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This article is cited in 1 scientific paper (total in 1 paper) Geodesic flow on an intersection of several confocal quadrics in G. V. Belozerov Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 7, DOI: https://doi.org/10.4213/sm9864 
Bibliographic databases:
    § 1. IntroductionEarth’s surface can well be approximated by an ellipsoid, namely, by an ellipsoid of revolution. In this connection the problem of the structure of shortest curves on an ellipsoid was among the most important problems in the 18th and 19th century, when geodesy was developing. Since shortest curves on smooth surfaces are geodesics, researchers of that time were to investigate the structure of the geodesic flow on an ellipsoid. Geodesics on a sphere are simple: these are cross-sections by planes passing through the centre of the sphere. Clairaut showed that the geodesic flow on an ellipsoid of revolution has an additional first integral, while Lagrange, Oriani and Bessel developed methods for the solution of triangles on it. Jacobi [1], [2] considered the most general case when an ellipsoid has different axes. Going over to elliptic coordinates and using the method of separation of variables he proved that the geodesic flow on an $n$-axial ellipsoid in $\mathbb R^n$ is integrable. The French researcher Chasles [3] showed that the tangent lines to all points on a fixed geodesic on a triaxial ellipsoid in $\mathbb R^3$ are also tangent to another quadric confocal to this ellipsoid. This result is now known as the Jacobi-Chasles theorem. Theorem 1 (Jacobi-Chasles theorem). At all points on a fixed geodesic on an ellipsoid in $n$-dimensional Euclidean space the tangent lines to it are also tangent to $n-2$ quadrics confocal with this ellipsoid, which are the same for all points on the geodesic. Remark 1. The parameters of these quadrics, together with the energy, form a system of functionally independent commuting (with respect to the standard Poisson bracket) first integrals. The reader can find a state-of-the-art proof of the Jacobi-Chasles theorem in Arnold’s book [4]. An illustration to this theorem is shown in Figure 1. The geodesic flow on a triaxial ellipsoid is an integrable Hamiltonian system with two degrees of freedom. The topology of the corresponding Liouville foliation was thoroughly examined by Bolsinov and Fomenko [5]. They showed that the geodesic flow on an ellipsoid is continuously orbitally equivalent to the Euler system with area constant equal to zero. (The definition of orbital equivalence and its invariants for systems with two degrees of freedom were presented in [6].) Zung [7] described the topology of the Liouville foliation for the geodesic flow on an $n$-axial ellipsoid in $\mathbb R^n$ with different semiaxes in a neighbourhood of singular points. Davison, Dullin and Bolsinov [8] described the singularities of the geodesic flow on a three-dimensional ellipsoid with equal middle semiaxes. Note that one consequence of Theorem 1 is that the billiard system bounded by an ellipsoid is integrable. In fact, consider an $n$-axial ellipsoid, and let its smallest semiaxis tend to zero. In the limit we obtain a domain bounded by an ellipsoid $\mathcal{E}$ of dimension ${n-1}$. Furthermore, the geodesic flow turns to the billiard system bounded by $\mathcal{E}$, and the parameters of caustics confocal with the original ellipsoid turn to the parameters of caustics confocal with $\mathcal{E}$. Moreover, billiard systems bounded by several confocal quadrics are also integrable. Planar billiards bounded by arcs of confocal quadrics were considered by Kozlov and Treshchev [9], Dragović and Radnović [10], and also Vedyushkina [11], [12]. Vedyushkina classified all locally planar topological billiards bounded by arcs of confocal ellipses and hyperbolae up to Liouville equivalence (see [13]). Fairly recently Kibkalo considered the question of the integrability of the billiard flow on the intersection of several confocal quadrics. He proved that if such an intersection has dimension two, then the corresponding geodesic flow is integrable. This result has turned out to hold for any intersection of several confocal quadrics. Moreover, the following result holds. Theorem 2 (Belozerov). Let $Q_1,\dots,Q_k$ be confocal nondegenerate quadrics of different types in $\mathbb R^n$, and let $Q=\bigcap_{i=1}^k Q_i$. Then The proof of this theorem takes § 5. Before reading it, we advise the reader to look at §§ 2–4: they underlie the proof of Theorem 2. Note that the above result is nontrivial: it is not a consequence of the classical Jacobi-Chasles theorem. In fact, consider coordinate surfaces in spherical coordinates. These are spheres, cones with vertex at the origin and axis $Oz$, and half-planes bounded by the $Oz$-axis. These sets are limiting cases of confocal quadrics in $\mathbb R^3$ for $a_1=a_2=a_3$. The intersection of the sphere $x_1^2+x_2^2+x_3^2=1$ and the cone $x_1^2+x_2^2-0.2x_3^2=0$ consists of two circles, which are not geodesics on the sphere or cone (Figure 2). That is, in the general case a geodesic on the intersection of several confocal quadrics is not a geodesic on any of these quadrics. We will see in §§ 4 and 5 that in elliptic coordinates the first integrals implied in part 1 of Theorem 2 depend only on the coordinates themselves and the squares of their velocities. Hence the geodesic billiards on intersections under consideration, with boundaries formed by confocal quadrics in the same family are also Liouville integrable, although in the piecewise smooth sense. Thus we obtain a new class of integrable billiards. By Kozlov’s theorem on topological obstructions to the integrability of the geodesic flow on a two-dimensional surface (see [14]) and in view of Theorem 2, a component of a two-dimensional compact intersection of several confocal nondegenerate quadrics is either homeomorphic to a 2-sphere or to a 2-torus. The same conclusion was previously made by Kibkalo. However, this argument does not work in the case when the intersection has dimension greater than $2$. Nevertheless, we can determine the homeomorphism class of a compact intersection of several confocal quadrics without mentioning the integrability of the corresponding geodesic flow but only using elliptic coordinates. Theorem 3. Let $Q_1,\dots,Q_k$ be confocal nondegenerate quadrics of different types in $\mathbb R^n$, and let $Q=\bigcap_{i=1}^k Q_i$ be compact. Then $Q$ is homeomorphic to a direct product of $k$ spheres. The dimensions of these spheres coincide with the numbers of free elliptic coordinates on $Q$ occurring in between pairs of successive fixed coordinates. This result was possibly known before, but the author could not find its statement in the literature, so we prove Theorem 3 in § 6 for completeness. Remark 2. The spheres mentioned in Theorem 3 can have dimension zero. (Recall that a zero-dimensional sphere is the disjoint union of two points.) Remark 3. Gitler and López de Medrano [15] showed that the intersection of the coaxial quadrics associated with moment-angle manifolds can be homeomorphic, under certain assumptions, to a direct sum of several direct products of spheres. As Bolsinov and Fomenko showed (see [5]), the geodesic flow on an ellipsoid is continuously orbitally equivalent to the Euler system on the Lie algebra $\mathfrak{so}(3)$. Generic orbits of the coadjoint actions on $\mathfrak{so}^*(3)$ are two-dimensional spheres $S^2$. It is well known that $\mathfrak{so}(4)\simeq\mathfrak{so}(3)\times\mathfrak{so}(3)$, and generic orbits of the coadjoint action on $\mathfrak{so}^*(4)$ are direct products of two 2-spheres $S^2\times S^2$. By Theorems 2 and 3 the geodesic flow on a compact intersection of several confocal quadrics is integrable, while the intersection itself if homeomorphic to a direct product of spheres. It is reasonable to believe that there exists an intersection of confocal quadrics the geodesic flow on which is continuously orbitally equivalent, or at least Liouville equivalent, to the Euler system on $\mathfrak{so}(4)$. This was conjectured by Fomenko, but has not been proved so far. The following observation plays a central role in the proof of Theorem 2: for a fixed family of confocal quadrics there exist smooth functionally independent functions $F_0,F_1,\dots,F_{n-1}$ defined on the whole of $T^*\mathbb R^n$ whose restrictions to any intersection of several confocal nondegenerate quadrics are commuting first integrals (with respect to the standard Poisson bracket) of the geodesic flow on this intersection. Using the method described by Kozlov in [16], in § 7 we find a sufficient condition on the potential $V$ in $\mathbb R^n$ that ensures the fulfillment of the above assumption. Theorem 4 (Belozerov). Let a family of confocal quadrics be fixed, and let $V\colon {\mathbb R^n\to\mathbb R}$ be a smooth function that satisfies the following system of differential equations in the elliptic coordinates: Then the problem of the motion of a point mass on the intersection of several nondegenerate confocal quadrics in this family in the presence of the potential $V$ is square integrable. Similarly to geodesic flows, geodesic billiard systems with potential $V$ from Theorem 4 that arise on an intersection of several confocal quadrics with other quadrics in this family forming the smooth faces of the boundary, are Liouville integrable in the piecewise smooth sense. AcknowledgementsThe author is grateful to V. A. Kibkalo, who posed the question, to A. T. Fomenko for his attention to this work and to N. A. Khotin for his assistance in research. § 2. Confocal quadrics and elliptic coordinatesIn this section we discuss the family of confocal quadrics and its properties, derive a formula connecting elliptic coordinates with Cartesian ones, and introduce some notation required in what follows. Definition. A family of confocal quadrics in $n$-dimensional Euclidean space is the set of quadrics defined by an equation where $a_1>a_2>\dots>a_n$ are fixed numbers and $\lambda$ is a real parameter. Remark 4. If $\lambda=a_i$ for some $i$, then the quadric corresponding to this value of the parameter is not defined. To define it we must multiply (2.1) by the product $\prod_j(a-\lambda_j)$ and then set $\lambda=a_i$ in the resulting equation. It is easy to see that the parameter equal to $a_i$ corresponds to the hyperplane $x_i=0$. Jacobi [2] showed that there are precisely $n$ confocal quadrics passing through an arbitrary point in $\mathbb R^n$ not lying on any hyperplane of the form $x_i=0$. Moreover, letting $\lambda_1<\dots<\lambda_n$ denote the parameters of these quadrics, we have $\lambda_1\in(-\infty,a_n)$, $\lambda_2\in(a_{n},a_{n-1})$, $\dots$, $\lambda_n\in(a_2,a_1)$. It turns out that the functions $\lambda_1,\dots,\lambda_n$ extend in a unique way to continuous functions on the whole of $\mathbb R^n$ which are smooth almost everywhere (except on the hyperplanes $x_i=0$, $i=1,\dots,n - 1$). The set of functions $(\lambda_1,\dots,\lambda_n)$ is called the elliptic coordinates associated with this family of confocal quadrics. The following result was proved by Jacobi; it describes the connection between the elliptic and Cartesian coordinates of a point. Proof. We prove this for $k=1$. The other cases can be treated similarly. We write equalities connecting the elliptic and Cartesian coordinates:
$$
\begin{equation}
\begin{cases} \dfrac{x_1^2}{a_1-\lambda_1}+\dfrac{x_2^2}{a_2-\lambda_1} +\dots+\dfrac{x_n^2}{a_n-\lambda_1}=1, \\ \dfrac{x_1^2}{a_1-\lambda_2}+\dfrac{x_2^2}{a_2-\lambda_2} +\dots+\dfrac{x_n^2}{a_n-\lambda_2}=1, \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \\ \dfrac{x_1^2}{a_1-\lambda_n}+\dfrac{x_2^2}{a_2-\lambda_n} +\dots+\dfrac{x_n^2}{a_n-\lambda_n}=1. \end{cases}
\end{equation}
\tag{2.3}
$$
For each $i$ we multiply the $i$th equation in (2.3) by $a_n-\lambda_i$. Then from the first $n-1$ equations we subtract the last one, after which for each $i<n$ we divide the $i$th equation by $\lambda_i\,{-}\,\lambda_n$. Suppressing the last equality we obtain the following system of $n-1$ equations for $n-1$ Cartesian coordinates:
$$
\begin{equation*}
\begin{cases} \dfrac{a_1-a_n}{(a_1-\lambda_1)(a_1-\lambda_n)}x_1^2 +\dots+\dfrac{a_{n-1}-a_n}{(a_{n-1}-\lambda_1)(a_{n-1}-\lambda_n)}x_{n-1}^2=1, \\ \dfrac{a_1-a_n}{(a_1-\lambda_2)(a_1-\lambda_n)}x_1^2 +\dots+\dfrac{a_{n-1}-a_n}{(a_{n-1}-\lambda_2)(a_{n-1}-\lambda_n)}x_{n-1}^2=1, \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \dots\dots\dots\dots\dots\dots\dots\dots \\ \dfrac{a_1-a_n}{(a_1-\lambda_{n-1})(a_1-\lambda_n)}x_1^2 +\dots+\dfrac{a_{n-1}-a_n}{(a_{n-1}-\lambda_{n-1}) (a_{n-1}-\lambda_n)}x_{n-1}^2=1. \end{cases}
\end{equation*}
\notag
$$
Now we eliminate the variables $x_{n-1},\dots,x_2$ successively in the same way as above. Then we obtain
$$
\begin{equation*}
\frac{\prod_{j\neq k}(a_k-a_j)}{\prod_{i=1}^n(a_k-\lambda_i)}x_k^2=1,
\end{equation*}
\notag
$$
which yields the required formula.
The proof is complete. For shorter expressions we introduce the notation
$$
\begin{equation*}
\Delta_k=\prod_{i=1}^n(a_k-\lambda_i), \qquad \Delta_k^j=\prod_{i\neq j}(a_k-\lambda_i)\quad\text{and} \quad \rho_k=\prod_{j\neq k}(a_k-a_j);
\end{equation*}
\notag
$$
$\sigma^m_{i_1,\dots,i_k}(b_1,\dots,b_n)$, where $m>0$, is the elementary symmetric polynomial of degree $m$ of the variables $\{b_1,\dots,b_n\}\setminus\{b_{i_1},\dots,b_{i_k}\}$; for $m=0$ this is identical one, and for $m=-1$ this is identical zero. Using this notation we can rewrite (2.2): We establish another result required in what follows. Proof. We write out the equations for the $i$th and $j$th elliptic coordinates:
$$
\begin{equation*}
\begin{cases} \dfrac{x_1^2}{a_1-\lambda_i}+\dfrac{x_2^2}{a_2-\lambda_i} +\dots+\dfrac{x_n^2}{a_n-\lambda_i}=1, \\ \dfrac{x_1^2}{a_1-\lambda_j}+\dfrac{x_2^2}{a_2-\lambda_j} +\dots+\dfrac{x_n^2}{a_n-\lambda_j}=1. \end{cases}
\end{equation*}
\notag
$$
Subtracting the second from the first we obtain
$$
\begin{equation*}
\frac{(\lambda_j-\lambda_i)x_1^2}{(a_1-\lambda_i)(a_1-\lambda_j)} +\dots+\frac{(\lambda_j-\lambda_i)x_n^2}{(a_n-\lambda_i)(a_n-\lambda_j)}=0.
\end{equation*}
\notag
$$
As $i\neq j$, we have $\lambda_i\neq\lambda_j$ in general. Dividing the above equation by $\lambda_j-\lambda_i$ we use (2.4). This completes the proof. This result has the following geometric interpretation: confocal quadrics are orthogonal at points of their intersection. This was also mentioned by Jacobi in [2]. § 3. Summation formulaeIn this section we prove several results on sums of certain special rational expressions. We use them in § 4, in going over to elliptic coordinates from Cartesian ones. Proof. Fix $s$ and consider the polynomial $P(z)$ defined by Interchanging the summation signs we use Vieta’s theorem:
It remains to observe that $P(a_k)=a_k^s$ for each $k$. Therefore, since $\mathrm{deg}\,P\leqslant n-1$, we have $P(z)\equiv z^s$. Comparing the coefficients of like powers in $P(z)$ we obtain the required equality. The proof is complete. Proof. On the left-hand side we extract $(-1)^{n+1}/\prod_i(a_i-b)$. Then we obtain
$$
\begin{equation*}
\begin{aligned} \, &\frac{(-1)^{n+1}}{(a_1-b)\dotsb(a_n-b)}\sum_{k=1}^n \frac{(b-a_1)\dotsb(b-a_{k-1})(b-a_{k+1})\dotsb (b-a_n)}{(a_k-a_1)\dotsb(a_k-a_{k-1})(a_{k}-a_{k+1}) \dotsb(a_k-a_n)} \\ &\qquad=\frac{(-1)^{n+1}}{(a_1-b)\dotsb(a_n-b)}Q(b), \end{aligned}
\end{equation*}
\notag
$$
where $Q(b)$ is a polynomial of degree $\leqslant n-1$ in $b$. However, it is not difficult to see that $Q(a_k)=1$ for $k=1,\dots,n$. Hence $Q(b)\equiv1$.
The proof is complete. Proof. Consider the polynomial $R(z)$ defined by
$$
\begin{equation*}
\begin{aligned} \, R(z)&=\sum_{m=0}^{n-1}(-1)^{m}z^{n-1-m} \\ &\qquad\times\sum_{k=1}^n\frac{(a_k-b_1)\dotsb(a_k-b_{l-1})(a_k-b_{l+1}) \dotsb(a_k-b_n)}{(a_k-a_1)\dotsb(a_k-a_{k-1})(a_{k}-a_{k+1}) \dotsb(a_k-a_n)}\sigma_k^m(a_1,\dots,a_n). \end{aligned}
\end{equation*}
\notag
$$
Changing the order of summation and using Vieta’s theorem we obtain
$$
\begin{equation*}
\begin{aligned} \, R(z)&=\sum_{k=1}^n\frac{(a_k-b_1)\dotsb(a_k-b_{l-1})(a_k-b_{l+1}) \dotsb(a_k-b_n)}{(a_k-a_1)\dotsb(a_k-a_{k-1})(a_{k}-a_{k+1})\dotsb (a_k-a_n)} \\ &\qquad\times\bigl((z-a_1)\dotsb(z-a_{k-1})(z-a_{k+1})\dotsb(z-a_n)\bigr). \end{aligned}
\end{equation*}
\notag
$$
Note that $R(a_k)=(a_k-b_1)\dotsb(a_k-b_{l-1})(a_k-b_{l+1})\dotsb(a_k-b_n)$ for all ${k=1,\dots,n}$. Since $R(z)$ is a polynomial of degree $\leqslant n-1$, it follows that
$$
\begin{equation*}
R(z)=(z-b_1)\dotsb(z-b_{l-1})(z-b_{l+1})\dotsb(z-b_n).
\end{equation*}
\notag
$$
Hence the coefficient of $z^{n-1-m}$ in $R(z)$ is $(-1)^m\sigma_l^m(b_1,\dots,b_n)$, which yields the required result.
The proof is complete. § 4. Inertial motion of a point in $ {\mathbb R}^n$Before turning to the proof of Theorem 2 we look at an auxiliary problem. Consider a point mass moving by inertia in $\mathbb R^n$ with Cartesian coordinates $(x_1,\dots,x_n)$. We find the parameters of the confocal quadrics in the family (2.1) that are tangent to its trajectory. Assume that at time $t$ the particle is at a point with coordinates $x_1,\dots,x_n$ and its velocity vector at this point is $(\dot{x}_1,\dots,\dot{x}_n)$. Then its trajectory is the straight line which can parametrically be described by
$$
\begin{equation*}
(x_1+\tau\dot{x}_1,\dots,x_n+\tau\dot{x}_n), \qquad \tau\in\mathbb{R}.
\end{equation*}
\notag
$$
To find the points of its intersection with the quadric with parameter $\mu$ we must solve the following quadratic equation with respect to $\tau$:
$$
\begin{equation*}
\frac{(x_1+\tau\dot{x}_1)^2}{a_1-\mu}+\dots +\frac{(x_n+\tau\dot{x}_n)^2}{a_n-\mu}=1.
\end{equation*}
\notag
$$
Note that the line is tangent to the quadric if and only if the discriminant of this (quadratic in $\tau$) equation is zero. This can be expressed as follows:
$$
\begin{equation}
\begin{aligned} \, \notag &\biggl(\frac{x_1\dot{x}_1}{a_1-\mu}+\dots+\frac{x_n\dot{x}_n}{a_n-\mu}\biggr)^2 \\ &\qquad =\biggl(\frac{\dot{x}_1^2}{a_1-\mu}+\dots+\frac{\dot{x}_n^2}{a_n-\mu}\biggr) \biggl(\frac{x_1^2}{a_1-\mu}+\dots+\frac{x_n^2}{a_n-\mu}-1\biggr). \end{aligned}
\end{equation}
\tag{4.1}
$$
Hence, to find quadrics tangent to the trajectory of the point mass we must solve equations (4.1) with respect to $\mu$. First we transform the equation (by multiplying out on both sides, taking all terms to the left-hand side and simplifying the expression by separating full squares):
$$
\begin{equation*}
\frac{\dot{x}_1^2}{a_1-\mu}+\dots+\frac{\dot{x}_n^2}{a_n-\mu} -\sum_{i<j}\frac{K_{i,j}^2}{(a_i-\mu)(a_j-\mu)}=0.
\end{equation*}
\notag
$$
Here $K_{i,j}=x_i\dot{x}_j-x_j\dot{x_i}$. We multiply both sides by $\prod_i(a_i-\mu)$:
$$
\begin{equation}
\sum_{k=1}^n \prod_{m\neq k} (a_m-\mu)\dot{x}_k^2-\sum_{i<j}\prod_{m\neq i,j} (a_m-\mu)K_{i,j}^2=0.
\end{equation}
\tag{4.2}
$$
We call the result the equation of tangency. It has degree $n - 1$. Let $F_{m}$ denote the coefficient of $2\cdot(-1)^{n-1-m}\mu^{n-m-1}$. Then we obtain the equalities
$$
\begin{equation}
F_m=\frac{1}{2}\sum_{k=1}^n\sigma_k^m(a_1,\dots,a_n)\dot{x}_k^2 -\frac{1}{2}\sum_{i<j}\sigma_{i,j}^{m-1}(a_1,\dots,a_n)K_{i,j}^2.
\end{equation}
\tag{4.3}
$$
Remark 5. All functions $F_m$ are smooth in the phase space of the dynamical system in question. The function $F_0$ is the total mechanical energy of the point mass. Now we examine the properties of the functions $F_m$ in (4.3). This is not so easy to do in Cartesian coordinates, because the conditions describing tangent confocal quadrics are closely connected with elliptic coordinates. Thus, we go over to elliptic coordinates. By Proposition 1, $x_k^2={\Delta_k}/{\rho_k}$. We differentiate this and divide by $2$:
$$
\begin{equation}
x_k\dot{x}_k=-\frac{1}{2\rho_k}\sum_{i=1}^n\Delta_k^i\dot{\lambda}_i.
\end{equation}
\tag{4.4}
$$
For compactness we do not write the summation sign below. Using (1.1) and (4.4) we obtain
$$
\begin{equation}
\dot{x}_k^2=\frac{(x_k\dot{x}_k)^2}{x_k^2} =\frac{1}{4\rho_k\Delta_k}\Delta_k^p\Delta_k^q\dot{\lambda}_p\dot{\lambda}_q.
\end{equation}
\tag{4.5}
$$
Substituting (1.1), (4.4) and (4.5) into (4.1) we remove parentheses and multiply the result by $4$:
$$
\begin{equation}
\begin{aligned} \, \notag &\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\beta}^q\dot{\lambda}_p\dot{\lambda}_q \\ &\qquad=\frac{\Delta_{\beta}}{\rho_{\alpha}\rho_{\beta}\Delta_{\alpha} (a_{\alpha}-\mu)(a_{\beta}-\mu)}\Delta_{\alpha}^p\Delta_{\alpha}^q \dot{\lambda}_p\dot{\lambda}_q-\frac{1}{\rho_{\alpha}\Delta_{\alpha} (a_{\alpha}-\mu)}\Delta_{\alpha}^p\Delta_{\alpha}^q\dot{\lambda}_p \dot{\lambda}_q. \end{aligned}
\end{equation}
\tag{4.6}
$$
Moving all terms to the left we simplify the resulting equation by collecting the coefficients of the products $\dot{\lambda}_p\dot{\lambda}_q$. Consider two cases, when $p\neq q$ and when $p=q$. 1) $p\neq q$. The coefficient of $\dot{\lambda}_p\dot{\lambda}_q$ has the following form:
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\beta}^q -\frac{\Delta_{\beta}}{\rho_{\alpha}\rho_{\beta}\Delta_{\alpha} (a_{\alpha}-\mu)(a_{\beta}-\mu)}\Delta_{\alpha}^p\Delta_{\alpha}^q \\ &\qquad+\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^q\Delta_{\beta}^p -\frac{\Delta_{\beta}}{\rho_{\alpha}\rho_{\beta}\Delta_{\alpha} (a_{\alpha}-\mu)(a_{\beta}-\mu)}\Delta_{\alpha}^q\Delta_{\alpha}^p \\ &\qquad+\frac{2}{\rho_{\alpha}\Delta_{\alpha}(a_{\alpha}-\mu)} \Delta_{\alpha}^p\Delta_{\alpha}^q. \end{aligned}
\end{equation*}
\notag
$$
We simplify the sum of the first two terms. We divide out $\Delta_\alpha^p\Delta_\beta^q$ and the denominator, simplify the expression in parentheses, after which we represent the rational expression with respect to $\mu$ as a difference of two fractions:
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\beta}^q -\frac{\Delta_{\beta}}{\rho_{\alpha}\rho_{\beta} \Delta_{\alpha}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\alpha}^q \\ &\qquad=\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\beta}^q \biggl(1-\frac{\Delta_{\beta}\Delta_{\alpha}^q}{\Delta_{\alpha} \Delta_{\beta}^q}\biggr) \\ &\qquad=\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\beta}^q \biggl(1-\frac{a_{\beta}-\lambda_{q}}{a_{\alpha}-\lambda_{q}}\biggr) \\ &\qquad=\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\beta}^q \frac{a_{\alpha}-a_{\beta}}{a_{\alpha}-\lambda_{q}} \\ &\qquad =\frac{\Delta_{\alpha}^p\Delta_{\beta}^q}{\rho_{\alpha} \rho_{\beta}(a_{\alpha}-\lambda_{q})}\biggl(\frac{1}{a_{\beta}-\mu} -\frac{1}{a_{\alpha}-\mu}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Now we multiply out, and then interchange $\alpha$ with $\beta$ in the second term. This yields
$$
\begin{equation*}
\frac{\Delta_{\beta}\Delta_{\alpha}}{\rho_{\alpha}\rho_{\beta}(a_{\beta}-\mu)} \biggl(\frac{1}{(a_{\alpha}-\lambda_{q})(a_{\alpha}-\lambda_{p}) (a_{\beta}-\lambda_{q})}-\frac{1}{(a_{\beta}-\lambda_{q}) (a_{\beta}-\lambda_{p})(a_{\alpha}-\lambda_{q})}\biggr).
\end{equation*}
\notag
$$
We treat the remaining two terms in a similar way:
$$
\begin{equation*}
\frac{\Delta_{\beta}\Delta_{\alpha}}{\rho_{\alpha}\rho_{\beta}(a_{\beta}-\mu)} \biggl(\frac{1}{(a_{\alpha}-\lambda_{p})(a_{\alpha}-\lambda_{q}) (a_{\beta}-\lambda_{p})}-\frac{1}{(a_{\beta}-\lambda_{p}) (a_{\beta}-\lambda_{q})(a_{\alpha}-\lambda_{p})}\biggr).
\end{equation*}
\notag
$$
The sum of these two expressions is
$$
\begin{equation*}
2\frac{\Delta_{\beta}\Delta_{\alpha}}{\rho_{\alpha}\rho_{\beta}(a_{\beta}-\mu)} \,\frac{a_{\beta}-a_{\alpha}}{(a_{\alpha}-\lambda_{p}) (a_{\alpha}-\lambda_{q})(a_{\beta}-\lambda_{p})(a_{\beta}-\lambda_{q})}.
\end{equation*}
\notag
$$
Representing $a_{\beta}-a_{\alpha}$ as $a_{\beta}-\lambda_p-a_{\alpha}+\lambda_p$, we sum the above expressions with respect to $\alpha$ and then with respect to $\beta$:
$$
\begin{equation*}
\begin{aligned} \, &2\frac{\Delta_{\beta}\Delta_{\alpha}}{\rho_{\alpha} \rho_{\beta}(a_{\beta}-\mu)}\, \frac{a_{\beta}-\lambda_p-a_{\alpha}+\lambda_p} {(a_{\alpha}-\lambda_{p})(a_{\alpha}-\lambda_{q})(a_{\beta}-\lambda_{p}) (a_{\beta}-\lambda_{q})} \\ &\qquad=2\frac{\Delta_{\beta}}{\rho_{\beta}(a_{\beta}-\mu) (a_{\beta}-\lambda_{p})(a_{\beta}-\lambda_{q})} \\ &\qquad\qquad\times \biggl((a_{\beta}-\lambda_{p})\frac{\Delta_{\alpha}} {\rho_{\alpha}(a_{\alpha}-\lambda_{p})(a_{\alpha}-\lambda_{q})} -\frac{\Delta_{\alpha}}{\rho_{\alpha}(a_{\alpha}-\lambda_{q})}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Using Propositions 1 and 2 we can simplify the result:
$$
\begin{equation*}
-2\frac{\Delta_{\beta}}{\rho_{\beta}(a_{\beta}-\mu) (a_{\beta}-\lambda_{p})(a_{\beta}-\lambda_{q})} =-2\frac{\Delta_{\beta}^p\Delta_{\beta}^q} {\rho_{\beta}\Delta_{\beta}(a_{\beta}-\mu)}.
\end{equation*}
\notag
$$
We see that the sum of the first four terms is equal to the fifth with minus sign. Hence the coefficient of $\dot{\lambda}_p\dot{\lambda}_q$ is zero for $p\neq q$. Consider the second case. 2) $p=q$. The coefficient of $\dot{\lambda}_p^2$ has the following form:
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\beta}^p-\frac{\Delta_{\beta}}{\rho_{\alpha} \rho_{\beta}\Delta_{\alpha}(a_{\alpha}-\mu)(a_{\beta}-\mu)} \Delta_{\alpha}^p\Delta_{\alpha}^p \\ &\qquad+\frac{1}{\rho_{\alpha}\Delta_{\alpha}(a_{\alpha}-\mu)} \Delta_{\alpha}^p\Delta_{\alpha}^p. \end{aligned}
\end{equation*}
\notag
$$
As in case 1), we can simplify the sum of the first two terms:
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\mu) (a_{\beta}-\mu)}\Delta_{\alpha}^p\Delta_{\beta}^p -\frac{\Delta_{\beta}}{\rho_{\alpha}\rho_{\beta}\Delta_{\alpha} (a_{\alpha}-\mu)(a_{\beta}-\mu)}\Delta_{\alpha}^p\Delta_{\alpha}^p \\ &\qquad =\frac{\Delta_{\alpha}^p\Delta_{\beta}^p}{\rho_{\alpha}\rho_{\beta} (a_{\alpha}-\lambda_{p})} \biggl(\frac{1}{a_{\beta}-\mu}-\frac{1}{a_{\alpha}-\mu}\biggr). \end{aligned}
\end{equation*}
\notag
$$
Multiplying out we consider the sum with respect to $\beta$ of the expressions coming second, in which we use (2.2):
$$
\begin{equation*}
\begin{aligned} \, &\frac{\Delta_{\alpha}^p\Delta_{\beta}^p}{\rho_{\alpha} \rho_{\beta}(a_{\alpha}-\lambda_{p})} \frac{1}{a_{\beta}-\mu}-\frac{\Delta_{\beta}^p\Delta_{\alpha}^p} {\rho_{\alpha}\rho_{\beta}(a_{\alpha}-\lambda_{p})} \frac{1}{a_{\alpha}-\mu} \\ &\qquad=\frac{\Delta^p_{\beta}}{\rho_{\beta}(a_{\beta}-\mu)} \biggl(\frac{\Delta_{\alpha}^p}{\rho_{\alpha}(a_{\alpha} -\lambda_{p})}\biggr)-\frac{\Delta_{\alpha}^p}{\rho_{\alpha}(a_{\alpha} -\lambda_{p})(a_{\alpha}-\mu)}\biggl(\frac{\Delta_{\beta}^p}{\rho_{\beta}}\biggr) \\ &\qquad=\frac{\Delta^p_{\beta}}{\rho_{\beta}(a_{\beta}-\mu)} \biggl(\frac{\Delta_{\alpha}^p}{\rho_{\alpha}(a_{\alpha}-\lambda_{p})}\biggr) -\frac{\Delta_{\alpha}^p}{\rho_{\alpha}(a_{\alpha}-\lambda_{p}) (a_{\alpha}-\mu)} \\ &\qquad=\frac{\Delta^p_{\beta}}{\rho_{\beta}(a_{\beta}-\mu)} \biggl(\frac{\Delta_{\alpha}^p}{\rho_{\alpha}(a_{\alpha}-\lambda_{p})}\biggr) -\frac{\Delta_{\alpha}^p\Delta_{\alpha}^p}{\rho_{\alpha}\Delta_{\alpha} (a_{\alpha}-\mu)}. \end{aligned}
\end{equation*}
\notag
$$
Thus, the coefficient of $\dot{\lambda}_p^2$ has the following form:
$$
\begin{equation*}
\frac{\Delta^p_{\beta}}{\rho_{\beta}(a_{\beta}-\mu)} \biggl(\frac{\Delta_{\alpha}^p}{\rho_{\alpha}(a_{\alpha}-\lambda_{p})}\biggr).
\end{equation*}
\notag
$$
Now we calculate the sum of the expressions $\dfrac{\Delta_{\alpha}^p}{\rho_{\alpha}(a_{\alpha}-\lambda_{p})}$. To do this we write the terms explicitly:
$$
\begin{equation*}
\sum_{\alpha=1}^n\frac{1}{a_{\alpha}-\lambda_p}\cdot \frac{(a_{\alpha}-\lambda_1)\dotsb(a_{\alpha}-\lambda_{p-1}) (a_{\alpha}-\lambda_{p+1})\dotsb(a_{\alpha}-\lambda_{n})} {(a_{\alpha}-a_1)\dotsb(a_{\alpha}-a_{\alpha-1}) (a_{\alpha}-a_{\alpha+1})\dotsb(a_{\alpha}-a_{n})}.
\end{equation*}
\notag
$$
We can represent each $a_{\alpha}-\lambda_i$ as $a_{\alpha}-\lambda_{p}+\lambda_p-\lambda_i$. Thus the numerator of each fraction is a polynomial of degree $\leqslant n-1$ in $a_{\alpha}-\lambda_p$. Since all denominators contain the factor $a_{\alpha}-\lambda_p$, using summation formula (3.1) we can simplify this sum and write it as
$$
\begin{equation*}
\sum_{a=1}^n\frac{1}{a_{\alpha}-\lambda_p}\cdot \frac{(\lambda_{p}-\lambda_1)\dotsb(\lambda_{p}-\lambda_{p-1}) (\lambda_{p}-\lambda_{p+1})\dotsb(\lambda_{p}-\lambda_{n})} {(a_{\alpha}-a_1)\dotsb(a_{\alpha}-a_{\alpha-1}) (a_{\alpha}-a_{\alpha+1})\dotsb(a_{\alpha}-a_{n})}.
\end{equation*}
\notag
$$
All fractions here have the same numerator, so using (3.2) we transform this expression into
$$
\begin{equation*}
\frac{(\lambda_{1}-\lambda_p)\dotsb(\lambda_{p-1}-\lambda_{p}) (\lambda_{p+1}-\lambda_{p})\dotsb(\lambda_{n}-\lambda_{p})} {(a_{1}-\lambda_p)(a_{2}-\lambda_{p})\dotsb(a_{n-1}-\lambda_{p}) (a_{n}-\lambda_{p})}.
\end{equation*}
\notag
$$
We denote this by $A_p$; then the coefficient of $\lambda_{p}$ is as follows:
$$
\begin{equation*}
A_p\sum_{\beta=1}^n\frac{\Delta_{\beta}^p}{\rho_{\beta}(a_{\beta}-\mu)}.
\end{equation*}
\notag
$$
As a result, we have reduced (4.1) to the following form:
$$
\begin{equation*}
\sum_{p=1}^{n}\biggl(\sum_{\beta=1}^n\frac{\Delta_{\beta}^p} {\rho_{\beta}(a_{\beta}-\mu)}\biggr)A_p\dot{\lambda}_p^2=0.
\end{equation*}
\notag
$$
We multiply this by $\prod_i(a_i-\mu)$:
$$
\begin{equation*}
\sum_{p=1}^{n}\biggl(\sum_{\beta=1}^n\frac{\Delta_{\beta}^p} {\rho_{\beta}}\prod_{i\neq\beta}(a_i-\mu)\biggr)A_p\dot{\lambda}_p^2=0.
\end{equation*}
\notag
$$
Since we have started by dividing the equation by $4$, it follows from the last relation that
$$
\begin{equation*}
F_{m}=\frac{1}{8}\sum_{p=1}^n \biggl(\sum_{\beta=1}^n\frac{\Delta_{\beta}^p}{\rho_{\beta}} \sigma_{\beta}^m(a_1,\dots,a_n)\biggr)A_p\dot{\lambda}_p^2.
\end{equation*}
\notag
$$
It remains to simplify the inner sum. To do this we use the summation formula (3.3):
$$
\begin{equation*}
\begin{aligned} \, &\sum_{\beta=1}^n\frac{\Delta_{\beta}^p}{\rho_{\beta}} \sigma_{\beta}^m(a_1,\dots,a_n) \\ &\qquad=\sum_{\beta=1}^n\frac{(a_k-\lambda_1)\dotsb(a_k-\lambda_{p-1}) (a_k-\lambda_{p+1})\dotsb(a_\beta-\lambda_n)}{(a_\beta-a_1) \dotsb(a_\beta-a_{\beta-1}) (a_{\beta}-a_{\beta+1})\dotsb(a_\beta-a_n)}\sigma_\beta^m(a_1,\dots,a_n) \\ &\qquad=\sigma_p^m(\lambda_1,\dots,\lambda_n). \end{aligned}
\end{equation*}
\notag
$$
Thus we obtain
$$
\begin{equation*}
F_m=\frac{1}{8}\sum_{p=1}^n \sigma_p^m(\lambda_1,\dots,\lambda_n) A_p\dot{\lambda}_p^2.
\end{equation*}
\notag
$$
Note that the expressions for the functions $F_m$ look simpler in elliptic coordinates. However, we will examine these functions from the standpoint of Hamiltonian mechanics. So we go over to the coordinate-momentum representation. We have already noticed that $F_0$ is the total mechanical energy of the point mass (see Remark 5). Set $\widehat{A}_p=A_{p}^{-1}$. By the definition of generalized momenta
$$
\begin{equation*}
p_i=\frac{\partial H}{\partial\dot{\lambda}_i}=\frac{1}{4}A_{i}\dot{\lambda}_i,
\end{equation*}
\notag
$$
which yields $\dot{\lambda}_i=4\widehat{A}_ip_i$. Therefore,
$$
\begin{equation}
F_{m}=2\sum_{i=1}^n \sigma_i^m(\lambda_1,\dots,\lambda_n)\widehat{A}_ip_i^2.
\end{equation}
\tag{4.7}
$$
We prove several results on the properties of the functions $F_i$. The first and most important property is as follows. Proof. First we calculate the partial derivatives involved in the formula (we set $\sigma(\lambda_1,\dots,\lambda_n)=\sigma$ for short):
$$
\begin{equation*}
\begin{gathered} \, \begin{split} \frac{\partial F_i}{\partial \lambda_k} &=2\sum_{m\neq k}\sigma^i_{m,k}\widehat{A}_mp_m^2 +2\sum_{m=1}^n\sigma^i_m\,\frac{\partial\widehat{A}_m}{\partial\lambda_k}p_m^2 \\ &=2\sigma_k^i\,\frac{\partial\widehat{A}_k}{\partial\lambda_k}p_k^2 +2\sum_{m\neq k}\biggl(\sigma^{i-1}_{m,k}\widehat{A}_m +\sigma^i_m\,\frac{\partial\widehat{A}_m}{\partial\lambda_k}\biggr)p_m^2, \end{split} \\ \frac{\partial F_i}{\partial p_k}=4\sigma_k^i\widehat{A}_kp_k. \end{gathered}
\end{equation*}
\notag
$$
Hence the expression $\dfrac{\partial F_i}{\partial \lambda_k}\,\dfrac{\partial F_j}{\partial p_k}-\dfrac{\partial F_j}{\partial \lambda_k}\,\dfrac{\partial F_i}{\partial p_k}$ is equal to
We claim that for all $m$ we have
$$
\begin{equation*}
(\sigma_k^j\sigma_{m,k}^{i-1}-\sigma_k^i\sigma_{m,k}^{j-1}) \widehat{A}_m+(\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j)\,\frac{\partial \widehat{A}_m}{\partial \lambda_k}=0.
\end{equation*}
\notag
$$
In fact, consider the two easily verifiable identities
$$
\begin{equation*}
\lambda_m\sigma_{m,k}^{i-1}+\sigma_{m,k}^{i}=\sigma_{k}^i\quad\text{and} \quad \lambda_k\sigma_{m,k}^{i-1}+\sigma_{m,k}^{i}=\sigma_{m}^i.
\end{equation*}
\notag
$$
We subtract the second from the first and divide the resulting identity by $\lambda_m-\lambda_k$. Then we obtain $\sigma_{m,k}^{i-1}=\dfrac{\sigma_{k}^i-\sigma_{m}^i}{\lambda_m-\lambda_k}$. In a similar way $\sigma_{m,k}^{j-1}=\dfrac{\sigma_{k}^j-\sigma_{m}^j}{\lambda_m-\lambda_k}$. Hence
$$
\begin{equation*}
\begin{aligned} \, &(\sigma_k^j\sigma_{m,k}^{i-1}-\sigma_k^i\sigma_{m,k}^{j-1}) \widehat{A}_m+(\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j)\, \frac{\partial \widehat{A}_m}{\partial \lambda_k} \\ &\qquad=\frac{\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j} {\lambda_k-\lambda_m}\widehat{A}_m+(\sigma_k^j\sigma_m^i -\sigma_k^i\sigma_m^j)\, \frac{\partial \widehat{A}_m}{\partial \lambda_k} \\ &\qquad=(\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j) \biggl(\frac{\widehat{A}_m}{\lambda_k-\lambda_m}+\frac{\partial \widehat{A}_m}{\partial \lambda_k}\biggr). \end{aligned}
\end{equation*}
\notag
$$
The equality $\dfrac{\widehat{A}_m}{\lambda_k-\lambda_m}+\dfrac{\partial \widehat{A}_m}{\partial \lambda_k}=0$ can be verified by easy calculation. Proposition 6 is proved. Now we can formulate Proposition 6 in terms of Poisson brackets. In the space of smooth functions on $T^*\mathbb R^n$ we consider the $n$ Poisson brackets $\{\,\cdot\,{,}\,\cdot\,\}_1,\dots,\{\,\cdot\,{,}\,\cdot\,\}_n$ defined by
$$
\begin{equation*}
\{F,G\}_i=\frac{\partial F}{\partial \lambda_i}\,\frac{\partial G}{\partial p_i}-\frac{\partial G}{\partial \lambda_i}\,\frac{\partial F}{\partial p_i} \quad \text{for } i=1,\dots,n.
\end{equation*}
\notag
$$
Each of these antisymmetric forms satisfies Leibniz’s identity, which is easy to verify. By Proposition 6 the functions $F_i$ commute pairwise with respect to each Poisson bracket $\{\,\cdot\,{,}\,\cdot\,\}_j$. This condition is strong and rather unusual. However, it perhaps went unnoticed by classical authors. This observation is key to the proof of Theorem 2. Recall that the standard Poisson bracket on the cotangent bundle of $\mathbb R^n$ has the form $\{\,\cdot\,{,}\,\cdot\,\}=\sum_{i=1}^n\{\,\cdot\,{,}\,\cdot\,\}_i$. This has an important consequence. Note that the converse result fails in general: the equality $\{F,G\}=0$ does not mean that $\{F,G\}_i=0$ for all $i=1,\dots,n$. Since the Hamiltonian of the dynamical system under consideration is equal to $F_0$, the functions $F_1,\dots,F_{n-1}$ are first integrals of the inertial motion of a point mass in $\mathbb R^n$. Hence the polynomial of tangency does not change along geodesics in $\mathbb R^n$ (that is, straight lines), which is quite obvious. The system of functions $F_0,\dots,F_{n-1}$ turns out to be functionally independent. This is ensured by the following result. Proof. We show that $\dfrac{D(F_0,\dots,F_{n-1})}{D(p_1,\dots,p_n)}\neq0$ almost everywhere. We have Therefore,
Note that the expression multiplying the determinant is distinct from zero almost everywhere. To the $j$th column of the matrix under the determinant sign we assign the polynomial Note that the determinant vanishes if and only if the polynomials $f_1(z),\dots,f_n(z)$ are linearly dependent. By Vieta’s theorem, for each $j$ we have
$$
\begin{equation*}
f_j(z)=(z-\lambda_1)\dotsb(z-\lambda_{j-1}) (z-\lambda_{j+1})\dotsb(z-\lambda_{n}).
\end{equation*}
\notag
$$
Let $\lambda_1,\dots,\lambda_n$ be pairwise distinct; then $f_j(\lambda_i)\neq0$ if and only if $i\neq j$. It immediately follows that for such $\lambda_1,\dots,\lambda_n$ the polynomials $f_1(z),\dots,f_n(z)$ are linearly independent. Since conditions of the form $\lambda_i=\lambda_j$ for $i\neq j$ describe a nullset in the phase space, the functions $F_0,\dots,F_{n-1}$ are functionally independent. The proof is complete. Thus the problem of the motion of a point mass is completely integrable, and its first integrals $F_0,\dots,F_{n-1}$ describe fully the tangency of the trajectory to a family of confocal quadrics. That the problem of an inertial motion of a point mass in $\mathbb R^n$ is integrable is well known. In fact, for first integrals one can take the momenta corresponding to Cartesian coordinates. However, it is important for us that our system of functionally independent commuting first integrals $F_0,\dots,F_{n-1}$ is related to tangency of confocal quadrics. Now we find the number of confocal quadrics tangent to the trajectory of the point mass in $\mathbb R^n$. To do this we establish an auxiliary result on separation of variables. Proposition 8 (formula of separation of variables). On the joint level $f_0,\dots,f_{n-1}$ of the first integrals $F_0,\dots,F_{n-1}$ the equations of motion have the following form: Proof. Consider the polynomial of tangency on the common level $(f_0,\dots,f_{n-1})$. Recall that $P(\mu_0)=0$ is and only if the trajectory is tangent to the quadric with parameter $\mu_0$. By Vieta’s theorem, using (4.7) we obtain Hence we conclude that for each $k=1,\dots,n$ we have
Using the connection between generalized momenta and velocities and also relations (4.9) we obtain
$$
\begin{equation*}
\dot{\lambda}_k=4\widehat{A}_jp_j=\pm\frac{2\sqrt{2}}{\prod_{i\neq k}(\lambda_i-\lambda_k)} \sqrt{(-1)^{n-1}\prod_{i=1}^n(a_i-\lambda_k)P(\lambda_k)}.
\end{equation*}
\notag
$$
The proof is complete. Note that in the formula of separation of variables the expression under the root sign is the polynomial in $\lambda_i$ equal to the product of $\prod_j(a_j-\lambda_i)$ and the polynomial of tangency. Using this observation we prove the following. Proposition 9. Let $f_0,\dots,f_{n-1}$ be a common level of the first integrals $F_0,\dots,F_{n-1}$. Then the polynomial $P(\mu)=\sum_{k=0}^{n-1}\mu^{n-1-k}(-1)^{k}f_k$ has $n-1$ real zeros taking account of multiplicities. Proof. We carry out the proof for odd $n$ (for even $n$ this is similar). Consider the motion equations on the common level $F_0=f_0,\dots,F_{n-1}=f_{n-1}$ (see (4.8)). They imply that for almost all tuples $f_0,\dots,f_{n-1}$ the intervals $(-\infty,a_n)$, $(a_n,a_{n-1}),\dots, (a_2,a_1)$ contains subintervals on which the polynomial takes positive values.
Note that $P(z)=\sum_{k=0}^{n-1}z^{n-1-k}(-1)^{k}f_k$ tends to $-\infty$ as $x\to-\infty$, while $Q(z)=\prod_{i=1}^n(a_i-z)$ tends to $+\infty$. Hence $V(z)\to-\infty$ as $x\to-\infty$. Since the interval $(-\infty,a_n)$ contains a subinterval with $V(z)>0$, the polynomial $V(z)$ has at least one zero on $(-\infty,a_n)$. Assume that there is $\varepsilon>0$ such that $V(z)< 0$ for $z \in (a_n,a_n + \varepsilon)$; then $P(z)$ has a zero on $(a_{n},a_{n-1})$. On the other hand, if $V(z)>0$ for $z\in(a_n,a_n+\varepsilon)$, then $P(z)$ has at least two zeros on $(-\infty,a_n)$. Thus, $P(z)$ has at least two zeros on the interval $(-\infty,a_{n-1})$. Proceeding by induction in a similar way we obtain the required result. The proof is complete. Thus, each straight line in $\mathbb R^n$ is tangent to at least $n-1$ confocal quadrics. Note that Chasles [3] showed that a straight line in $\mathbb R^3$ is tangent to precisely two confocal quadrics. We summarize this section by the following statement. Corollary 2. The trajectory of a point mass moving in $\mathbb{R}^n$ by inertia is tangent to $n -1$ confocal quadrics. The parameters of these quadrics and the energy of the particle form a system of $n$ functionally independent pairwise commuting first integrals. § 5. Proof of the main theorem Proof of Theorem 2. Let $Q_1,\dots,Q_k$ be confocal nondegenerate quadrics of different types in $\mathbb R^n$, and let $Q=\bigcap_{i=1}^k Q_i$. By fixing $Q_i$ we fix the elliptic coordinate with some index $c_i$, and vice versa (by fixing an elliptic coordinate we fix a quadric in the family of confocal quadrics). As all the quadrics $Q_i$ are nondegenerate and have different types, $Q$ can be defined as a surface on which the $k$ elliptic coordinates $\lambda_{c_1},\dots,\lambda_{c_k}$ are fixed. The momenta corresponding to these coordinates vanish on $Q$, while the other elliptic coordinates form a system of coordinates on this intersection.
The function $H=F_0$ is the Hamiltonian of the geodesic flow on $Q$, so that by Proposition 6 the functions $F_0,\dots,F_{n-1}$ are first integrals of this system. In fact, to restrict some $F_j$ to a quadric $Q_i$ we fix the corresponding elliptic coordinate and set its momentum equal to zero. This does not affect the statement of Proposition 6. Hence the equation of tangency (4.2) is preserved along each geodesic on $Q$. By Proposition 9 this equation always has $n-1$ roots. Some of these roots are known: these are the parameters of the quadrics $Q_1,\dots,Q_k$. The other $n-k-1$ roots are the parameters of the quadrics tangent (in a nontrivial way) to all tangent lines to the geodesic in question. To complete the proof we observe that the functions $F_0,\dots,F_{n-k-1}$ are functionally independent on $Q$. This can be verified as in Proposition 7. Theorem 2 is proved. § 6. Topology of compact intersections of confocal quadricsIn this section we prove Theorem 3 on the homeomorphism classes of compact intersections of several nondegenerate confocal quadrics. First of all note that by Kozlov’s theorem on topological obstructions to integrability (see [14]) the geodesic flow on a compact closed orientable two-dimensional analytic surface has an additional analytic first integral if and only if this surface is a 2-sphere $S^2$ or a 2-torus $T^2$. Note that if the intersection of $n-2$ confocal quadrics in $\mathbb R^n$ is compact, then by Kozlov’s result and Theorem 2 a connected component of this intersection is homeomorphic to the 2-sphere $S^2$ or the 2-torus $T^2$. This was previously pointed out by Kibkalo. However, this approach cannot be applied to a larger number of degrees of freedom. First we look at a simpler problem. We find the homeomorphism classes of compact intersections of pairs of confocal quadrics in $\mathbb R^4$. An intersection is compact if and only if one quadric is an ellipsoid. On the other hand this is equivalent to fixing the elliptic coordinate $\lambda_1$. A triaxial ellipsoid in $\mathbb R^4$ is glued from two equal domains, bounded by ellipsoids, along their boundary. Moreover, the elliptic coordinates $\lambda_2$, $\lambda_3$ and $\lambda_4$ on a triaxial ellipsoid in $\mathbb R^4$ correspond to the elliptic coordinates $\lambda_1'$, $\lambda_2'$ and $\lambda_3'$ in the domain bounded by a two-dimensional ellipsoid in $\mathbb R^3$. Since we examine compact intersections, three cases of fixed coordinates are possible: 1) $\lambda_1$ and $\lambda_2$; 2) $\lambda_1$ and $\lambda_3$; 3) $\lambda_1$ and $\lambda_4$. Consider case 1). Fixing the elliptic coordinate $\lambda_1'$ inside the two-dimensional ellipsoid we obtain a surface homeomorphic to the 2-sphere $S^2$ (Figure 3, a). Hence by gluing two such domains along the boundary we see that the intersection of two confocal nondegenerate quadrics in $\mathbb R^4$ corresponding to coordinates $\lambda_1$ and $\lambda_2$ is homeomorphic to a disjoint union of two spheres, that is, to the direct product $S^0\times S^2$ of a zero-dimensional and a two-dimensional sphere. In case 2), in the domain inside a two-dimensional ellipsoid we must fix the elliptic coordinate $\lambda_2'$. Then we obtain a surface homeomorphic to a cylinder (Figure 3, b). When two such domains are glued, the cylinders glue into a 2-torus $T^2=S^1\times S^1$. In case 3), fixing $\lambda_3'$ in the domain bounded by a two-dimensional ellipsoid we obtain a disjoint union of two discs (Figure 3, c). In gluing two copies of such a domain four discs are transformed into two 2-spheres, that is, a direct product of spheres $S^2\times S^0$. Thus a compact intersection of two nondegenerate confocal quadrics in $\mathbb R^4$ is always homeomorphic to a direct product of two spheres. Note that the dimensions of these spheres are equal to the number of nonfixed elliptic coordinates between the two fixed ones and to the number of the nonfixed coordinates following the fixed quadric with the greatest index. Now we go over to the general case. To avoid confusion in notation we assume that
$$
\begin{equation*}
a_1<\dots<a_n,\qquad \lambda_1\in(-\infty,a_1),\quad \lambda_2\in(a_1,a_2),\quad \dots,\quad \lambda_n\in(a_{n-1},a_n).
\end{equation*}
\notag
$$
Assume that the quadrics $Q_1,\dots,Q_k$ are arranged in the increasing order of their parameters. Let $c_i$ be the index of the elliptic coordinate corresponding to $Q_i$. By assumption $c_1<\dots<c_k$. Set $c_{k+1}=n+1$. Let $Q=\bigcap_{i=1}^kQ_i$ be compact: this is equivalent to one of the quadrics $Q_i$ being an ellipsoid. In its turn, this is equivalent to fixing the coordinate $\lambda_1$ on $Q$, so that $c_1=1$. We divide the indices of the elliptic coordinates into the following $k$ groups:
$$
\begin{equation*}
I_1=\{c_1,\dots,c_2-1\},\quad I_2=\{c_2,\dots,c_3-1\},\quad \dots,\quad I_k=\{c_k,\dots,c_{k+1}-1\}.
\end{equation*}
\notag
$$
Note that these sets are disjoint and exhaust all integers from $1$ to $n$. Moreover, precisely one coordinate in each group is fixed on $Q$. Let $m_j=c_{j+1}-c_j-1$ be the cardinality of $I_j$ minus $1$. Theorem 3 claims that $Q$ is homeomorphic to the direct product of spheres $S^{m_1}\times\dots\times S^{m_k}$. Now we show this. Proof of Theorem 3. Recall that elliptic coordinates, regarded as functions of a point in the space $\mathbb R^n$ with Cartesian coordinates $(x_1,\dots,x_n)$, are continuous. By Proposition 1 we have equalities (2.2). For each $\alpha=1,\dots,k$ and $i\in I_\alpha$ we write the corresponding equality for $x_i$:
$$
\begin{equation}
x_i^2=\frac{\prod_{j=1}^n(a_i-\lambda_j)}{\prod_{j\neq i}(a_i-a_j)}= \frac{\prod_{j\in I_\alpha}(a_i-\lambda_j)}{\prod_{j\in I_{\alpha},j\neq i}(a_i-a_j)}\, \frac{\prod_{j\notin I_\alpha}(a_i-\lambda_j)}{\prod_{j\notin I_{\alpha}}(a_i-a_j)}.
\end{equation}
\tag{6.1}
$$
We denote the second factor in this product by $B_i(\lambda)$. Note that the first factor is the right-hand side of a connection formula of type (2.2) for the elliptic coordinates corresponding to the family of confocal quadrics
$$
\begin{equation*}
\frac{z_1^2}{a_{c_\alpha}-\lambda}+\frac{z_2^2}{a_{c_{\alpha}+1} -\lambda}+\dots+\frac{z_{m_i+1}^2}{a_{c_{\alpha+1}-1}-\lambda}=1.
\end{equation*}
\notag
$$
Hence the first factor in (6.1) is nonnegative, and therefore $B_i(\lambda)\geqslant0$ for each $i$.
We show that all the expressions $B_i(\lambda)$ are positive on $Q$. It is sufficient to show that $B_i(\lambda)\neq 0$ on this intersection. Let $B_i(\lambda)=0$ for some $i$; then there exists $j\notin I_\alpha$ such that $\lambda_j=a_i$. Since only the two elliptic coordinates $\lambda_i$ and $\lambda_{i+1}$ can assume the value $a_i$, we have $j=i + 1$. As $j\notin I_\alpha$, $j$ belongs to $I_{\alpha+1}$ and is the least element in this set. The coordinates with smallest indices in the sets $I_{k}$ are fixed on $Q$, so that $\lambda_j$ is fixed. Now, since $Q_1,\dots,Q_k$ are nondegenerate quadrics, on $Q$ we have $\lambda_{i+1}\in(a_{i},a_{i+1})$. Hence $\lambda_j\neq a_i$. Thus all the $B_{i}(\lambda)$ are positive on the intersection under consideration. Now consider $\mathbb R^n$ with Cartesian coordinates $(y_1,\dots,y_n)$, and let $f$ be the map of $Q$ defined by the formulae
$$
\begin{equation*}
y_i=\frac{x_i}{\sqrt{B_i(\lambda)}}\quad\text{for all } i=1,\dots,n.
\end{equation*}
\notag
$$
Since $B_i(\lambda)>0$ on $Q$, $f$ is well defined. Moreover, as elliptic coordinates are continuous functions of the point, the map $f$ is continuous. In addition, for each $\alpha=1,\dots,k$ and all $i\in I_\alpha$
$$
\begin{equation*}
y_i^2=\frac{\prod_{j\in I_\alpha}(a_i-\lambda_j)}{\prod_{j\in I_{\alpha},j\neq i}(a_i-a_j)}.
\end{equation*}
\notag
$$
We show that $f$ is injective. In fact, if we know all the coordinates $y_i$ of a point, then we recover the elliptic coordinates of its preimage in a unique way. Further, as, in addition, coordinates of a point and the corresponding coordinates of its image have the same signs, we can recover the $f$-preimage of the point $(y_1,\dots,y_n)$. Since $Q$ is compact and $f$ is a continuous bijection, $Q$ is homeomorphic to $f(Q)$. We find the homeomorphism class of $f(Q)$. Since $y_{c_i},\dots,y_{c_{i+1}-1}$ are connected with $\lambda_{c_i},\dots,\lambda_{c_{i+1}-1}$ in just the same way as elliptic coordinates are connected with Cartesian ones, by fixing a quadric $Q_i$ we obtain a constraint of the form
$$
\begin{equation}
\frac{y_{c_i}^2}{a_{c_i}-\lambda_{c_i}}+\dots +\frac{y_{c_{i+1}-1}^2}{a_{c_{i+1}-1}-\lambda_{c_i}}=1.
\end{equation}
\tag{6.2}
$$
The surface in $\mathbb R^{m_i+1}(y_{c_i},\dots,y_{c_{i+1}-1})$ defined by equation (6.2) is an $m_i$-dimensional ellipsoid, which is homeomorphic to the sphere $S^{m_i}$.
Thus, $f(Q)$ is the intersection of the quadrics (6.2) for $i=1,\dots,k$. Note that each variable $y_i$ occurs just once in all these equations, so that $f(Q)$ is homeomorphic to the direct product of spheres $S^{m_1}\times\dots \times S^{m_k}$. Theorem 3 is proved. Remark 6. In the above proof we did not use the fact that the corresponding geodesic flow is integrable. § 7. Systems with potentialProposition 6 was a key to the proof of Theorem 3. In this section we describe a class of integrable systems satisfying all the assumptions of this proposition. After that we prove Theorem 4. Consider the motion of a point mass in $\mathbb R^n$ under the action of the force field with potential $V(x_1,\dots,x_n)$. The total mechanical energy of this system is equal to $G_0=F_0+V$. Following Kozlov [16] we seek additional first integrals $G_1,\dots,G_{n-1}$, complementing the Hamiltonian $G_0$ in the following form: $G_i=F_i+f_i$, where $i=1,\dots,n-1$ and the functions $f_1,\dots,f_{n-1}$ depend only on the space variables. Remark 7. Kozlov used this method to find additional first integrals of a billiard system with potential inside an ellipse. Using this approach one need not take account of reflections, that is, one can look for first integrals in the problem without reflections. If we find the functions $f_i$, then by Proposition 7 the functions $G_0,\dots,G_{n-1}$ will be functionally independent. Now, functions $G_i$ are first integrals of the system in question if and only if $\{G_i,G_0\}=0$ for each $i=1,\dots,n-1$, that is,
$$
\begin{equation*}
0=\{G_i,G_0\}=\{F_i+f_i,\,F_0+V\}=\{F_i,V\}-\{F_0,f_i\}.
\end{equation*}
\notag
$$
We write these equations in terms of the elliptic coordinates:
$$
\begin{equation*}
4\sum_{j=1}^n \sigma^i_j\widehat{A}_jp_j\,\frac{\partial V}{\partial \lambda_j}-4\sum_{j=1}^n \widehat{A}_jp_j\,\frac{\partial f_i}{\partial \lambda_j}=0.
\end{equation*}
\notag
$$
This is equivalent to the following system of partial differential equations:
$$
\begin{equation*}
\frac{\partial f_i}{\partial \lambda_j}=\sigma^i_j\,\frac{\partial V}{\partial \lambda_j}\quad\text{for all } i\ \text{and}\ j.
\end{equation*}
\notag
$$
This system is solvable if and only if the mixed derivatives are well defined, that is,
$$
\begin{equation*}
\frac{\partial}{\partial \lambda_k}\biggl(\sigma^i_j\,\frac{\partial V}{\partial \lambda_j}\biggr)=\frac{\partial}{\partial \lambda_j}\biggl(\sigma^i_k\,\frac{\partial V}{\partial \lambda_k}\biggr), \qquad i=1,\dots,n-1, \quad j,k=1,\dots,n.
\end{equation*}
\notag
$$
Writing out derivatives of a product and using the equality $\sigma^i_j-\sigma^i_k=(\lambda_j-\lambda_k)\sigma^i_{jk}$ we reduce this system of partial differential equations to the form (1.1). Thus, if a smooth function $V$ satisfies equations (1.1), then functions $f_1,\dots,f_n$ exist, and therefore the system has $n$ functionally independent first integrals. This turns out to be a strong condition. If a smooth function $V$ satisfies (1.1), then the functions $G_0,\dots,G_{n-1}$ satisfy Proposition 6 as can easily be verified. Repeating the argument in the proof of Theorem 2 we see that if a smooth function $V$ satisfies (1.1), then the problem of the motion of a point mass on the intersection of several nondegenerate confocal quadrics under the action of the potential $V$ is square integrable. Theorem 4 is proved. Note that a Hooke potential with centre at the origin satisfies the conditions in Theorem 4. The potential $V=\sum_{i=1}^n\alpha_i/x_i^2$ discovered by Kozlov is also integrable. (Here the $\alpha_i$ are arbitrary real constants.) Although this potential does not satisfy the conditions in Theorem 4, one can find the functions $f_i$ for it too. 
Citation in format AMSBIB
\Bibitem{Bel23} |
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