# Integrable Hamiltonian Systems on Complex Lie Groups

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*V. Jurdjevic*

This paper is a study of the elastic problems on
simply connected manifolds \(M_n\) whose orthonormal frame
bundle is a Lie group \(G\). Such manifolds, called the space
forms in the literature on differential geometry, are classified and
consist of the Euclidean spaces \(\mathbb{E}^n\), the
hyperboloids \(\mathbb{H}^n\), and the spheres \(S^n\),
with the corresponding orthonormal frame bundles equal to the
Euclidean group of motions \(\mathbb{E}^n\rtimes
SO_n(\mathbb{R})\), the rotation group
\(SO_{n+1}(\mathbb{R})\), and the Lorentz group
\(SO(1,n)\).

The manifolds \(M_n\)
are treated as the symmetric spaces \(G/K\) with \(K\)
isomorphic with \(SO_n(R)\). Then the Lie algebra
\(\mathfrak{g}\) of \(G\) admits a Cartan decomposition
\(\mathfrak{g}=\mathfrak{p}+\mathfrak{k}\) with
\(\mathfrak{k}\) equal to the Lie algebra of \(K\) and
\(\mathfrak{p}\) equal to the orthogonal comlement
\(\mathfrak{k}\) relative to the trace form. The elastic
problems on \(G/K\) concern the solutions \(g(t)\) of a
left invariant differential systems on \(G\)
\[\frac{dg}{dt}(t)=g(t)(A_0+U(t)))\] that minimize the
expression \(\frac{1}{2}\int_0^T (U(t),U(t))\,dt\) subject
to the given boundary conditions \(g(0)=g_0\),
\(g(T)=g_1\), over all locally bounded and measurable
\(\mathfrak{k}\) valued curves \(U(t)\) relative to a
positive definite quadratic form \((\, , \,)\) where
\(A_0\) is a fixed matrix in \(\mathfrak{p}\).

These variational problems fall in two classes, the
Euler-Griffiths problems and the problems of Kirchhoff. The
Euler-Griffiths elastic problems consist of minimizing the integral
\[\tfrac{1}{2}\int_0^T\kappa^2(s)\,ds\] with \(\kappa
(t)\) equal to the geodesic curvature of a curve \(x(t)\)
in the base manifold \(M_n\) with \(T\) equal to the
Riemannian length of \(x\). The curves \(x(t)\) in this
variational problem are subject to certain initial and terminal
boundary conditions. The elastic problems of Kirchhoff is more
general than the problems of Euler-Griffiths in the sense that the
quadratic form \((\, , \,)\) that defines the functional to be
minimized may be independent of the geometric invariants of the
projected curves in the base manifold. It is only on two dimensional
manifolds that these two problems coincide in which case the solutions
curves can be viewed as the non-Euclidean versions of L. Euler
elasticae introduced in 174.

Each elastic problem defines the appropriate
left-invariant Hamiltonian \(\mathcal{H}\) on the dual
\(\mathfrak{g}^*\) of the Lie algebra of \(G\) through
the Maximum Principle of optimal control. The integral curves of the
corresponding Hamiltonian vector field \(\vec{\mathcal{H}}\)
are called the extremal curves.

The paper is
essentially concerned with the extremal curves of the Hamiltonian
systems associated with the elastic problems. This class of
Hamiltonian systems reveals a remarkable fact that the Hamiltonian
systems traditionally associated with the movements of the top are
invariant subsystems of the Hamiltonian systems associated with the
elastic problems.

The paper is divided into two parts. The first part of
the paper synthesizes ideas from optimal control theory, adapted to
variational problems on the principal bundles of Riemannian spaces,
and the symplectic geometry of the Lie algebra \(\mathfrak{g},\)
of \(G\), or more precisely, the symplectic structure of the
cotangent bundle \(T^*G\) of \(G\).

The second part of the paper is devoted to the solutions
of the complexified Hamiltonian equations induced by the elastic
problems. The paper contains a detailed discussion of the algebraic
preliminaries leading up to \(so_n(\mathbb{C})\), a natural
complex setting for the study of the left invariant Hamiltonians on
real Lie groups \(G\) for which \(\mathfrak{g}\) is a
real form for \(so_n(\mathbb{C})\). It is shown that the
Euler-Griffiths problem is completely integrable in any dimension with
the solutions the holomorphic extensions of the ones obtained by
earlier P. Griffiths. The solutions of the elastic problems of
Kirchhoff are presented in complete generality on
\(SO_3(\mathbb{C})\) and there is a classification of the
integrable cases on \(so_4(\mathbb{C})\) based on the criteria
of Kowalewski-Lyapunov in their study of the mechanical tops.
Remarkably, the analysis yields essentially only two integrables cases
analogous to the top of Lagrange and the top of Kowalewski. The paper
ends with the solutions of the integrable complex Hamiltonian systems
on the \(SL_2(\mathbb{C})\times SL_2(\mathbb{C})\), the
universal cover of \(SO_4(\mathbb{C})\).

#### Table of Contents

# Table of Contents

## Integrable Hamiltonian Systems on Complex Lie Groups

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Cartan decomposition and the Generalized elastic problems 1928 free
- Chapter 3. The Maximum Principle and the Hamiltonians 2534
- Chapter 4. The left-invariant symplectic form 3544
- Chapter 5. Symmetries and the conservation laws 5261
- Part 2. Complex Lie groups and complex Hamiltonians 5968
- Chapter 6. Complexified elastic problems 5968
- Chapter 7. Complex elasticae of Euler and its n-dimensional extensions 7786
- Chapter 8. Cartan algebras, root spaces and extra integrals of motion 92101
- Chapter 9. Elastic Curves for the case of Lagrange 98107
- Chapter 10. Elastic Curves for the case of Kowalewski 108117
- Bibliography 132141