The complex geometry of the Lagrange top.

*(English)*Zbl 0972.37042The Lagrange top is one of the most classical examples of integrable system. Let \(\Omega =(\Omega_1,\Omega_2,\Omega_3)\) denote the angular velocity expressed in body-coordinates. Due to symmetry of the body there is an additional integral of motion \(H_4=\Omega_3\), which makes the Hamiltonian system of the Lagrange top Liouville integrable.

Let \(T_h\) be a complex invariant level set of the Lagrange top. Consider the complexified group of rotation \(\mathbb{C}^*\sim \mathbb{C}/2\pi i \mathbb{Z}\) defined by the flow of the Hamiltonian vector field generated by \(H_4\). For generic \(h_i\) the complex invariant level set \(T_h\) is biholomorphic to \(\mathbb{C}^2/\Lambda\), where \(\Lambda \subset\mathbb{C}^2\) is a rank three lattice.

The starting point of the present article is the observation that, generically, the algebraic manifold \(T_h\) is not isomorphic to a direct product of the curve \(T_h/\mathbb{C}^*\) and \(\mathbb{C}^*\) (although as a topological manifold it is). The generic invariant level set \(T_h\) is an extension of an elliptic curve \(C\sim \text{Jac}(C)\) by \(\mathbb{C}^*\) and the flow is projected on this curve into a well defined linear flow.

In this paper an algebraic description of the Lagrange top is given. The authors deduced, by making use of a Baker-Akhiezer function, simple explicit formulae for the general solution of the Lagrange top. The authors study reality conditions on the complex solutions. Besides the usual real structure of the Lagrange top given by complex conjugation there is a second natural real structure induced by the eigenvalue map of the corresponding Lax pair representation. It turns out that these two structures coincide on \(\text{Jac}(C)\) but are different on \(\mathbb{C}^2/\Lambda\sim T_h\). The corresponding real level sets are described. This makes clear the relation between the real structure of the curve \(C\), its Jacobian \(\text{Jac}(C)\) and the real level set \(H_h^{\mathbb{R}}\).

Let \(T_h\) be a complex invariant level set of the Lagrange top. Consider the complexified group of rotation \(\mathbb{C}^*\sim \mathbb{C}/2\pi i \mathbb{Z}\) defined by the flow of the Hamiltonian vector field generated by \(H_4\). For generic \(h_i\) the complex invariant level set \(T_h\) is biholomorphic to \(\mathbb{C}^2/\Lambda\), where \(\Lambda \subset\mathbb{C}^2\) is a rank three lattice.

The starting point of the present article is the observation that, generically, the algebraic manifold \(T_h\) is not isomorphic to a direct product of the curve \(T_h/\mathbb{C}^*\) and \(\mathbb{C}^*\) (although as a topological manifold it is). The generic invariant level set \(T_h\) is an extension of an elliptic curve \(C\sim \text{Jac}(C)\) by \(\mathbb{C}^*\) and the flow is projected on this curve into a well defined linear flow.

In this paper an algebraic description of the Lagrange top is given. The authors deduced, by making use of a Baker-Akhiezer function, simple explicit formulae for the general solution of the Lagrange top. The authors study reality conditions on the complex solutions. Besides the usual real structure of the Lagrange top given by complex conjugation there is a second natural real structure induced by the eigenvalue map of the corresponding Lax pair representation. It turns out that these two structures coincide on \(\text{Jac}(C)\) but are different on \(\mathbb{C}^2/\Lambda\sim T_h\). The corresponding real level sets are described. This makes clear the relation between the real structure of the curve \(C\), its Jacobian \(\text{Jac}(C)\) and the real level set \(H_h^{\mathbb{R}}\).