##
**Spinning tops. A course on integrable systems.**
*(English)*
Zbl 0867.58034

Cambridge Studies in Advanced Mathematics. 51. Cambridge: Cambridge Univ. Press. 139 p. (1996).

Suppose a Hamiltonian dynamical system \(\mathcal H\) can be written in the Lax form \(\dot A = [A,B(A)]\) and suppose moreover that this Lax form can be rewritten
\[
\dot A_\lambda=[A_\lambda,B_\lambda(A_\lambda)]
\]
where the entries of \(A_\lambda\) and \(B_\lambda\) are Laurent polynomials in \(\lambda\). \(\lambda\) is called the spectral parameter. Consider the algebraic curve \(C\) defined by
\[
\text{det}(A_\lambda-\mu I_d)= 0.
\]
The coefficients of this equation are constants of motion for \(\mathcal H\) depending on the level values of integrals \(H_i\) of \(\mathcal H\).

Consider next the eigenvector \(V_i(\lambda,\mu)\) associated to an eigenvalue \(\mu\) of \(A_\lambda\) (with simple spectrum). One obtains a complex line bundle over \(C\) when \(\lambda\) varies. This line bundle is indexed by the level of the integrals \(H_i\). If \(T_C\) denotes the set of matrices \(A_\lambda\) for fixed levels, the mapping \(\phi_C:T_C\to \text{Pic}(C)\) is well defined. \(\text{Pic}(C)\) is the Picard group obtained from the divisors which are identified with complex line bundles and called the eigenvector mapping.

In the book under review the author considers specific systems, i.e., spinning tops and the periodic Toda lattice. She presents classical approaches which study their dynamics, but the emphasis is on in describing the properties of the spectral curve \(C\) and the eigenvector mapping of these systems. From these properties, information on the level sets of the original dynamical system can be obtained; the information is topological, algebraic, concerns linearization (in the Hamiltonian sense) of the solutions, regularity of the levels. In the examples under consideration, the level sets are described themselves by the intersection of algebraic surfaces. If the flows are complete (true in the compact case of spinning tops), they are Liouville tori. So, the book applies to specific systems an approach which brings together two disciplines: Riemann surface theory and Hamiltonian dynamics.

The introduction reminds in an intuitive fashion the basic definitions and properties of Hamiltonian dynamics, defines the eigenvector mapping, explains what can be expected from it, and presents the contents of the chapters of the book.

In chapter one, the basic equations of the three-dimensional rigid body dynamics with a fixed point are presented. They are written directly in the Lax form using the adjoint action of SO(3) on its Lie algebra. The variables represent the orientation of the top and the momentum (6 reals). Two Casimir functions (the norm of the vector characterizing the orientation which is constant, the momentum with respect to the vertical) provide two integrals. Then, four independent remaining variables belong to a phase space diffeomorphic to the symplectic manifold \(T(S^2)\) on which complete integrability occurs. The Hamiltonian \(H\) gives a third integral and a fourth one \(K\) whose existence and form depends on the repartition of mass in the top leads to several cases studied in more detail in the remaining chapters: In the Euler-Poinsot case (leading to the free rigid body situation; cf. chapter four), the center of gravity \(G\) coincides with the fixed point of the top; in the Lagrange case (also called the symmetric top; cf. chapter two), \(G\) belongs to an axis of revolution and the eigenvalues corresponding to the two other axes coincide; in the Kovalevskaya case (cf. chapter three), the eigenvalues of the inertia matrix are proportional to 2, 2, 1 and \(G\) belongs to the plane corresponding to the largest eigenvalue(s). The Goryachev-Chaplygin case (cf. also chapter three) is presented. In the first two cases, it is shown by computation that \(H\) and \(K\) commute.

The classical treatment of the equations (i.e., skillful computation or (and) geometric reasoning) describes 1) in the Lagrange case, how Liouville tori appear as circle bundles over the connected component of an elliptic curve which allows to linearize the solution; 2) in the Kovalevskaya case, how the flow is linearized via the apparition of a hyperelliptic genus two curve \(\kappa\) which moreover allows to describe potential critical values \((H,K)\) of the momentum mapping; 3) in the free rigid body case, (where one can restrict the study to the dynamics describing the momentum and \(K\) plays a role similar to the two Casimirs from the other cases), how to linearize the flow and to characterize the regular levels (including bifurcation diagram) in terms of an elliptic curve \(V\); \(V_\mathbb{R}\) has two connected components which are ovals; 4) in the Euler-Poinsot case, the regular levels (including bifurcation diagram) as two Liouville tori.

For the first three cases just considered, spectral curve and eigenvector mapping are constructed. From \(C\), involutivity of \(H\) and \(K\) is obtained in many cases via residue computations which are not shown. Moreover, \(\phi_C\) is linearizing in the sense that \(\phi_C (A_\lambda)\) belongs to a line in \(\text{Pic}(C)\) due to a general result (Reyman, Griffiths). It is shown that the smoothness of \(C\) (or a closely related curve (denoted \(C'\)) for the four dimensional free rigid body and in the Kovalevskaya case) implies regularity of the momentum mapping (and the converse holds in the Lagrange case but not in the Kovalevskaya case), which allows to draw curves in the \((H, K)\) plane where singularity has to occur from multiple zeros of \(C\). One has in the respective cases just mentioned 1) \(C\) has genus one, \(C_\mathbb{R}=\emptyset\) and \(\phi_C\) defines a connected double covering of \(\frac{T_C}{\rho}\), (\(\rho\) represents the effect of a rotation along the principal axis on \(A_\lambda\)) onto one of the two components of \((\text{Pic}^4(C))_{\mathbb{R}}\) if \(C\) is smooth. 2) The Lax pair with spectral parameter is obtained from work of Bobenko, Semenov, Tian-Shanski and justified. \(C\) is smooth if and only if \(\kappa\) is smooth which implies that a) its genus is 5, b) \(\phi_{C'}\) is an isomorphism onto its image in \(\text{Pic}^4(C')\) which preserves real structures; from the real part of a characterization of this image, one can count the Liouville tori depending on the values of \(H\), \(K\). This characterization, and the justification for the corresponding given bifurcation diagrams can be found in a publication of the author and R. Silhol [Compos. Math. 87, No. 2, 153-229 (1993; Zbl 0774.58012)]. 3) The lax pair with spectral parameter is obtained from work of Manakov. From \(C\), one gets that its genus is one and \(C_\mathbb{R}\) has two connected components. The eigenvector mapping can now be regarded as \(V\to \text{Pic}^1(C)\simeq C\); it is an algebraic mapping and a group homomorphism with kernel \(\mathbb{Z}/2\times \mathbb{Z}/2\) so that \(C\simeq V\) which implies that \(C\) and \(V\) are smooth at the same time. Moreover, there is a real isomorphism between \(C\) and \(\text{Jac}(V)\) (the Jacobian of \(V\)), and the two components of \(V_\mathbb{R}\) are mapped by \(\phi_C\) into one component of \(C_\mathbb{R}\). The bifurcation diagram having already been easily obtained, the four-dimensional case is investigated and one obtains (without information on the number of Liouville tori) via the same methods and the use of a computer package a bifurcation diagram. Results due to Haine concerning \(\phi_C\) (whose image is in \(\text{Pic}^6(C))\) are stated.

Chapter five is devoted to the periodic Toda lattice which is not a spinning top. Now, the levels are not compact. \(C\) is a genus \(n\) hyperelliptic curve. The eigenvector mapping is computed for \(n=2\) and bifurcation diagrams are drawn using information from \((\text{Jac}(C))_\mathbb{R}\).

Appendix one explains coadjoint orbits, their symplectic structure and when it is possible to identify a Lie algebra and its dual.

Appendix two is devoted to the Adler-Kostant-Symes theorem and its application to Lax equations with spectral parameter. This allows to synthesize Lax equations which are Hamilton’s equation of a given invariant function, here \(H\) or \(K\), being obtained via a residue computation from the coefficients of the equation of \(C\). Moreover, the associated Hamiltonian vector fields commute. Appendix three defines the eigenvector mapping, its tangent mapping using tools from homological algebra, and gives conditions under which the eigenvector mapping is linearizing. This appendix is by no means elementary.

Appendix four defines Riemann surfaces, the Jacobian, the Picard group and states the Riemann-Hurwitz formula used for genus computation. An interesting section describes real structures on the Jacobian. It is possible to relate the number of connected components of \(X_\mathbb{R}\) to those of \((\text{Jac}(X))_\mathbb{R}\) (\(X\) is a Riemann surface). In the genus one case, \(\text{Jac}(X)\) is classified as well as its real structure. Of course, since the motions are real, one has to pay special attention to the real part of the objects constructed from \(C\).

Appendix five defines Prym varieties and considers duality between Pryms.

There are several typographical mistakes or imprecisions of minor importance. Page 41, the first coordinate of the intersection point of the two branches of the discriminant is positive for \(c>2\) so that the bifurcation diagram has to be corrected. Page 66, the branching points \(b^2_1\), \(b^2_3\) are real but one of the branching points \(b^2_2\) or \(H'/p^2\) is complex according to the sign of their difference leading to two connected components of \(X_\mathbb{R}\) (the author states that all the branching points are real). Page 67, the pole of the first (resp. second) coordinate of the eigenvector is \(P_2\) (resp. \(P_1\)) (and not \(P_1\) (resp. \(P_2\))). Page 75, arrow \(a\) goes to a forbidden region and should go where arrow \(c\) is directed. Arrow \(c\) should go to the top part of the allowable region.

Nevertheless, the book is written with great care and therefore should be read carefully. None of the read sentences is gratuitous. Actually, they collaborate to form a harmonious balance between traditional methods and modern ones, generality and care for the specificity of each problem under consideration, abstraction and physical intuition. The style is entertaining but the ideas are rigorously exposed. It is also very dense which has the advantage of stimulating the intelligence of the reader at the price of his time. A consequence is also that many computations are left to the reader. The chapters can be read largely independently but it is a good idea to look at chapter one first. They can really be appreciated by reading the appendices which are very useful. Moreover, the appendices themselves may require further reading (We recommend the book of D. Mumford [‘Curves and their Jacobians’ (1975; Zbl 0316.14010)] to build an intuition on Riemann surfaces, Jacobians,etc.). The references (89) span more than a century from top mathematicians.

The author mentions (related?) mysterious points: Is there a natural way of constructing a Lax pair with spectral parameter? Sometimes the curves obtained via the classical methods appear also in the treatment with spectral Lax equations; why?

The reviewer has been delighted by the book and strongly recommends it to those who have a keen interest in handling applied problems with theoretical tools.

Consider next the eigenvector \(V_i(\lambda,\mu)\) associated to an eigenvalue \(\mu\) of \(A_\lambda\) (with simple spectrum). One obtains a complex line bundle over \(C\) when \(\lambda\) varies. This line bundle is indexed by the level of the integrals \(H_i\). If \(T_C\) denotes the set of matrices \(A_\lambda\) for fixed levels, the mapping \(\phi_C:T_C\to \text{Pic}(C)\) is well defined. \(\text{Pic}(C)\) is the Picard group obtained from the divisors which are identified with complex line bundles and called the eigenvector mapping.

In the book under review the author considers specific systems, i.e., spinning tops and the periodic Toda lattice. She presents classical approaches which study their dynamics, but the emphasis is on in describing the properties of the spectral curve \(C\) and the eigenvector mapping of these systems. From these properties, information on the level sets of the original dynamical system can be obtained; the information is topological, algebraic, concerns linearization (in the Hamiltonian sense) of the solutions, regularity of the levels. In the examples under consideration, the level sets are described themselves by the intersection of algebraic surfaces. If the flows are complete (true in the compact case of spinning tops), they are Liouville tori. So, the book applies to specific systems an approach which brings together two disciplines: Riemann surface theory and Hamiltonian dynamics.

The introduction reminds in an intuitive fashion the basic definitions and properties of Hamiltonian dynamics, defines the eigenvector mapping, explains what can be expected from it, and presents the contents of the chapters of the book.

In chapter one, the basic equations of the three-dimensional rigid body dynamics with a fixed point are presented. They are written directly in the Lax form using the adjoint action of SO(3) on its Lie algebra. The variables represent the orientation of the top and the momentum (6 reals). Two Casimir functions (the norm of the vector characterizing the orientation which is constant, the momentum with respect to the vertical) provide two integrals. Then, four independent remaining variables belong to a phase space diffeomorphic to the symplectic manifold \(T(S^2)\) on which complete integrability occurs. The Hamiltonian \(H\) gives a third integral and a fourth one \(K\) whose existence and form depends on the repartition of mass in the top leads to several cases studied in more detail in the remaining chapters: In the Euler-Poinsot case (leading to the free rigid body situation; cf. chapter four), the center of gravity \(G\) coincides with the fixed point of the top; in the Lagrange case (also called the symmetric top; cf. chapter two), \(G\) belongs to an axis of revolution and the eigenvalues corresponding to the two other axes coincide; in the Kovalevskaya case (cf. chapter three), the eigenvalues of the inertia matrix are proportional to 2, 2, 1 and \(G\) belongs to the plane corresponding to the largest eigenvalue(s). The Goryachev-Chaplygin case (cf. also chapter three) is presented. In the first two cases, it is shown by computation that \(H\) and \(K\) commute.

The classical treatment of the equations (i.e., skillful computation or (and) geometric reasoning) describes 1) in the Lagrange case, how Liouville tori appear as circle bundles over the connected component of an elliptic curve which allows to linearize the solution; 2) in the Kovalevskaya case, how the flow is linearized via the apparition of a hyperelliptic genus two curve \(\kappa\) which moreover allows to describe potential critical values \((H,K)\) of the momentum mapping; 3) in the free rigid body case, (where one can restrict the study to the dynamics describing the momentum and \(K\) plays a role similar to the two Casimirs from the other cases), how to linearize the flow and to characterize the regular levels (including bifurcation diagram) in terms of an elliptic curve \(V\); \(V_\mathbb{R}\) has two connected components which are ovals; 4) in the Euler-Poinsot case, the regular levels (including bifurcation diagram) as two Liouville tori.

For the first three cases just considered, spectral curve and eigenvector mapping are constructed. From \(C\), involutivity of \(H\) and \(K\) is obtained in many cases via residue computations which are not shown. Moreover, \(\phi_C\) is linearizing in the sense that \(\phi_C (A_\lambda)\) belongs to a line in \(\text{Pic}(C)\) due to a general result (Reyman, Griffiths). It is shown that the smoothness of \(C\) (or a closely related curve (denoted \(C'\)) for the four dimensional free rigid body and in the Kovalevskaya case) implies regularity of the momentum mapping (and the converse holds in the Lagrange case but not in the Kovalevskaya case), which allows to draw curves in the \((H, K)\) plane where singularity has to occur from multiple zeros of \(C\). One has in the respective cases just mentioned 1) \(C\) has genus one, \(C_\mathbb{R}=\emptyset\) and \(\phi_C\) defines a connected double covering of \(\frac{T_C}{\rho}\), (\(\rho\) represents the effect of a rotation along the principal axis on \(A_\lambda\)) onto one of the two components of \((\text{Pic}^4(C))_{\mathbb{R}}\) if \(C\) is smooth. 2) The Lax pair with spectral parameter is obtained from work of Bobenko, Semenov, Tian-Shanski and justified. \(C\) is smooth if and only if \(\kappa\) is smooth which implies that a) its genus is 5, b) \(\phi_{C'}\) is an isomorphism onto its image in \(\text{Pic}^4(C')\) which preserves real structures; from the real part of a characterization of this image, one can count the Liouville tori depending on the values of \(H\), \(K\). This characterization, and the justification for the corresponding given bifurcation diagrams can be found in a publication of the author and R. Silhol [Compos. Math. 87, No. 2, 153-229 (1993; Zbl 0774.58012)]. 3) The lax pair with spectral parameter is obtained from work of Manakov. From \(C\), one gets that its genus is one and \(C_\mathbb{R}\) has two connected components. The eigenvector mapping can now be regarded as \(V\to \text{Pic}^1(C)\simeq C\); it is an algebraic mapping and a group homomorphism with kernel \(\mathbb{Z}/2\times \mathbb{Z}/2\) so that \(C\simeq V\) which implies that \(C\) and \(V\) are smooth at the same time. Moreover, there is a real isomorphism between \(C\) and \(\text{Jac}(V)\) (the Jacobian of \(V\)), and the two components of \(V_\mathbb{R}\) are mapped by \(\phi_C\) into one component of \(C_\mathbb{R}\). The bifurcation diagram having already been easily obtained, the four-dimensional case is investigated and one obtains (without information on the number of Liouville tori) via the same methods and the use of a computer package a bifurcation diagram. Results due to Haine concerning \(\phi_C\) (whose image is in \(\text{Pic}^6(C))\) are stated.

Chapter five is devoted to the periodic Toda lattice which is not a spinning top. Now, the levels are not compact. \(C\) is a genus \(n\) hyperelliptic curve. The eigenvector mapping is computed for \(n=2\) and bifurcation diagrams are drawn using information from \((\text{Jac}(C))_\mathbb{R}\).

Appendix one explains coadjoint orbits, their symplectic structure and when it is possible to identify a Lie algebra and its dual.

Appendix two is devoted to the Adler-Kostant-Symes theorem and its application to Lax equations with spectral parameter. This allows to synthesize Lax equations which are Hamilton’s equation of a given invariant function, here \(H\) or \(K\), being obtained via a residue computation from the coefficients of the equation of \(C\). Moreover, the associated Hamiltonian vector fields commute. Appendix three defines the eigenvector mapping, its tangent mapping using tools from homological algebra, and gives conditions under which the eigenvector mapping is linearizing. This appendix is by no means elementary.

Appendix four defines Riemann surfaces, the Jacobian, the Picard group and states the Riemann-Hurwitz formula used for genus computation. An interesting section describes real structures on the Jacobian. It is possible to relate the number of connected components of \(X_\mathbb{R}\) to those of \((\text{Jac}(X))_\mathbb{R}\) (\(X\) is a Riemann surface). In the genus one case, \(\text{Jac}(X)\) is classified as well as its real structure. Of course, since the motions are real, one has to pay special attention to the real part of the objects constructed from \(C\).

Appendix five defines Prym varieties and considers duality between Pryms.

There are several typographical mistakes or imprecisions of minor importance. Page 41, the first coordinate of the intersection point of the two branches of the discriminant is positive for \(c>2\) so that the bifurcation diagram has to be corrected. Page 66, the branching points \(b^2_1\), \(b^2_3\) are real but one of the branching points \(b^2_2\) or \(H'/p^2\) is complex according to the sign of their difference leading to two connected components of \(X_\mathbb{R}\) (the author states that all the branching points are real). Page 67, the pole of the first (resp. second) coordinate of the eigenvector is \(P_2\) (resp. \(P_1\)) (and not \(P_1\) (resp. \(P_2\))). Page 75, arrow \(a\) goes to a forbidden region and should go where arrow \(c\) is directed. Arrow \(c\) should go to the top part of the allowable region.

Nevertheless, the book is written with great care and therefore should be read carefully. None of the read sentences is gratuitous. Actually, they collaborate to form a harmonious balance between traditional methods and modern ones, generality and care for the specificity of each problem under consideration, abstraction and physical intuition. The style is entertaining but the ideas are rigorously exposed. It is also very dense which has the advantage of stimulating the intelligence of the reader at the price of his time. A consequence is also that many computations are left to the reader. The chapters can be read largely independently but it is a good idea to look at chapter one first. They can really be appreciated by reading the appendices which are very useful. Moreover, the appendices themselves may require further reading (We recommend the book of D. Mumford [‘Curves and their Jacobians’ (1975; Zbl 0316.14010)] to build an intuition on Riemann surfaces, Jacobians,etc.). The references (89) span more than a century from top mathematicians.

The author mentions (related?) mysterious points: Is there a natural way of constructing a Lax pair with spectral parameter? Sometimes the curves obtained via the classical methods appear also in the treatment with spectral Lax equations; why?

The reviewer has been delighted by the book and strongly recommends it to those who have a keen interest in handling applied problems with theoretical tools.

Reviewer: A.Akutowicz (Berlin)

### MSC:

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

70E15 | Free motion of a rigid body |

30F99 | Riemann surfaces |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |