G. V. Belozerov, A. T. Fomenko, “Generalized Jacobi–Chasles theorem in non-Euclidean spaces”, Sb. Math., 215:9 (2024), 1159

Processing math: 19%

Sbornik: Mathematics

RUS ENG

JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PACKAGE AMSBIB

JavaScript is disabled in your browser. Please switch it on to enable full functionality of the website

	General information
	Latest issue
	Forthcoming papers
	Archive
	Impact factor
	Guidelines for authors
	License agreement
	Submit a manuscript

	Search papers
	Search references

	RSS
	Latest issue
	Current issues
	Archive issues
	What is RSS

Mat. Sb.:
Year:
Volume:
Issue:
Page:
	Find

Personal entry:
Login:
Password:
	Save password
	Enter
	Forgotten password?
	Register

Sbornik: Mathematics, 2024, Volume 215, Issue 9, Pages 1159–1181
DOI: https://doi.org/10.4213/sm10099e (Mi sm10099)

Generalized Jacobi–Chasles theorem in non-Euclidean spaces

G. V. Belozerov^a, A. T. Fomenko^ab

^a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

English version PDF (618 kB) HTML full-text Russian version article

References:

PDF

HTML

DOI: https://doi.org/10.4213/sm10099e

Abstract: The classical Jacobi–Chasles theorem states that tangent lines to a geodesic curve on an

$n$ -axial ellipsoid in

$n$ -dimensional Euclidean space are also tangent, along with this ellipsoid, to

$n-2$ quadrics confocal with it, which are the same for all points on this geodesic. This result ensures the integrability of the geodesic flow on the ellipsoid. As recent results due to Belozerov and Kibkalo show, a similar theorem also holds for an arbitrary intersection of confocal quadrics in Euclidean space. In the present paper it is shown that the geodesic flow on an intersection of several confocal quadrics in a pseudo-Euclidean space

$\mathbb R^{p,q}$ or on a constant curvature space is integrable. As a consequence, a similar result is established for confocal billiards on such intersections. It is also shown that in codimension 2 the last result cannot be extended to surfaces not locally isometric to a space of constant curvature.
Bibliography: 15 titles.

Keywords: geodesic flow, integrable system, confocal quadrics, elliptic coordinates, Jacobi–Chasles theorem.

Funding agency	Grant number
Russian Science Foundation	22-71-10106
This research was carried out at Lomonosov Moscow State University and supported by the Russian Science Foundation under grant no. 22-71-10106, https://rscf.ru/en/project/22-71-10106/.

Received: 27.03.2024 and 19.04.2024

Bibliographic databases:

Document Type: Article

MSC: 37D40, 51N35, 53C22

Language: English

Original paper language: Russian

§ 1. Introduction

By the classical Jacobi–Chasles theorem the geodesic flow on a $n$ -axial ellipsoid in $n$ -dimensional Euclidean space is integrable. Moreover, the tangent line at each point of a geodesic curve on the ellipsoid is also tangent to $n-2$ quadrics confocal with the ellipsoid and the same for all points on this curve (see [1]–[3]). For a modern proof of the Jacobi–Chasles theorem, see [4].

Recall that a family of confocal quadrics in the $n$ -dimensional Euclidean space $\mathbb R^n$ is the set of quadrics defined by the equation

$\begin{equation} \frac{x_1^2}{a_1-\lambda}+\dots+\frac{x_n^2}{a_n-\lambda}=1, \end{equation} \tag{1.1}$

where

$a_1<\dots<a_n$ are fixed numbers and

$\lambda$ is a real parameter.

Recently Kibkalo considered the question of integrability for the geodesic flow on an intersection of several confocal quadrics. He showed that if such an intersection has dimension 2, then this system is integrable. In fact, a more general result holds.

Theorem 1 (Belozerov [5]). Let $Q_1,\dots,Q_k$ be nondegenerate confocal quadrics of different types in $\mathbb R^n$ , and let $Q=\bigcap_{i=1}^k Q_i$ . Then

(1) the geodesic flow on $Q$ is quadratically integrable;
(2) along with $Q_1,\dots,Q_k$ , tangent lines at points of a geodesic curve on $Q$ are also tangent to $n-k-1$ quadrics confocal with them, which are the same for all points on this curve.

Remark 1. Theorem 1 is nontrivial, that is, it is not a consequence of the classical Jacobi–Chasles theorem. In fact, a geodesic curve on the intersection of confocal quadrics $Q_1,\dots,Q_k$ is not a geodesic on any $Q_i$ in the general case. For example consider the limit of a family of confocal quadrics in $\mathbb R^3$ as $a_2,a_3\to a_1$ (where first we take the limit with respect to $a_2$ and then with respect to $a_3$ ). It is easy to show that in the limit the elliptic coordinates become the spherical ones. Hence confocal quadrics turn to level surfaces of spherical coordinates: spheres, cones and half-planes. However, the intersection of a cone and a sphere with the same centre consists of two circles (Figure 1), which are not geodesic curves on the sphere or on the cone.

Figure 1.A sphere and a cone with the common centre intersect in curves which are not geodesics on the sphere or cone.

Remark 2. Theorem 1 is of interest staring with ${n=4}$ . On the plane the phenomenon cannot be observed. In $\mathbb R^3$ , for $k=1$ we have the classical Jacobi–Chasles theorem, while for $k=2$ there is nothing to see again because the intersection of two confocal quadrics in $\mathbb R^3$ has dimension one. A thorough proof of Theorem 1 was presented in [6].

Remark 3. The intersection of several confocal quadrics in $\mathbb R^n$ is diffeomorphic to a Cartesian product of the form $\mathbb R^{k_0}\times S^{k_1}\times\dots\times S^{k_m}$ (where $S^{k_i}$ is a sphere of dimension $k_i$ ). Here the integers $m,k_0,\dots,k_m$ can be defined as follows. Let $i_1\leqslant\dots\leqslant i_m$ be the indices of elliptic coordinates that are fixed on the intersection of confocal quadrics, and set $i_0=0$ and $i_{m+1}=n+1$ . Then $k_j=i_{j+1}-i_j-1$ for $j=0,\dots,m$ . The corresponding proof was also presented in [6].

This paper is devoted to discovering analogues of the generalized Jacobi–Chasles theorem in non-Euclidean spaces, namely, in the Minkowski spaces $\mathbb R^{p,q}$ and in spaces of constant sectional curvature in two-dimensional directions: spheres, Lobachevsky spaces and projective spaces.

An analogue of the classical Jacobi–Chasles theorem (on the integrability of the geodesic flow on an ellipsoid in $\mathbb R^{p,q}$ ) was proved by Khesin and Tabachnikov [7]. In particular, it follows from their result that the geodesic flow on an ellipsoid in $\mathbb R^{p,q}$ , and the classical billiard inside this ellipsoid are integrable Hamiltonian systems. The Liouville foliations for the geodesic flow on an ellipsoid in $\mathbb R^{2,1}$ and for the billiard system inside an ellipse on the Minkowski plane were considered by Dragović and Radnović [8]. Karginova [9], [10] investigated the topology of the Liouville foliation for billiards bounded by confocal quadrics on the Minkowski plane. Also note the paper [11] by Dragović and Radnović, where they examined the structure of the elliptic coordinates in $\mathbb R^{p,q}$ and periodic trajectories of pseudo-Euclidean billiards bounded by confocal quadrics.

The generalized Jacobi–Chasles theorem turns out to hold in $\mathbb R^{p,q}$ ; we devote § 3 to its proof. But before that, in § 2 we investigate the structure of the family of confocal quadrics in $\mathbb R^{p,q}$ and demonstrate a few of its important properties.

It often occurs that when a result holds in Euclidean and pseudo-Euclidean spaces, an analogue of it can be stated for spaces of constant curvature. Based on this observation, Fomenko conjectured that the generalized Jacobi–Chasles theorem holds for spheres $S^n$ , projective spaces $\mathbb R P^n$ and Lobachevsky spaces $L^n$ of any dimension (endowed with the standard Riemannian metrics) and perhaps also for the quotients of $S^n$ and $L^n$ by discrete subgroups of their isometry groups. In § 4 we show that this conjecture holds for $S^n$ , $L^n$ and $\mathbb R P^n$ . This turns out to be a consequence of the generalized Jacobi–Chasles theorem for Euclidean and pseudo-Euclidean spaces.

After that, in § 5, on the basis of the analogues of the generalized Jacobi–Chasles theorem that we establish, we show that billiard systems on intersections of confocal quadrics, in domains bounded by a finite number of other confocal quadrics, are integrable too. Note that for a billiard on a sphere, in a domain bounded by a cone with the same centre as this sphere, its integrability was shown in the book [12] by Kozlov and Treshchev. For Lobachevsky spaces a similar result is due to Bolotin [13].

Apart from this, in § 5 we show that in dimension $2$ a manifold such that each small circular billiard on it is quadratically integrable is locally isometric to a 2-sphere, the Lobachevsky plane, or the Euclidean plane, that is, to a space of constant (positive, negative, or zero) curvature. This suggests that the generalized Jacobi–Chasles theorem can only be true on spaces of that have a constant sectional curvature in two-dimensional directions.

Acknowledgement

The authors are grateful to V. N. Zav’yalov for his attention to this work.

§ 2. Confocal quadrics and elliptic coordinates in the Minkowski space ${\mathbb R}^{p,q}$

Recall that the Minkowski space $\mathbb R^{p,q}(x_1,\dots,x_p,x_{p+1},\dots,x_{p+q})$ is the real vector space of dimension $n=p+q$ with variables $(x_1,\dots,x_n)$ that is endowed with a pseudo-scalar product $\langle\,\cdot\,{,}\,\cdot\,\rangle$ which can be calculated in these variables by the formula

$\begin{equation*} \langle v, w\rangle=\sum_{i=1}^p v_i w_i-\sum_{i=p+1}^n v_i w_i\quad \forall\, v=(v_1,\dots,v_n), \quad w=(w_1,\dots,w_n)\in \mathbb R^{p,q}. \end{equation*} \notag$

Below we call

$(x_1,\dots,x_n)$ the Cartesian coordinates.

Since a pseudo-scalar product is not positive definite in general, all vectors in $\mathbb R^{p,q}$ are usually divided into three classes: timelike, spacelike and isotropic ones. Recall that a vector ${v\in \mathbb{R}^{p,q}}$ is said to be spacelike if $\langle v,v \rangle>0$ , timelike if $\langle v,v \rangle<0$ and isotropic if $\langle v,v \rangle=0$ .

Definition 1. By a family of confocal quadrics in the Minkowski space $\mathbb R^{p,q}(x_1,\dots,x_{n})$ we mean a family of quadrics given by the equations

$\begin{equation} \frac{x_1^2}{a_1-\lambda}+\dots+\frac{x_p^2}{a_p-\lambda}+\frac{x_{p+1}^2}{b_{1}+\lambda} +\dots+\frac{x_{n}^2}{b_{q}+\lambda}=1, \end{equation} \tag{2.1}$

where

$0<a_1<\dots<a_p$ and

$0<b_1<\dots<b_q$ are fixed numbers and

$\lambda$ is a real parameter.

Remark 4. If $\lambda=a_i$ or $\lambda=-b_j$ for some $i$ or $j$ , then the corresponding quadric is not defined. For its definition, first of all we must multiply (2.1) by the product $\prod_i(a-\lambda_j)\prod_j(b+\lambda_j)$ and then set $\lambda=a_i$ in the resulting expression. It is easy to see that the quadric corresponding to the value $a_i$ of the parameter is the hyperplane $x_i=0$ .

We establish a few important properties of confocal quadrics in $\mathbb R^{p,q}$ .

Proposition 1. (1) Each point in $\mathbb R^{p,q}$ lies on $n$ or $n-2$ confocal quadrics, counting multiplicities. The parameter of at least $n-2$ of these lie on the intervals $[a_i,a_{i+1}]$ and $[-b_{j+1},-b_j]$ , where $i=1,\dots,p-1$ and $j=1,\dots,q-1$ .

(2) If a point in $\mathbb R^{p,q}$ lies on a quadric with parameter

• $\lambda\in(-\infty,-b_{q})$ ,
• $\lambda\in(-b_{1},a_1)$ ,
• $\lambda\in(a_{p},+\infty)$ ,

then there is another quadric (taking account of multiplicity) through this point, whose parameter lies on the same interval.

Proof. We begin with part (1) and prove it for a point

$P=(x_1,\dots,x_n)$ in general position, when all of its coordinates are distinct from zero. Consider the function

$\begin{equation*} f_P(\lambda)=\frac{x_1^2}{a_1-\lambda}+\dots+\frac{x_p^2}{a_p-\lambda} +\frac{x_{p+1}^2}{b_{1}+\lambda}+\dots+\frac{x_{p+q}^2}{b_{q}+\lambda}. \end{equation*} \notag$

For a point in

$\mathbb R^{4,4}$ its graph is shown in Figure 2.

Figure 2.The graph of $f_P(\lambda)$ for a point in $\mathbb R^{4,4}$ .

Note that the right-hand limits of this functions at the points $a_i$ are equal to $-\infty$ , while the left-hand limits are equal to $+\infty$ . Hence by the intermediate value theorem for continuous functions, on each interval $(a_i,a_{i+1})$ , $i= 1,\dots,p-1$ , the function $f_{P}(\lambda)$ takes the value $1$ at least once. In a similar way, on each interval $(-b_{i+1},-b_i)$ , where $i=1,\dots,q-1$ , $f_{P}(\lambda)$ takes the value $1$ at least once.

Hence $f_P(\lambda)$ takes the value $1$ at least at $n- 2$ points. This is the same as to say that at least $n-2$ confocal quadrics pass through $P$ . It remains to observe that $\prod_{i=1}^p(a_i-\lambda)\prod_{j=1}^q(b_j+\lambda)(f_P(\lambda)-1)$ is a polynomial of degree $n$ . Thus, the equation $f_P(\lambda)=1$ has at most $n$ real roots counting multiplicities. This completes the proof of part (1).

Next we turn to part (2). Assume that a quadric with parameter $\lambda_0\in(-b_1,a_1)$ passes through $P$ . Then generically there exists $\lambda'\in(-b_1,a_1)$ such that ${f_P(\lambda')<1}$ . Without loss of generality we can assume that $\lambda'>\lambda_0$ . In approaching the endpoints of the interval $(-b_1,a_1)$ the function $f_P(\lambda)$ tends to $+\infty$ , so by the intermediate value theorem for continuous functions there exists $\lambda''\in(-\lambda',a_1)$ such that $f_P(\lambda'')=1$ . The other cases are treated similarly.

The proof is complete.

Proposition 1 prompts us to distinguish the confocal quadrics with parameters on the intervals $(-\infty,-b_{q})$ , $(-b_{1},a_1)$ and $(a_{p},+\infty)$ .

Definition 2. We say that a quadric in the family (2.1) with parameter

$\begin{equation*} \lambda\in(-\infty,-b_{q})\cup(-b_{1},a_1)\cup(a_{p},+\infty) \end{equation*} \notag$

is unique.

Proposition 2. At a point of intersection of two confocal quadrics in $\mathbb R^{p,q}$ their tangent planes are orthogonal (in the sense of $\mathbb R^{p,q}$ ).

Proof. Let two confocal quadrics with parameters

$\lambda_1$ and

$\lambda_2$ intersect at

$P=(x_1,\dots,x_n)$ . Then the following system of equalities holds:

$\begin{equation*} \begin{cases} \dfrac{x_1^2}{a_1-\lambda_1}+\dots+\dfrac{x_p^2}{a_p-\lambda_1}+\dfrac{x_{p+1}^2}{b_{1}+\lambda_1} +\dots+\dfrac{x_{p+q}^2}{b_{q}+\lambda_1}=1, \\ \dfrac{x_1^2}{a_1-\lambda_2}+\dots+\dfrac{x_p^2}{a_p-\lambda_2}+\dfrac{x_{p+1}^2}{b_{1}+\lambda_2} +\dots+\dfrac{x_{p+q}^2}{b_{q}+\lambda_2}=1. \end{cases} \end{equation*} \notag$

From the first equality we subtract the second and divide the result by

$\lambda_1-\lambda_2$ :

$\begin{equation} \frac{x_1^2}{(a_1-\lambda_1)(a-\lambda_2)}+\dots-\frac{x_{p+q}^2}{(b_{q}+\lambda_1)(b_{q}+\lambda_2)}=0. \end{equation} \tag{2.2}$

This is equivalent to the condition that the normals to the quadrics with parameters

$\lambda_1$ and

$\lambda_2$ are orthogonal at

$P$ .

The proof is complete.

Let $A$ and $B$ denote the ordered tuples of coefficients $(a_1,\dots,a_p)$ and $(b_1,\dots,b_q)$ , respectively. We denote the subset of $\mathbb R^{p,q}$ such that precisely $n$ confocal quadrics (counting multiplicities) pass through each point in it by $D(A,B)$ .

Remark 5. By Proposition 1 precisely $n$ quadrics (counting multiplicities) pass through each point on an ellipsoid in the family (2.1). Hence $D(A,B)$ is nonempty.

For each point ${P\in D(A,B)}$ we denote by $\lambda_1\leqslant\dots\leqslant\lambda_n$ the parameters of confocal quadrics passing through $P$ .

Definition 3. The functions $\lambda_1,\dots,\lambda_n$ on the set $D(A,B)$ are called the elliptic coordinates in the Minkowski space $\mathbb R^{p,q}$ .

Note that we can define elliptic coordinates at each point in $\mathbb R^{p,q}$ as the set of solutions of (2.1); however, by Proposition 1 this equation can also have $n-2$ real roots $\lambda_1\leqslant\dots\leqslant\lambda_{n-2}$ and two complex conjugate roots $\lambda_{n-1},\lambda_n$ . In this case we call the set of functions $\lambda_i$ pseudoelliptic coordinates: as $\lambda_{n-1}$ and $\lambda_n$ are complex, they are not coordinates in $\mathbb R^{p,q}$ . However, $\lambda_{n-1}$ and $\lambda_{n}$ are complex conjugate, so we can replace them by $z_{n-1}=\lambda_1+\lambda_2$ and $z_n=i(\lambda_1-\lambda_2)$ , respectively. Then the set of functions $\lambda_1,\dots,\lambda_{n-2},z_{n-1},z_n$ forms a smooth regular system of coordinates (almost everywhere).

Definition 4. By the generalized elliptic coordinates we mean the elliptic coordinates in $D(A,B)$ and the pseudoelliptic coordinates in the remaining part of $\mathbb R^{p,q}$ .

For convenience we introduce unified notation. We denote $-b_1,\dots,-b_q$ by $a_{p+1},\dots,a_{n}$ , respectively. Then the family of confocal quadrics is defined by

$\begin{equation*} \sum_{i=1}^{p}\frac{x_i^2}{a_i-\lambda}-\sum_{i=p+1}^{n}\frac{x_i^2}{a_i-\lambda}=1. \end{equation*} \notag$

Set

$\begin{equation*} s_i= \begin{cases} 1 &\text{for } i=1,\dots,p, \\ -1 &\text{for } i=p+1,\dots,n. \end{cases} \end{equation*} \notag$

The next statement establishes a relation between the Cartesian coordinates $(x_1,\dots,x_n)$ and the generalized elliptic coordinates $(\lambda_1,\dots,\lambda_n)$ .

Proposition 3. The Cartesian coordinates $(x_1,\dots,x_n)$ and the generalized elliptic coordinates $(\lambda_1,\dots,\lambda_n)$ are related by the equalities

$\begin{equation} x_i^2=s_i\frac{\prod_{j=1}^n(a_i-\lambda_j)}{\prod_{j\neq i}^n(a_i-a_j)}. \end{equation} \tag{2.3}$

Proof. We make the substitution

$\widetilde{x}_j=i x_j$ (here

$i$ is the imaginary unit) for

$j=p+1,\dots,n$ . Then the equation of a family of confocal quadrics in

$\mathbb R^{p,q}(x_1,\dots,x_n)$ turns to the equation of a family of conformal quadrics in the Euclidean space

$\mathbb R^n(x_1,\dots,x_p,$

$\widetilde{x}_{p+1},\dots,\widetilde{x}_n)$ .

It remains to use the formulae linking the elliptic and Cartesian coordinates in Euclidean space (see, for instance, [2] or [6]) and then make the inverse change of variables.

The proof is complete.

Remark 6. Explicit formulae for the relation between elliptic and Cartesian coordinates were discovered by Jacobi [2].

In what follows we widely use the elliptic coordinates. In this connection we introduce some notation to simplify dealing with formulae:

$\begin{equation*} \Delta_k=\prod_{j=1}^n(a_k-\lambda_j)\quad\text{and} \quad \rho_k=\prod_{j\neq k}(a_k-a_j). \end{equation*} \notag$

We let

$\sigma^m_{i_1,\dots,i_k}(a_1,\dots,a_n)$ denote the elementary symmetric polynomial of degree

$m$ in the variables

$\{a_1,\dots,$

$a_n\}\backslash\{a_{i_1},\dots,a_{i_k}\}$ , where we set that for

$m=0$ the polynomial is identically equal to one and for

$m=-1$ it is zero.

Note that by using this notation we can state Proposition 2 as follows.

Proposition 4. For all $i,j=1,\dots, n$ , $i\neq j$ ,

$\begin{equation} \sum_{k=1}^n\frac{\Delta_k}{\rho_k(a_k-\lambda_i)(a_k-\lambda_j)}=0. \end{equation} \tag{2.4}$

Proof. First we note that formulae (2.3) linking the elliptic and Cartesian coordinates in

$\mathbb R^{p,q}$ can be expressed as follows:

$\begin{equation} x_i^2=s_i\frac{\Delta_i}{\rho_i}. \end{equation} \tag{2.5}$

It remains to observe that for all

$i\neq j$ we have formula (2.2), which assumes the required form, provided that we take (2.5) into account.

The proof is complete.

In the next section, using the information obtained about confocal quadrics and elliptic coordinates in Minkowski spaces $\mathbb R^{p,q}$ , we state and prove an analogue of the generalized Jacobi–Chasles theorem.

§ 3. Generalized Jacobi–Chasles theorem in pseudo-Euclidean spaces

Below we consider geodesic flows on submanifolds of $\mathbb R^{p,q}$ . In the motion along a geodesic curve the velocity vector preserves its length, so we can divide all geodesics, in accordance with the length of the velocity vector on them, into timelike, spacelike and isotropic ones.

Theorem 2 (Belozerov and Fomenko). Let $Q_1,\dots,Q_k$ be distinct nondegenerate confocal quadrics in $\mathbb R^{p,q}$ , and let $Q=\bigcap_{i=1}^k Q_i\neq\varnothing$ . Then

(1) the geodesic flow on $Q$ is quadratically integrable;
(2) along with $Q_1,\dots,Q_k$ , the tangent line at each point on a timelike or spacelike geodesic curve on $Q$ is tangent to some other $n-k-1$ or $n-k-3$ quadrics confocal with $Q$ , which are the same for all points on this curve.

Proof. We carry out the proof in several steps. First we look at an auxiliary problem. Namely, by examining the inertial motion of a point mass in

$\mathbb R^{p,q}$ we find the parameters of confocal quadrics in the family (2.1) that are tangent to the straight trajectory of the point mass. We will see that these parameters are roots of a polynomial with functional coefficients defined on

$T\mathbb R^{p,q}$ . At the next step, by passing to the elliptic coordinates, we analyse the properties of these functions and prove a formula for separating variables. Finally, on the basis of the properties and formulae established we prove the assertion of the theorem.

Step 1. Consider the straight line through a point ${P=(x_1,\dots,x_n)}$ in $\mathbb R^{p,q}$ in the direction $v=(\dot{x}_1,\dots,\dot{x}_n)$ . Note that each straight line is the trajectory of a point mass moving by inertia in $\mathbb R^{p,q}$ . To find the points of intersection of this line and the quadric with parameter $\mu$ we must solve the following quadratic equation (with respect to $\tau$ ):

$\begin{equation*} \sum_{i=1}^ns_i\frac{(x_i+\tau\dot{x}_i)^2}{a_i-\mu}=1. \end{equation*} \notag$

The line is tangent to the quadric if and only if the discriminant of this equation vanishes. This can also be written as

$\begin{equation} \biggl(\sum_{i=1}^ns_i\frac{x_i\dot{x}_i}{a_i-\mu}\biggr)^2=\biggl(\sum_{i=1}^ns_i\frac{\dot{x}_i^2}{a_i-\mu}\biggr) \biggl(\sum_{i=1}^n s_i\frac{x_i^2}{a_i-\mu}-1\biggr). \end{equation} \tag{3.1}$

Hence to find quadrics tangent to the straight trajectory of the point mass we must solve equation (3.1) with respect to

$\mu$ .

We transform this equation by removing parentheses on both sides, moving all terms to the left-hand side and selecting full squares:

$\begin{equation*} \sum_{i=1}^{n}s_i\frac{\dot{x}_i^2}{a_i-\mu}-\sum_{i<j}\frac{s_is_jK_{i,j}^2}{(a_i-\mu)(a_j-\mu)}=0. \end{equation*} \notag$

Here

$K_{i,j}=x_i\dot{x}_j-x_j\dot{x_i}$ . We multiply both sides by

$\prod_i(a_i-\mu)$ :

$\begin{equation} \sum_{k=1}^n s_i\prod_{m\neq k} (a_m-\mu)\dot{x}_k^2-\sum_{i<j}s_is_j\prod_{m\neq i,j}(a_m-\mu)K_{i,j}^2=0. \end{equation} \tag{3.2}$

We call this the tangency equation, and we call the left-hand polynomial the tangency polynomial. We denote the latter by

$W(\mu)$ . Generically, it has degree

$n-1$ . Let

$F_{m}$ denote the coefficient of

$2\cdot(-1)^{n-1-m}\mu^{n-m-1}$ in

$W(\mu)$ . Then we obtain the equalities

$\begin{equation} F_m=\frac{1}{2}\sum_{k=1}^ns_i\sigma_k^m(a_1,\dots,a_n)\dot{x}_k^2 -\frac{1}{2}\sum_{i<j}s_is_j\sigma_{i,j}^{m-1}(a_1,\dots,a_n)K_{i,j}^2. \end{equation} \tag{3.3}$

Remark 7. All functions $F_m$ are smooth in $T\mathbb R^{p,q}$ . Furthermore, $F_0$ is the kinetic energy of the point mass in $\mathbb R^{p,q}$ .

Now we examine the properties of the coefficients of the tangency polynomial. They are not easy to see in Cartesian variables. However, as $W(\mu)$ was obtained in the study of the tangency of confocal quadrics, it looks reasonable to express it in terms of the elliptic coordinates.

We replace $x_{p+1},\dots,x_n$ by $i\widetilde{x}_{p+1},\dots,i\widetilde x_n$ (here $i$ is the imaginary unit). Then $W(\mu)$ is transformed into the tangency polynomial of a family of confocal quadrics in the Euclidean space $\mathbb R^n(x_1,\dots,x_p,\widetilde{x}_{p+1},\dots,\widetilde{x}_n)$ . In accordance with the derivation of the analogous formula in [6], we can express the coefficient $F_m$ as follows:

$\begin{equation*} F_m=\frac{1}{8}\sum_{p=1}^n \sigma_p^m(\lambda_1,\dots,\lambda_n) A_p\dot{\lambda}_p^2, \end{equation*} \notag$

where

$\begin{equation*} A_p=\frac{(\lambda_{1}-\lambda_p)\dotsb (\lambda_{p-1}-\lambda_{p})(\lambda_{p+1}-\lambda_{p})\dotsb (\lambda_{n}-\lambda_{p})}{(a_{1}-\lambda_p)(a_{2}-\lambda_{p}) \dotsb(a_{n-1}-\lambda_{p})(a_{n}-\lambda_{p})}. \end{equation*} \notag$

Note that in the generalized elliptic coordinates the functions $F_m$ have simpler expressions. Since we investigate their properties from the standpoint of Hamiltonian mechanics, we can go over to a representation in terms of coordinates and momenta.

We have already noted that $F_0$ is the kinetic energy of the point mass (see Remark 7). Set $\widehat{A}_p=A_{p}^{-1}$ . By the definition of generalized momenta

$\begin{equation*} p_i=\frac{\partial H}{\partial\dot{\lambda}_i}=\frac{1}{4}A_{i}\dot{\lambda}_i, \end{equation*} \notag$

so that

$\dot{\lambda}_i=4\widehat{A}_ip_i$ . Therefore,

$\begin{equation} F_{m}=2\sum_{i=1}^n \sigma_i^m(\lambda_1,\dots,\lambda_n)\widehat{A}_ip_i^2. \end{equation} \tag{3.4}$

Step 2. As shown above, the functions $F_i$ are smooth on the phase space of the system under consideration. On the cotangent bundle of $\mathbb R^{p,q}$ with the variables $(\lambda_1,\dots,\lambda_n,p_1,\dots,p_n)$ we define $n$ Poisson brackets by

$\begin{equation*} \{f,g\}_i=\frac{\partial f}{\partial \lambda_i} \frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial \lambda_i} \quad\forall\, i=1,\dots,n, \quad\forall\, f,g\in C^{\infty}(T^*\mathbb R^{p,q}). \end{equation*} \notag$

Definition 5. The bracket $\{\,\cdot\,{,}\,\cdot\,\}_i$ is called the $i$ th partial Poisson bracket.

Remark 8. The canonical Poisson bracket $\{\,\cdot\,{,}\,\cdot\,\}$ on $T^*\mathbb R^{p,q}$ is the sum of all partial brackets:

$\begin{equation*} \{\,\cdot\,{,}\,\cdot\,\}=\sum_{i=1}^n\{\,\cdot\,{,}\,\cdot\,\}_i. \end{equation*} \notag$

As in the case of Euclidean space, the following important result holds.

Proposition 5. The functions $F_i$ commute with respect to all partial Poisson brackets.

Proof. We calculate the derivatives of two functions

$F_i$ and

$F_j$ that are involved in their

$k$ th partial bracket; for brevity we do not indicate the arguments of elementary symmetric polynomials, that is, we write

$\sigma$ in place of

$\sigma(\lambda_1,\dots,\lambda_n)$ :

$\begin{equation*} \begin{gathered} \, \begin{split} \frac{\partial F_i}{\partial \lambda_k} &=2\sum_{m\neq k}\sigma^i_{m,k}\widehat{A}_mp_m^2+2\sum_{m=1}^n\sigma^i_m\, \frac{\partial\widehat{A}_m}{\partial\lambda_k}p_m^2 \\ &=2\sigma_k^i\, \frac{\partial\widehat{A}_k}{\partial\lambda_k}p_k^2+2\sum_{m\neq k}\biggl(\sigma^{i-1}_{m,k}\widehat{A}_m+\sigma^i_m\, \frac{\partial\widehat{A}_m}{\partial\lambda_k}\biggr)p_m^2, \end{split} \\ \frac{\partial F_i}{\partial p_k}=4\sigma_k^i\widehat{A}_kp_k. \end{gathered} \end{equation*} \notag$

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

$\begin{equation*} 8\widehat{A}_kp_k\cdot\sum_{m\neq k}\biggl((\sigma_k^j\sigma_{m,k}^{i-1}-\sigma_k^i\sigma_{m,k}^{j-1})\widehat{A}_m +(\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j)\, \frac{\partial \widehat{A}_m}{\partial \lambda_k}\biggr)p_m^2. \end{equation*} \notag$

We show that for all $m$ we have the equality

$\begin{equation*} (\sigma_k^j\sigma_{m,k}^{i-1}-\sigma_k^i\sigma_{m,k}^{j-1})\widehat{A}_m +(\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j)\, \frac{\partial \widehat{A}_m}{\partial \lambda_k}=0. \end{equation*} \notag$

For the proof we write 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 equality from the first and divide by

$\lambda_m-\lambda_k$ . This yields

$\sigma_{m,k}^{i-1}=\dfrac{\sigma_{k}^i-\sigma_{m}^i}{\lambda_m-\lambda_k}$ . In a similar way

$\sigma_{m,k}^{j-1}=\dfrac{\sigma_{k}^j-\sigma_{m}^j}{\lambda_m-\lambda_k}$ . Therefore,

$\begin{equation*} \begin{aligned} \, &(\sigma_k^j\sigma_{m,k}^{i-1}-\sigma_k^i\sigma_{m,k}^{j-1})\widehat{A}_m +(\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j)\, \frac{\partial \widehat{A}_m}{\partial \lambda_k} \\ &\qquad=\frac{\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j}{\lambda_k-\lambda_m}\widehat{A}_m +(\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j)\, \frac{\partial \widehat{A}_m}{\partial \lambda_k} \\ &\qquad=(\sigma_k^j\sigma_m^i-\sigma_k^i\sigma_m^j) \biggl(\frac{\widehat{A}_m}{\lambda_k-\lambda_m}+\frac{\partial \widehat{A}_m}{\partial \lambda_k}\biggr). \end{aligned} \end{equation*} \notag$

The equality $\dfrac{\widehat{A}_m}{\lambda_k-\lambda_m}+\dfrac{\partial \widehat{A}_m}{\partial \lambda_k}=0$ can be verified by simple calculations.

The proof is complete.

We will see below that Proposition 5 is crucial for the proof of Theorem 2.

By the above remark, as a consequence, we obtain the following remarkable fact.

Corollary 1. The functions $F_i$ commute with respect to the standard Poisson bracket.

The system $F_0,\dots,F_{n-1}$ also turns out to be functionally independent. This is ensured by the following statement.

Proposition 6. The functions $F_i$ are functionally independent.

Proof. We show that the Jacobian

$\begin{equation*} \frac{D(F_0,\dots,F_{n-1})}{D(p_1,\dots,p_n)} \end{equation*} \notag$

is distinct from zero almost everywhere. We have

$\begin{equation*} \frac{\partial F_i}{\partial p_j}=4\sigma_j^i\widehat{A}_jp_j. \end{equation*} \notag$

Therefore,

$\begin{equation*} \frac{D(F_0,\dots,F_{n-1})}{D(p_1,\dots,p_n)}=4^n \begin{vmatrix} \sigma^0_1 & \dots & \sigma^0_n\\ \vdots & \ddots & \vdots\\ \sigma^{n-1}_1 & \dots & \sigma^{n-1}_n\\ \end{vmatrix}\prod_{j=1}^n\widehat{A}_jp_j. \end{equation*} \notag$

Note that the product after the determinant is distinct from zero almost everywhere. We show that the determinant itself has this property too. With the $j$ th column of the matrix under the sign of determinant we associate the polynomial

$\begin{equation*} f_j(z)=\sum_{i=0}^{n-1}z^{n-1-i}(-1)^i\sigma_j^i. \end{equation*} \notag$

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

$j$

$\begin{equation*} f_j(z)=(z-\lambda_1)\dotsb(z-\lambda_{j-1})(z-\lambda_{j+1})\dotsb(z-\lambda_{n}). \end{equation*} \notag$

Assume that $\lambda_1,\dots,\lambda_n$ are pairwise distinct; then $f_j(\lambda_i)\neq0$ if and only if $i\neq j$ . Hence it is straightforward 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 $i\neq j$ define a nullset in the phase space, $F_0,\dots,F_{n-1}$ are functionally independent.

The proof is complete.

Hence the problem of the inertial motion of a point mass in $\mathbb R^{p,q}$ is completely integrable and its first integrals $F_0,\dots,F_{n-1}$ describe fully the tangency of a trajectory to the family of confocal quadrics.

That the problem of the motion of a point mass in $\mathbb R^{p,q}$ is integrable is well known, but it is important for us that the above system of functionally independent commuting first integrals $F_0,\dots,F_{n-1}$ has relation to tangency with confocal quadrics.

Now we find the number of confocal quadrics tangent to a line in $\mathbb R^{p,q}$ . To do this we prove an auxiliary result on separation of variables.

Proposition 7 (separation of variables). On a joint level set $f_0,\dots,f_{n-1}$ of the first integrals $F_0,\dots,F_{n-1}$ the equations of motion take the following form:

$\begin{equation} \dot{\lambda}_k=\pm\frac{2\sqrt{2}}{\prod_{i\neq k}(\lambda_k-\lambda_i)} \sqrt{-\prod_{i=1}^n(\lambda_k-a_i)\biggl(\sum_{k=0}^{n-1}\lambda_k^{n-1-k}(-1)^{k}f_k\biggr)}. \end{equation} \tag{3.5}$

Proof. Consider the tangency polynomial

$\begin{equation*} W(\mu)=2\sum_{k=0}^{n-1}\mu^{n-1-k}(-1)^{n-1-k}f_k \end{equation*} \notag$

on a joint level set

$(f_0,\dots,f_{n-1})$ . Recall that

$W(\mu_0)=0$ if and only if the trajectory is tangent to the quadric with parameter

$\mu_0$ . By Vieta’s theorem, using (3.4) we obtain

$\begin{equation*} W(\mu)=4\sum_{k=1}^n(\lambda_1-\mu)\dotsb(\lambda_{k-1}-\mu)(\lambda_{k+1}-\mu)\dotsb(\lambda_n-\mu)\widehat{A}_kp_k^2. \end{equation*} \notag$

Hence for each

$k=1,\dots,n$ we have

$\begin{equation} W(\lambda_k)=4\prod_{i=1}^n(a_i-\lambda_k)p_k^2. \end{equation} \tag{3.6}$

Using the connection between generalized momenta and velocities, from (3.6) we obtain

$\begin{equation*} \dot{\lambda}_k=4\widehat{A}_jp_j=\pm\frac{2}{\prod_{i\neq k}(\lambda_k-\lambda_i)} \sqrt{\prod_{i=1}^n(a_i-\lambda_k)W(\lambda_k)}. \end{equation*} \notag$

The proof is complete.

Now using Proposition 7 we establish the following fundamental fact.

Proposition 8. Let $f_0,\dots,f_{n-1}$ be a joint level set of the integrals $F_0,\dots,F_{n-1}$ , and assume that $f_0\neq0$ . Then the polynomial

$\begin{equation*} W(\mu)=2\sum_{k=0}^{n-1}\mu^{n-1-k}(-1)^{n-k-1}f_k \end{equation*} \notag$

has

$n-1$ or

$n-3$ real zeros counting multiplicities.

Proof. We prove the proposition in the generic case: we assume that all zeros of the tangency polynomial

$W(\mu)$ are different. By Proposition 1 there are at least

$n-2$ confocal quadrics (counting multiplicities) through each point in

$\mathbb R^{p,q}$ . Moreover, each interval

$[a_i,a_{i+1}]$ or

$[a_{j+1},a_j]$ , where

$i=1,\dots,p-1$ and

$j=1,\dots,q-1$ , contains the parameter of at least one of these quadrics. In other words, at least one generalized elliptic coordinate of this point ‘lives’ on each of the intervals. Hence, generically, the polynomial

$\begin{equation*} V(\mu)=-\prod_{i=1}^n(\mu-a_i)\biggl(\sum_{k=0}^{n-1}\mu^{n-1-k}(-1)^{k}f_k\biggr) \end{equation*} \notag$

takes positive values on some subintervals of these intervals.

Since $f_0\neq0$ , the degree of the tangency polynomial is the greatest possible, that is, $n-1$ . Let $N$ be the number of zeros of $W(\mu)$ on the interval $(a_1,a_p)$ . By the above argument this interval contains at least $p-1$ distinct subintervals on which the tangency polynomial is positive; moreover, their endpoints are distinct and are zeros of $V(\mu)$ on the interval $[a_1,a_p]$ . Hence $2(p-1)\leqslant p+N$ . Therefore, $N\geqslant p-2$ . In other words, $W(\mu)$ has at least $p-2$ zeros on $(a_1,a_p)$ .

In a similar way, the interval $(a_{p+1},a_n)$ contains at least $q-2$ zeros of $W(\mu)$ . Thus, $W(\mu)$ has at least $n-4$ real zeros. Since its degree is $n-1$ , the number of real zeros of $W(\mu)$ is either $n-3$ or $n-1$ .

The proof is complete.

Since the functions $F_i$ (equal to the coefficients of the tangency polynomial) commute pairwise with respect to the partial Poisson brackets, the zeros of the tangency polynomial also have this property.

Note that this result is in fact an analogue of a well-known theorem of Chasles [3] and was established earlier by Khesin and Tabachnikov [7].

Now using the properties of the tangency polynomial and its coefficients established at this step we prove the assertions of Theorem 2.

Step 3. Let $Q_1,\dots, Q_k$ be nondegenerate confocal quadrics in $\mathbb R^{p,q}$ . Then on their intersection $Q=\bigcap_{i=1}^kQ_i$ (assuming that $Q\neq\varnothing$ ) precisely $k$ generalized elliptic coordinates $\lambda_{i_1},\dots,\lambda_{i_k}$ are fixed, and no restrictions are imposed on the other coordinates.

On $Q$ we consider the pseudo-Riemannian metric induced from the ambient space $\mathbb R^{p,q}$ . It defines a geodesic flow on the cotangent bundle of $Q$ . As the metric is induced from the ambient space, the energy of the resulting system is equal to the energy of a point mass moving freely in $\mathbb R^{p,q}$ .

Note that by Proposition 5 the functions $F_i$ are first integrals of the geodesic flow on $Q$ . Indeed, $Q$ is defined by the relations $\lambda_{i_1}=c_1$ , $\dots$ , $\lambda_{i_k}=c_k$ , where the $c_i$ are some fixed numbers; the momenta corresponding to these coordinates must vanish. Now, the canonical Poisson bracket $\{\,\cdot\,{,}\,\cdot\,\}_Q$ on $Q$ is the sum of the partial Poisson brackets with respect to free elliptic coordinates, that is,

$\begin{equation*} \{\,\cdot\,{,}\,\cdot\,\}_Q=\sum_{j\neq i_1,\dots,i_k}\{\,\cdot\,{,}\,\cdot\,\}_j. \end{equation*} \notag$

Since the

$F_i$ commute with respect to all partial brackets, they also commute with respect to

$\{\,\cdot\,{,}\,\cdot\,\}_Q$

Hence the functions $F_i$ are quadratic first integrals of the geodesic flow on $Q$ . Furthermore, using the arguments from the proof of Proposition 6 it is easy to see that $F_0,\dots,F_{n-k-1}$ are functionally independent on $T^{*}Q$ . This proves assertion (1) of Theorem 2.

To prove part (2) it remains to use Proposition 8.

The proof of Theorem 2 is complete.

The following interesting question arises: when is the number of tangent quadrics equal to $n-1-k$ ? We answer it for the part of $Q$ occurring in $D(A,B)$ . Moreover, we assume that at least one unique quadric participates in the intersection $Q$ .

Definition 6. We say that an intersection of nondegenerate confocal quadrics $Q_1,\dots,Q_k$ is unique if at least one $Q_i$ is unique.

Recall that a quadric is unique if its parameter lies in the union of the intervals $\mathrm{I}=(-\infty,-a_{n})$ , $\mathrm{II}=(-a_{p+1},a_1)$ and $\mathrm{III}=(a_p,+\infty)$ . Note that by Proposition 1 two unique quadrics have a nontrivial intersection if and only if their parameters lie on the same interval $\mathrm{I}$ , $\mathrm{II}$ or $\mathrm{III}$ . Using this observation we can divide the unique intersections of quadrics into three classes.

Definition 7. We call the intersection of confocal quadrics $Q_1,\dots,Q_k$ a unique intersection of type $\mathrm{I}$ ( $\mathrm{II}$ or $\mathrm{III}$ ) if the parameter of some $Q_i$ lies on the interval $\mathrm{I}$ (respectively, $\mathrm{II}$ or $\mathrm{III}$ ).

The unique quadrics define three subsets of $\mathbb R^{p,q}$ :

• $D_\mathrm{I}=\{x\in\mathbb R^{p,q}\mid x\text{ lies on a unique quadric with parameter } \lambda\in \mathrm{I}\}$ ;
• $D_{\mathrm{II}}=\{x\in\mathbb R^{p,q}\mid x\text{ lies on a unique quadric with parameter } \lambda\in \mathrm{II}\}$ ;
• $D_{\mathrm{III}}=\{x\in\mathbb R^{p,q}\mid x\text{ lies on a unique quadric with parameter } \lambda\in \mathrm{III}\}$ .

Proposition 9. Let $Q_1,\dots,Q_k$ be distinct nondegenerate confocal quadrics in $\mathbb R^{p,q}$ , and let $Q=\bigcap_{i=1}^kQ_i\neq\varnothing$ .

1. Let $Q$ be a unique intersection of type $\mathrm{I}$ or $\mathrm{II}$ . Then, along with $Q_1,\dots,Q_k$ , the tangent lines at all points of a timelike geodesic on $Q$ are also tangent to $n-k-1$ other quadrics confocal with them, which are the same at all points on this geodesic.

2. Let $Q$ be a unique intersection of type $\mathrm{II}$ or $\mathrm{III}$ . Then, along with $Q_1,\dots,Q_k$ , the tangent lines at all points of a spacelike geodesic on $Q$ are also tangent to $n-k-1$ other quadrics confocal with them, which are the same for all points on this geodesic.

Proof. We use the same method as in the proof of Proposition 8. Consider an inertial motion of a point mass in the set

$D_{\mathrm{II}}$ . By definition each point in

$D_{\mathrm{II}}$ lies on at least one quadric in the family (2.1) with parameter on the interval

$(a_{p+1},a_1)$ . By Proposition 7 this interval contains a subinterval on which the polynomial

$\begin{equation*} V(\mu)=-\prod_{i=1}^n(\mu-a_i) \biggl(\sum_{k=0}^{n-1}\mu^{n-1-k}(-1)^{k}f_k\biggr) \end{equation*} \notag$

is positive.

In the generic case the intervals $(a_i,a_{i+1})$ and $(a_{j+1},a_{j})$ , where $i=1,\dots,p-1$ and $j=p+1,\dots,n-1$ , have the analogous property. Using the arguments from the proof of Proposition 8 we conclude that the tangency polynomial has at least $n-2$ real zeros on $(a_{n},a_1)$ , counting multiplicities. It remains to observe that for $f_0\neq0$ the tangency polynomial has degree $n-1$ , so that a timelike or spacelike straight line passing through the set $D_{\mathrm{II}}$ is tangent to precisely $n-1$ quadric in the family (2.1). Hence any nonisotropic geodesic on a unique intersection of type $\mathrm{II}$ of confocal quadrics $Q_1,\dots,Q_k$ is also tangent to some other $n-k-1$ confocal quadrics. Thus for unique intersections of type $\mathrm{II}$ everything is proved.

When the motion proceeds in $D_\mathrm{I}$ and $f_0<0$ (so that the trajectory is timelike), the polynomial $V(\mu)$ takes negative values for $\mu$ close to $-\infty$ . However, one of the elliptic coordinates in this domain ‘lives’ on $(-\infty,a_{p+1})$ . Hence $V(\mu)$ must have at least one zero on this interval. From the estimates in Proposition 8 for the number of zeros we conclude that the tangency polynomial has only real zeros. However, this reasoning does not hold for spacelike geodesics (that is, for $f_0>0$ ). In the domain $D_{\mathrm{III}}$ the situation is the opposite one: spacelike trajectories are tangent to a full set of quadrics, while timelike ones do not necessarily have this property.

The proof is complete.

§ 4. Generalized Jacobi–Chasles theorem in constant curvature spaces

We see that the generalized Jacobi–Chasles theorem holds in Euclidean and pseudo-Euclidean spaces alike. Based on this result, Fomenko conjectured that there must exist an analogue of this theorem for spaces of constant sectional curvature, namely, spheres, projective spaces and Lobachevsky spaces equipped with the standard metrics. This conjecture turns out to be true indeed. Moreover, this is a consequence of the generalized Jacobi–Chasles theorem for Euclidean spaces (see Theorem 1) and Theorem 2.

In the Euclidean space $\mathbb R^{n+1}(x_0,\dots,x_n)$ consider a family of confocal quadrics given by

$\begin{equation} \frac{x_0^2}{a_0-\lambda}+\frac{x_1^2}{a_1-\lambda}+\dots+\frac{x_{n}^2}{a_{n}-\lambda}=1. \end{equation} \tag{4.1}$

We assume that all the

$a_i$ are positive and arranged in increasing order:

$0<a_1<\dots<a_n$ . For an arbitrary

$t\in[0,1]$ consider the new family of confocal quadrics

$\begin{equation} \frac{x_0^2}{ta_{0}-\lambda}+ \frac{x_1^2}{ta_1-\lambda}+\dots+\frac{x_{n}^2}{ta_n-\lambda}=1. \end{equation} \tag{4.2}$

Note that for

$t=1$ this is the original family, while for

$t=0$ (4.2) becomes the equation of concentric spheres with centre at the origin. It is quite clear that these spheres are limits of confocal ellipsoids, but what happens to other confocal quadrics in approaching this degeneracy?

Fix a point $P=(x_0,\dots,x_n)$ . Let $\lambda_0(t)\leqslant\dots\leqslant\lambda_{n}(t)$ be the elliptic coordinates of this point in the family (4.2). Since ellipsoids in (4.1) retract to spheres, asymptotically, $\lambda_0(t)=-R^2+o(1)$ for some $R$ as $t\to0$ . For $i=1,\dots,n$ we have $ta_{i-1}\leqslant\lambda_i(t)\leqslant ta_i$ , and the interval $[ta_{i-1},ta_{i}]$ contracts to zero, so we have the asymptotic formula $\lambda_i(t)=\mu_it+o(t)$ as $t\to0$ for some $\mu_i$ , $i=1,\dots,n$ .

Now we find relations for the $\mu_i$ . To do this we write down the formulae connecting the elliptic and Cartesian coordinates:

$\begin{equation} \begin{aligned} \, \notag x_i^2&=\frac{\prod_{j=0}^n (ta_i-\lambda_j(t))}{\prod_{j\neq i} (ta_i-ta_j)} \\ &=\frac{(ta_{i}+R^2+o(t))\prod_{j=1}^n (ta_{i}-\mu_jt+o(t))}{\prod_{j\neq i} (ta_i-ta_j)} =R^2\frac{\prod_{j=1}^n (a_{i}-\mu_j)}{\prod_{j\neq i} (a_i-a_j)}. \end{aligned} \end{equation} \tag{4.3}$

Proposition 10. The quantities $\mu_i$ are the parameters of cones in the one-parameter family

$\begin{equation} \frac{x_0^2}{a_0-\mu}+\dots+\frac{x_n^2}{a_n-\mu}=0 \end{equation} \tag{4.4}$

that pass through the point in question.

Proof. For

$j=1,\dots,n$ let us calculate

$\sum_{i=0}^n\dfrac{x_i^2}{a_i-\mu_j}$ . To do this we substitute in formulae (4.3):

$\begin{equation*} \begin{aligned} \, \sum_{i=0}^n\frac{x_i^2}{a_i-\mu_j} &=R^2\sum_{i=0}^n\frac{\prod_{k\neq j} (a_{i}-\mu_k)}{\prod_{k\neq i} (a_i-a_k)} \\ &=R^2\sum_{i=0}^n \frac{\sum_{k=0}^{n-1}a_i^k(-1)^{n-1-k}\sigma^{n-1-k}_j(\mu_1,\dots,\mu_n)}{\prod_{k\neq i} (a_i-a_k)}. \end{aligned} \end{equation*} \notag$

Changing the order of summation in this expression, we use summations formulae (for instance, see [3] or [6]), which show that this expression is identically equal to zero.

The proof is complete.

We see that the limit of the family (4.2) as $t\to0$ consists of a family of concentric spheres and the one-parameter family of cones (4.4). Furthermore, there are precisely $n$ cones of the form (4.4) (counting multiplicities) through each point in $\mathbb R^{n+1}$ . Thus, the following definition of a family of confocal quadrics on a sphere looks reasonable.

Definition 8. By a family of confocal quadrics on the sphere $S_R^n$ defined by $x_0^2+\dots+x_n^2=R^2$ we mean the intersections of the sphere with the one-parameter family of cones (4.4).

A family of confocal quadrics on a sphere and its stereographic projection are shown in Figure 3. Note that a family of confocal quadrics on a sphere (a one-parameter family of cones) arises in [14].

Figure 3.(a) A family of confocal quadrics on a sphere. Dotted lines correspond to degenerate quadrics and bullets to foci of the family. (b) The stereographic projection of confocal quadrics onto the plane.

Remark 9. The stereographic projection takes quadrics on a sphere to plane curves of order $4$ . In fact, the stereographic projection of $S^2_R$ onto the plane with Cartesian coordinates $(u,v)$ is expressed by the formulae

$\begin{equation*} x=\frac{2R^2u}{u^2+v^2+R^2},\qquad y=\frac{2R^2v}{u^2+v^2+R^2}\qquad\text{and}\qquad z=R\frac{u^2+v^2-R^2}{u^2+v^2+R^2}. \end{equation*} \notag$

Substituting them into the equation of a cone in the one-parameter family (2.5), we obtain a curve of order

$4$ .

We have thus defined confocal quadrics on spheres. Now we define confocal quadrics in the Lobachevsky space. Recall that the Lobachevsky space $L_R^n$ is the connected component of the pseudo-sphere $x_0^2+ \dots+x_{n-1}^2-x_n^2=-R^2$ that lies in the half-space $x_n>0$ .

We proceed in a similar way. In $\mathbb R^{n,1}(x_0,\dots,x_n)$ consider the family of confocal quadrics

$\begin{equation*} \frac{x_0^2}{a_0-\lambda}+\frac{x_1^2}{a_1-\lambda}+\dots+\frac{x_{n-1}^2}{a_{n-1}-\lambda}+\frac{x_{n-1}^2}{b+\lambda}=1. \end{equation*} \notag$

As before, we assume that

$0<a_1<\dots<a_{n-1}$ and

$b>0$ . Again, for an arbitrary

$t\in[0,1]$ we introduce the family of confocal quadrics

$\begin{equation} \frac{x_0^2}{ta_0-\lambda}+\frac{tx_1^2}{a_1-\lambda}+\dots+\frac{x_{n-1}^2}{ta_{n-1}-\lambda}+\frac{x_{n-1}^2}{tb+\lambda}=1 \end{equation} \tag{4.5}$

and look at its limit as

$t\to0$ . We obtain concentric spheres and pseudo-spheres (in the sense of

$\mathbb R^{n,1}$ ) with centre at the origin and the one-parameter family of cones

$\begin{equation} \frac{x_0^2}{a_0-\mu}+\frac{x_1^2}{a_1-\mu}+\dots+\frac{x_{n-1}^2}{a_{n-1}-\mu}+\frac{x_{n-1}^2}{b+\mu}=0. \end{equation} \tag{4.6}$

Definition 9. By a family of confocal quadrics in the Lobachevsky space $L_R^n$ we mean the intersections of $L_R^n$ with the one-parameter family of cones (4.6).

As in the spherical case, there are precisely $n$ confocal quadrics through each point on the pseudosphere.

A family of confocal quadrics on the Lobachevsky plane $L^2_R$ and its standard stereographic projection onto the Poincaré model (plane disc) are shown in Figure 4. As in the spherical case, the stereographic projection takes confocal quadrics to plane curves of order $4$ .

Figure 4.(a) A family of confocal quadrics on the Lobachevsky plane. (b) The stereographic projection of confocal quadrics onto the plane (Poincaré model). Dotted lines indicate degenerate quadrics. The bullets are at the foci of the family.

We equip the spheres $S^n_R$ and Lobachevsky spaces $L^n_R$ with the standard metrics of constant sectional curvature induced by the standard embedding in the Euclidean space $\mathbb R^{n+1}$ or the $(n+1)$ -dimensional pseudo-Euclidean space $\mathbb R^{n,1}$ of index $1$ . We denote these manifold by the same symbol $M^n_R$ and assume that the dimension $n$ is at least $3$ .

Theorem 3 (Belozerov and Fomenko). Let $Q_1,\dots,Q_k$ be nondegenerate confocal quadrics of different types in the space $M^n_R$ , and let $Q=\bigcap_{i=1}^k Q_i$ . Then

(1) the geodesic flow on $Q$ is quadratically integrable;
(2) along with $Q_1,\dots,Q_k$ , the tangent geodesic curves in $M^n_R$ to all points on a geodesic curve on $Q$ are tangent to other $n-k-1$ quadrics confocal with $Q_1,\dots,Q_k$ , the same for all points on this geodesic.

Proof. The manifolds

$M^n_R$ and confocal quadrics on them are obtained by a limit transition from families of confocal quadrics in the Euclidean space

$\mathbb R^n$ and the pseudo-Euclidean space

$\mathbb R^{n,1}$ , in which the generalized Jacobi-Chasles theorem is valid (see Theorems 1 and 2). Hence a similar theorem must also hold in

$M^n_R$ . However, first, in the statement of Theorem 2 the number of tangent quadrics is either

$n-3-k$ or

$n-1-k$ . Second, in both theorems tangent curves ‘live’ in the ambient space. It remains to fill in these gaps.

First of all, we show that an arbitrary geodesic curve on the Lobachevsky plane is tangent to $n-1$ quadrics.

Let ${P\in L_R^n}$ . After taking the limit with respect to $t$ in (4.6) the equation of the family of confocal quadrics takes the form

$\begin{equation*} x_0^2+\dots+x_{n-1}^2-x_n^2=-\mu. \end{equation*} \notag$

Hence one elliptic coordinate of the point

$P$ becomes equal to

${R^2>0}$ . Thus, for sufficiently small

$t$ this point has an elliptic coordinate larger than

$a_n$ because it must approach

$R^2$ as

$t\to0$ . So for all small

$t$ the point

$P$ lies in the domain

$D_{\mathrm{III}}$ defined in terms of the corresponding family of confocal quadrics.

Now we observe that each geodesic curve on $L^n_R$ is spacelike. In fact, the metric on $L^n_R$ is positive definite, so each tangent vector to the Lobachevsky space is spacelike. By Proposition 9 a spacelike geodesic curve on a unique intersection of type $\mathrm{III}$ of nondegenerate confocal quadrics $Q_1,\dots,Q_k$ is also tangent to some other $n-k-1$ confocal quadrics. Thus, we have established the first lacking result. Now we turn to the second.

Note that if a tangent line to $S_R^n$ is tangent to a cone $Q$ with the same centre as the sphere, then the geodesic curve on the sphere in the direction of this line is tangent to the curve $Q\cap S_R^n$ (Figure 5). This can be proved on the basis of the fact that if a straight line is tangent to a cone, then the plane through this line and the centre of the cone is also tangent to the cone. A similar result holds in Lobachevsky spaces.

Figure 5.An illustration of the fact that a central cross-section of the standard 2-sphere is tangent to a cone with centre at the origin (a) if and only if the corresponding great circle on the sphere is tangent to the curve equal to the trace of the cone (b).

Thus the proof of Theorem 3 is complete.

Remark 10. In Theorem 3 we considered a metric on the intersection of several confocal quadrics that is induced from the ambient space $M^n_R$ . The geodesic flow on the intersection was defined in terms of just this metric.

Note that antipodal points on $S^n_R$ have equal elliptic coordinates. Using this observation, we can transfer Theorem 3 to the projective spaces $\mathbb R P^n$ equipped with the standard metric of constant sectional curvature. Thus we arrive at the following corollary.

Corollary 2. The result of Theorem 3 is also valid if in place of $M^n_R$ one considers the projective space $\mathbb R P^n$ equipped with the standard metric of constant sectional curvature.

§ 5. Quadratically integrable billiards on two-dimensional Riemannian manifolds

In this section we show that in dimension $2$ the Jacobi–Chasles theorem only holds for spaces of constant Gaussian curvature. However, first we make a few observations.

In Theorem 2 the first integrals $F_i$ are sums of the squares of momenta with coefficients depending on the generalized elliptic coordinates. Hence the billiard system in a domain in $\mathbb R^{p,q}$ ( $S^n_R$ or $L^n_R$ ) which is bounded by a finite number of confocal quadrics is integrable, and the functions $F_i$ are first integrals for it (which do not depend on the parameters of boundary quadrics). Moreover, the following theorem holds.

Theorem 4 (Belozerov and Fomenko). Let $M$ be one of the following spaces: $S^n_R$ , $\mathbb R P^n$ , $L^n_R$ , Euclidean $\mathbb R^{n}$ , $\mathbb R^{p,q}$ . Also let $Q_1,\dots,Q_k$ be nondegenerate confocal quadrics of different types in $M$ , let $Q=\bigcap_{i=1}^k Q_i$ , and let $D$ be a domain in $Q$ bounded by a finite number of quadrics confocal with $Q_i$ that has polyhedral angles $\pi/2$ at corners of the boundary. Then

(1) the geodesic billiard on $D$ is quadratically integrable in the piecewise sense and its first integrals are independent of the parameters of boundary quadrics;
(2) along with $Q_1,\dots,Q_k$ , the tangent geodesics in $M$ to all points of a billiard trajectory in $D$ are tangent to $n-k-1$ other quadrics confocal with $Q_1,\dots,Q_k$ , which are the same for all points on this trajectory.

In what other spaces $M$ , apart from the above list, is the assertion of Theorem 4 also valid? We answer this question for $n=2$ . Note that by letting the parameters $a_i$ approach one another, we let the elliptic coordinate system degenerate. One limiting degenerate case is a semi-geodesic coordinate system, which can on the other hand be defined on any manifold. Our investigations are based on this observation.

Consider a two-dimensional Riemannian manifold $(M^2,g)$ . The metric $g$ defines the geodesic distance $\rho_g$ between points in $M^2$ .

Definition 10. The geodesic circle (disc) on $M^2$ with centre $P$ and radius $\varepsilon$ is the set

$\begin{equation*} \begin{gathered} \, S_\varepsilon(P)=\{Q\in M^2|\rho_g(P,Q)=\varepsilon\} \\ (\text{respectively},\ K_\varepsilon(P)=\{Q\in M^2|\rho_g(P,Q)\leqslant\varepsilon\}). \end{gathered} \end{equation*} \notag$

Let $\varepsilon$ be sufficiently small so that the exponential map at the point $P$ is injective and smooth in a small neighbourhood of the geodesic disc $K_\varepsilon(P)$ . Then ${\partial K_{\varepsilon}P=S_\varepsilon(P)}$ . Consider the classical billiard inside the disc $K_\varepsilon(P)$ , that is, the following dynamical system: a unit point mass moves along geodesics in $K_\varepsilon(P)$ at a velocity constant in modulus and is reflected absolutely elastically off the boundary $S_\varepsilon(P)$ .

Definition 11. We say that a Riemannian manifold $(M^2,g)$ is quadratically integrable in the sense of Jacobi and Chasles if in a small neighbourhood of each point $P\in M^2$ the geodesic flow of the metric $g$ has an additional first integral $F_P$ which is quadratic in momenta and is a first integral of the classical billiard in each sufficiently small geodesic disc with centre $P$ .

By Theorem 4 the sphere $S^2_R$ , the Lobachevsky space $L^2_R$ and the Euclidean plane with standard metrics on them are quadratically integrable in the sense of Jacobi and Chasles. It turns out that these examples exhaust (in a certain sense) the list of quadratically integrable surfaces in the sense of Jacobi and Chasles. More precisely, the following result is true.

Theorem 5 (Belozerov and Fomenko). A two-dimensional Riemannian manifold $(M^2,g)$ is quadratically integrable in the sense of Jacobi and Chasles if and only if it is locally isometric to a constant curvature space.

Proof. One part of the assertion is obvious. We prove the second. Let

$P\in M^2$ . Consider the semigeodesic coordinates

$(r,\varphi)$ in a neighbourhood of

$P$ . In them the metric tensor has the following expression:

$\begin{equation*} g=dr^2+B(r,\varphi)\,d\varphi^2, \end{equation*} \notag$

where

$B$ is a positive function in a punctured neighbourhood of

$P$ . Hence the energy of the point mass has the following expression in terms of coordinates and momenta:

$\begin{equation*} H=\frac{1}{2}(p_r^2+\widetilde{B}p_\varphi^2). \end{equation*} \notag$

Here

$\widetilde{B}=1/B$ .

Let $F_P=F$ be a quadratic first integral, additional to $H$ , of the billiard system in a small neighbourhood of $P$ . As it is preserved by reflections, the coefficient of $p_r\cdot p_\varphi$ must be zero. In other words,

$\begin{equation*} F=\frac{1}{2}(A_1(r,\varphi)p_r^2+A_2(r,\varphi)p_\varphi^2). \end{equation*} \notag$

Since $F$ is a first integral of the billiard, it is also a first integral of the geodesic flow. Hence the Poisson bracket of $H$ and $F$ must vanish:

$\begin{equation*} \begin{aligned} \, 0&=\{H,F\}=\frac{\partial H}{\partial p_r}\frac{\partial F}{\partial r}+\frac{\partial H}{\partial p_\varphi}\frac{\partial H}{\partial \varphi}-\frac{\partial F}{\partial p_r}\frac{\partial H}{\partial r}-\frac{\partial F}{\partial p_\varphi}\frac{\partial H}{\partial \varphi} \\ &=\frac{p_r}{2}\biggl(\frac{\partial A_1}{\partial r}\, p_r^2+\frac{\partial A_2}{\partial r}\, p_\varphi^2\biggr)+ \frac{\widetilde{B}p_\varphi}{2}\biggl(\frac{\partial A_1}{\partial \varphi}\, p_r^2+\frac{\partial A_2}{\partial \varphi}\, p_\varphi^2\biggr) \\ &\qquad -\frac{A_1p_r}{2}\frac{\partial \widetilde{B}}{\partial r}\, p_\varphi^2-\frac{A_2p_\varphi}{2}\frac{\partial\widetilde{B}}{\partial \varphi}\, p_\varphi^2. \end{aligned} \end{equation*} \notag$

This equality is equivalent to the following system of partial differential equations:

$\begin{equation*} \frac{\partial A_1}{\partial r}=0 , \qquad \frac{\partial A_2}{\partial r}=A_1\,\frac{\partial \widetilde{B}}{\partial r} , \qquad \widetilde{B}\,\frac{\partial A_1}{\partial \varphi}=0 , \qquad \widetilde{B}\,\frac{\partial A_2}{\partial \varphi}= A_2\,\frac{\partial\widetilde{B}}{\partial \varphi}. \end{equation*} \notag$

From the first and second equations we conclude that

$A_1$ is a constant. Subtracting the function

$A_1 H$ from

$F$ we obtain a quadratic first integral independent of

$H$ and with coefficient of

$p_r^2$ equal to zero. Hence we can assume that

$A_1=0$ . Taking this into account, from the second equation we see that

$A_2$ is independent of

$r$ . It remains to use the fourth equation. Since

$A_2$ is independent of

$r$ , the logarithmic derivative of

$B$ with respect to

$\varphi$ is also independent of

$r$ . Therefore,

$\log \widetilde{B}=u(\varphi)+v(r)$ for some smooth functions

$u$ and

$v$ . Thus,

$B=1/\widetilde{B}=f^2(r)h^2(\varphi)$ for some smooth

$f$ and

$g$ , so that

$\begin{equation*} g=dr^2+f^2(r)h^2(\varphi)\,d\varphi^2. \end{equation*} \notag$

We move

$h$ under the sign of differential, so that from

$\varphi$ we proceed to the variable

$\displaystyle\varphi' =\int h(\varphi)\,d\varphi$ . In the variables

$(r,\varphi')$ the metric tensor has the form

$\begin{equation} g=dr^2+f^2(r)\,d\varphi'^2. \end{equation} \tag{5.1}$

Hence

$M^2$ has a constant Gaussian curvature in small geodesic discs with centre

$P$ . Since the point

$P$ can be arbitrary, we conclude that the Gaussian curvature of

$M$ is constant.

It is well known (for instance, see [15]) that on a surface with metric tensor of the form (5.1) the Gaussian curvature can be calculated by the formula

$\begin{equation*} K=-\frac{f''(r)}{f(r)}. \end{equation*} \notag$

Since the Gaussian curvature of

$M^2$ is constant, we obtain a differential equation for

$f$ . Furthermore, we have the initial condition

$f(0)=0$ . Three cases are possible:

$K=0$ ,

$K=R^2$ or

$K=-R^2$ . In the first case

$f(r)=\alpha r$ , which means that

$M^2$ is locally isometric to the plane. In the second

$f(r)=\beta \cos(R\cdot r)$ , that is,

$M^2$ is locally isometric to the sphere of radius

$R$ . In the third case

$f(r)=\gamma \cosh(R\cdot r)$ and

$M^2$ is locally isometric to the Lobachevsky plane of radius

$R$ .

Now the proof of Theorem 5 is complete.



Bibliography

1.	C. G. J. Jacobi, “Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution”, J. Reine Angew. Math., 1839:19 (1839), 309–313
2.	C. G. J. Jacobi, Vorlesungen über Dynamik, Gesammelte Werke, Supplementband, 2. rev. Ausg., G. Reimer, Berlin, 1884, viii+300 pp.
3.	M. Chasles, “Sur les lignes géodésiques et les lignes de courbure des surfaces du second degré”, J. Math. Pures Appl., 11 (1846), 5–20
4.	V. I. Arnold, Mathematical methods of classical mechanics, 3rd ed., Nauka, Moscow, 1989, 472 pp. ; English transl. of 1st ed., Grad. Texts in Math., 60, Springer-Verlag, New York–Heidelberg, 1978, x+462 pp.
5.	G. V. Belozerov, “Integrability of a geodesic flow on the intersection of several confocal quadric”, Dokl. Math., 107:1 (2023), 1–3
6.	G. V. Belozerov, “Geodesic flow on an intersection of several confocal quadrics in $\mathbb{R}^n$ ”, Sb. Math., 214:7 (2023), 897–918
7.	B. Khesin and S. Tabachnikov, “Pseudo-Riemannian geodesics and billiards”, Adv. Math., 221:4 (2009), 1364–1396
8.	V. Dragović and M. Radnović, “Topological invariants for elliptical billiards and geodesics on ellipsoids in the Minkowski space”, J. Math. Sci. (N.Y.), 223:6 (2017), 686–694
9.	E. E. Karginova, “Liouville foliation of topological billiards in the Minkowski plane”, J. Math. Sci. (N.Y.), 259:5 (2021), 656–675
10.	E. E. Karginova, “Billiards bounded by arcs of confocal quadrics on the Minkowski plane”, Sb. Math., 211:1 (2020), 1–28
11.	V. Dragović and M. Radnović, “Bicentennial of the Great Poncelet Theorem (1813–2013): current advances”, Bull. Amer. Math. Soc. (N.S.), 51:3 (2014), 373–445
12.	V. V. Kozlov and D. V. Treshchev, Billiards. A genetic introduction to the dynamics of systems with impacts, Transl. Math. Monogr., 89, Amer. Math. Soc., Providence, RI, 1991, viii+171 pp.
13.	S. V. Bolotin, “Integrable billiards on surfaces of constant curvature”, Math. Notes, 51:2 (1992), 117–123
14.	P. M. Morse and H. Feshbach, Methods of theoretical physics, v. 1, 2, McGraw-Hill Book Co., Inc., New York–Toronto–London, 1953, xl+1978 pp.
15.	S. Kobayashi and K. Nomizu, Foundations of differential geometry, v. I, II, Intersci. Tracts Pure Appl. Math., 15, Interscience Publishers John Wiley & Sons, Inc., New York–London–Sydney, 1963, 1969, xi+329 pp., xv+470 pp.

Citation: G. V. Belozerov, A. T. Fomenko, “Generalized Jacobi–Chasles theorem in non-Euclidean spaces”, Sb. Math., 215:9 (2024), 1159–1181

Citation in format AMSBIB

\Bibitem{BelFom24}

\by G.~V.~Belozerov, A.~T.~Fomenko

\paper Generalized Jacobi--Chasles theorem in non-Euclidean spaces

\jour Sb. Math.

\yr 2024

\vol 215

\issue 9

\pages 1159--1181

\mathnet{http://mi.mathnet.ru/eng/sm10099}

\crossref{https://doi.org/10.4213/sm10099e}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4837038}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2024SbMat.215.1159B}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001375658800003}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85212525525}

Linking options:

https://www.mathnet.ru/eng/sm10099

https://doi.org/10.4213/sm10099e

https://www.mathnet.ru/eng/sm/v215/i9/p30

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Statistics & downloads:
Abstract page:	281
Russian version PDF:	5
English version PDF:	10
Russian version HTML:	12
English version HTML:	84
References:	22
First page:	10

Что такое QR-код?

Registration to the website

Logotypes

§ 1. Introduction

Acknowledgement

§ 2. Confocal quadrics and elliptic coordinates in the Minkowski space Rp,q {\mathbb R}^{p,q}

§ 3. Generalized Jacobi–Chasles theorem in pseudo-Euclidean spaces

§ 4. Generalized Jacobi–Chasles theorem in constant curvature spaces

§ 5. Quadratically integrable billiards on two-dimensional Riemannian manifolds

Bibliography

§ 2. Confocal quadrics and elliptic coordinates in the Minkowski space ${\mathbb R}^{p,q}$