigr) + a^2\gamma_2\gamma_3\bigr)\, \partial_4 \\ &\!\quad- 2\bigl((M_1^2 + J2^2)M_1M_3-a(2(\gamma_1M_1+\gamma_2M_2)M_3 - \gamma_3(M_1^2 - M_2^2)) \,{+}\,a^2\gamma_1\gamma_3\bigr)\, \partial_5 \\ &\!\quad+ 2a\bigl(\gamma_2(M_1^2 - M_2^2) - 2\gamma_1M_1M_2\bigr) \, \partial_6. \end{aligned}
\end{equation*}
\notag
$$
The vector field $X$ is bi-Hamiltonian [64], [65], and therefore, we know several absolute invariants of the vector field $X$ in the Poisson bivector space. The following solutions of the invariance equation are new and are not available in the existing literature. 4.1. Invariant tensor fields of type $(2,0)$Substituting the tensor field $P$ of type $(2,0)$ with components of the form (1.12) into the equation $\mathcal L_XP=0$ and equating to zero the coefficients with different degrees of the variables $\gamma$ and $M$, we obtain a system of 7164 algebraic equations for 3024 parameters. Solving the equations thus obtained via computer algebra systems, we obtain absolute tensor invariants, among which, unlike the Lagrange case, there are no symmetric tensor fields. Among the symmetric bivectors thus obtained, only five satisfy the Jacobi identity, and three of them are compatible with the canonical Poisson bivector $P_{\mathrm{c}}$ on the Lie–Poisson algebra $e^*(3)$. The first of the invariant Poisson bivectors is expressed in terms of the first integrals and the canonical bivector $P_{\mathrm{c}}=P_3$ (see (3.2)):
$$
\begin{equation*}
P_1=(c_1f_1+ c_2 f_2+c_3)P_{\mathrm{c}},\qquad c_i\in\mathbb R.
\end{equation*}
\notag
$$
The components of the second invariant Poisson bivector have the form
$$
\begin{equation*}
\begin{aligned} \, P_2^{12} &= 2(f_2\gamma_3-f_1M_3),\qquad P_2^{13}= f_1M_2- 2f_2\gamma_2, \\ P_2^{14} &=\gamma_1M_2 -\gamma_2 M_1)M_3+\frac{a\gamma_2 \gamma_3}{2},\qquad P_2^{15}=\frac{\gamma_2f_3}{2} - f_2M_3, \\ P_2^{16} &=(\gamma_3M_2 -\gamma_2M_3)M_3 -\frac{(f_3-a\gamma_1)\gamma_2}{2},\qquad P_2^{23}=2f_2\gamma_1-f_1M_1, \\ P_2^{24} &=f_2M_3- \frac{f_3\gamma_3}{2},\qquad P_2^{25}=(\gamma_1M_2 - \gamma_2M_1)M_3+\frac{a\gamma_2\gamma_3}{2}, \\ P_2^{26} &=\frac{f_3\gamma_1}{2}+(\gamma_1M_3 -\gamma_3 M_1)M_3-\frac{a\gamma_2^2}{2}, \qquad P_2^{34}=\frac{(M_1^2 + M_2^2)\gamma_2}{2} +\gamma_3M_2M_3, \\ P_2^{35} &=(\gamma_1M_3 -\gamma_3 M_1)M_3-\frac{f_3\gamma_1}{2}+\frac{a(\gamma_1^2 + \gamma_3^2)}{2}, \\ P_2^{36} &=(\gamma_2M_1 - \gamma_1M_2)M_3-\frac{a\gamma_2\gamma_3}{2},\qquad P_2^{45}=\frac{a\gamma_3M_1 - (M_1^2 + M_2^2)M_3}{2}, \\ P_2^{46} &=-\frac{f_3M_2}{2},\qquad P_2^{56}=\frac{f_3M_1}{2} -f_2. \end{aligned}
\end{equation*}
\notag
$$
This Poisson bivector satisfies the relations
$$
\begin{equation*}
P_2 \, df_1= f_1X,\qquad P_2 \, df_2=f_2X,\qquad P_2 \, df_3 = f_3X + \frac{Y}{4},\qquad P_2 \, df_4=f_4X,
\end{equation*}
\notag
$$
from which we can find the Casimir functions
$$
\begin{equation*}
C_1=\frac{f_2}{f_1}\quad\text{and}\quad C_2=\frac{f_4}{f_1},\quad\text{so that }\ P_2\, dC_{1,2}=0.
\end{equation*}
\notag
$$
The components of the third Poisson bivector are equal to
$$
\begin{equation*}
\begin{aligned} \, P_3^{12} &=-2(\gamma_1^2 + \gamma_2^2)M_3,\qquad P_3^{13}=f_1M_2 - 2\gamma_2\gamma_3M_3, \\ P_3^{14} &=(\gamma_3M_2 -\gamma_1 M_3)M_1 +\gamma_1M_2M_3 -\frac{a\gamma_2\gamma_3}{2}, \\ P_3^{15} &= \gamma_3(M_2^2 + 2M_3^2) -f_2M_3+a\gamma_1\gamma_3,\qquad P_3^{15}=-\gamma_2M_3^2 -\frac{a\gamma_1\gamma_2}{2}, \\ P_3^{23} &=-f_1M_1 + 2\gamma_1\gamma_3M_3,\qquad P_3^{24}=f_2M_3-(M_1^2 - 2M_3^2)\gamma_3, \\ P_3^{25} &=\gamma_1 M_2M_3-\gamma_3 M_1M_2 -\gamma_2M_1 M_3 +\frac{a\gamma_2\gamma_3}{2}, \qquad P_3^{26}=\gamma_1M_3^2 -\frac{a\gamma_2^2}{2}, \\ P_3^{34} &=\frac{\gamma_2(2M_1^2 + 2M_3^2 + a\gamma_1)}{2}-(\gamma_1M_1 +\gamma_3M_3)M_2 , \\ P_3^{35} &=(\gamma_2M_2 -\gamma_3 M_3)M_1 - \gamma_1(M_2^2 + M_3^2)-\frac{a(\gamma_1^2 - \gamma_3^2)}{2}, \qquad\! P_3^{36}=-\frac{a\gamma_2\gamma_3}{2}, \\ P_3^{45} &=M_3^3 + \frac{a\gamma_1M_3}{2}, \qquad P_3^{46}=-M_2M_3^2 -\frac{ a\gamma_2M_1}{2},\qquad P_3^{56}=M_1M_3^2 +\frac{ a\gamma_1M_1}{2}. \end{aligned}
\end{equation*}
\notag
$$
This Poisson bivector is also compatible with the canonical bivector $P_{\mathrm{c}}$. Multiplying this bivector by the 1-forms $df_1,\dots,df_4$, we obtain
$$
\begin{equation*}
P_3 \, df_1=f_1X,\quad P_3 \, df_2=\frac{1}{2}f_2X,\quad P_3 \, df_3= \frac{1}{2}f_3X + \frac{1}{4}Y,\quad P_3 \, df_4=f_4X -\frac{1}{2}f_3Y.
\end{equation*}
\notag
$$
As Casimir functions we can take
$$
\begin{equation*}
C_1=\frac{f_2}{\sqrt{f_1}}\quad\text{and}\quad C_2=\frac{f_3^2-f_4}{f_1},\quad\text{so that }\ P_3\, d{C}_{1,2}=0.
\end{equation*}
\notag
$$
For completeness, we also give two invariant Poisson bivectors which are not compatible with the canonical bivector $P_{\mathrm{c}}$. The components of the first one have the form
$$
\begin{equation*}
\begin{aligned} \, P_4^{12} &=2\gamma_3(\gamma_1M_1 + \gamma_2M_2)-4(\gamma_1^2 +\gamma_2^2)M_3, \\ P_4^{13} &=2(\gamma_1^2 + \gamma_3^2)M_2-2\gamma_2 (\gamma_1M_1+\gamma_3M_3), \\ P_4^{14} &=\gamma_3 M_1M_2 -2(\gamma_2M_1-\gamma_1M_2)M_3, \\ P_4^{15} &=\gamma_3(M_2^2 + a\gamma_1)-2 (\gamma_1M_1+\gamma_2M_2)M_3 , \\ P_4^{16} &=(\gamma_3M_2 \,{-}\, 2\gamma_2M_3)M_3-a\gamma_1\gamma_2,\qquad\! P_4^{23}{=}\,2\gamma_1(\gamma_2M_2 \,{+}\, 2\gamma_3M_3)\,{-}\,2(\gamma_2^2 \,{+}\, \gamma_3^2)M_1, \\ P_4^{24} &= 2(\gamma_1M_1 +\gamma_2M_2)M_3-\gamma_3M_1^2, \\ P_4^{25} &=2(\gamma_1M_2-\gamma_2M_1)M_3 - \gamma_3(M_1M_2 - a\gamma_2), \\ P_4^{26} &=(2\gamma_1M_3-\gamma_3M_1)M_3-a\gamma_2^2,\qquad P_4^{34}=2\gamma_3M_2M_3 + (\gamma_2M_1 -\gamma_1M_2)M_1, \\ P_4^{35} &=\gamma_3(a\gamma_3 - 2M_1M_3) + (\gamma_2M_1 \,{-}\,\gamma_1M_2)M_2,\qquad\! P_4^{36}{=}\,(\gamma_2M_1 \,{-}\,\gamma_1M_2)\,{-}\,a\gamma_2\gamma_3, \\ P_4^{45} &=\frac{a\gamma_3M_1}{2} -(M_1^2 + M_2^2) M_3,\qquad P_4^{46}= -\frac{a\gamma_2M_1}{2}-M_2M_3^2, \\ P_4^{56} &=M_1 M_3^2-\frac{a(\gamma_2M_2+\gamma_3M_3)}{2}. \end{aligned}
\end{equation*}
\notag
$$
The second Poisson invariant bivector, which is incompatible with $P_{\mathrm{c}}$, is expressed in terms of $P_4$ and $P_{\mathrm{c}}$ as follows: The Poisson bivectors $P_4$ and $P_5$ have ranks two and four, respectively. In this case, we have
$$
\begin{equation*}
P_5\, df_1= 2f_1X,\qquad P_5\, df_2=\frac{3}{2}f_2X,\qquad P_5\, df_3=\frac{1}{2}f_3X,\qquad P_5\, df_4= 2f_4X -\frac{f_3}{2}Y,
\end{equation*}
\notag
$$
and as Casimir functions of the bivector $P_5$ we can take the functions
$$
\begin{equation*}
C_1=\frac{f_2^{2/3}}{f_1^{1/2}},\qquad C_2=\frac{f_3^2}{f_1^{1/2}},\qquad P_5\, dC_{1,2}=0.
\end{equation*}
\notag
$$
This completes the list of Poisson invariant tensors that we have obtained in our mathematical experiment. Proposition 7. For the Kovalevskaya system, the set of solutions of the invariance equation in the Poisson bivector space with components of the form (1.12) consists of the above five bivectors $P_1,\dots,P_5$. The proof in a direct solution of the equation of invariance via substitution (1.12). The divergence of the vector fields $X$ and $Y$ is zero, and hence the tensor product
$$
\begin{equation*}
P_6=\mathcal E\, df_1\wedge df_2\wedge df_3\wedge df_4
\end{equation*}
\notag
$$
of the completely antisymmetric unit tensor $\mathcal E$ (see (1.9)) with the product of four invariant 1-forms $df_i$ is an invariant second-rank Poisson bivector, whose components are inhomogeneous polynomials of degree 6 in the variables $(\gamma,M)$. 4.2. Invariant tensor fields of type $(3,0)$Substituting the tensor field $T$ of type $(3,0)$ with components of the form (1.12) into the invariance equation (1.2) and equating to zero the coefficients with different degrees of the variables $\gamma$ and $M$, we obtain a system of 43953 algebraic equations for 18144 unknowns. Solving the equations thus obtained in this way via computer algebra systems, we obtain the solution $T=P_{\mathrm{c}}X$, and, unlike the Lagrange case, there are no other solutions. To get an analogue of the solution (3.4) for the Lagrange case, we have to use, in the Kovalevskaya case, the higher order polynomial anzats for the components of tensor field $T$ of type $(3,0)$. We will not present this solution for brevity. Next, as in the Lagrange case, we proceed with the solution of the equation of relative invariance (3.5), which involves a linear combination of the first integrals $H=\sum_k e_kf_k$, where the unknown coefficients $e_k\in \mathbb R$ are obtained in the process of solution. Proposition 8. In the Kovalevskaya case, in the space of tensor fields of type $(3,0)$ with constant components, there is only one solution For a proof, one should solve a system of algebraic equations using various computer algebra systems. Thus, we proved that, for the equations of motion of a rigid body in the Euler, Lagrange, and Kovalevskaya cases, there exists an antisymmetric numerical tensor field $T$ of type $(3,0)$, which allows us to rewrite the original vector field in the form
$$
\begin{equation*}
X=T\omega,\qquad\mathcal L_x\omega=0, \quad d\omega=0,
\end{equation*}
\notag
$$
where $\omega$ is an invariant differential form of type $(0,2)$.
§ 5. ConclusionsFor a divergence-free vector field $X$, the completely antisymmetric unit tensor fields of valences $(0,n)$ and $(n,0)$ are basic tensor invariants. If this vector field also has first integral $f_1,\dots,f_{k}$, we can construct a set of tensor invariants also in the space of differential fpr,s (2.2) and in the space of multivector fields (1.23). Our mathematical experiment was aimed at finding all tensor invariants of the equations of motion of a rigid body in the Euler, Lagrange, and Kovalevskaya cases with special form of the components (1.12). The main result of the experiment is that, for the Euler, Lagrange, and Kovalevskaya systems, in the space of differential forms, all the solutions of the invariance equation (1.2) obtained above can be constructed from the basic invariant $\Omega$ and first integrals by differentiation and tensor multiplication. In the space of multivector fields, for the same systems, we found several absolute and conditional invariants, which, at least for the moment, cannot be constructed from the basic invariants via standard tensor operations. We emphasise again that we are talking about global tensor invariants uniquely defined on the whole phase space, since locally, near Liouville tori, there is no difference between the number of solutions in the spaces of covariant and contravariant tensor fields [21].
Citation in format AMSBIB
\Bibitem{Tsi25} |
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