Equivariant Transversality Theory Applied to Hamiltonian Relative Equilibria Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik, Informatik und Naturwissenschaften Fachbereich Mathematik der Universität Hamburg vorgelegt von Mara Sommerfeld aus Hamburg Hamburg, 2018 Als Dissertation angenommen vom Fachbereich Mathematik der Universität Hamburg Vorsitzender der Prüfungskommision: Prof. Dr. Janko Latschev Gutachter: Prof. Dr. Reiner Lauterbach Prof. Dr. James Montaldi Datum der Disputation: 12. Februar 2018 Contents 1 Relative equilibria in symmetric Hamiltonian systems 13 1.1 Hamiltonian dynamics . 13 1.2 Group actions . 15 1.3 Momentum maps . 21 2 Local theory 27 2.1 Equivariant Darboux theorem . 27 2.2 Marle-Guillemin-Sternberg normal form . 28 2.3 Bundle equations . 33 2.4 Splitting Lemma reduction . 36 3 Linear theory 41 4 Free actions 51 4.1 Bifurcation at non-regular momentum values . 52 4.2 Transverse relative equilibria . 55 5 Continuous isotropy 67 5.1 Bifurcation theory perspective . 68 5.2 Representations . 75 5.2.1 Bifurcation equation for representations . 75 5.2.2 Implications for groups of rank 1 . 82 5.2.3 Equivariant Weinstein-Moser theorem . 85 5.3 Some results derived from the bundle equations . 91 6 Equivariant transversality approach 97 6.1 Equivariant transversality theory . 98 6.1.1 Definition of equivariant transversality . 100 6.1.2 Higher order version . 103 6.2 Application to bifurcation theory . 109 6.3 Transverse relative equilibria with continuous isotropy . 116 6.4 Representations . 127 6.4.1 Torus representations . 128 6.4.2 Representations of connected compact groups . 137 6.5 Examples . 148 6.6 Application to Birkhoff normal forms . 151 7 Prospects 153 3 4 CONTENTS A Thom-Mather transversality theorem 155 A.1 C∞- and Whitney C∞-topology . 155 A.2 Transversality to Whitney stratified subsets . 158 Introduction Motivation The existence of symmetries is a fundamental assumption of theoretical me- chanics. In the classical Newton model, the laws of motion do not depend on a par- ticular choice of a linear coordinate system. Therefore, they are unchanged by translations, rotations, and reflections. These transformations generate the Eu- clidean group E(3), the group of isometries of 3-dimensional Euclidean space R3. The motion of n particles within R3 is described by a dynamical system given by a vector field X on the tangent space T R3n = R3n × R3n. The diago- nal E(3)-action on R3 induces an action of E(3) on the tangent space, and the invariance of the laws of motion is reflected by the fact that the flow ϕX of the vector field X is E(3)-equivariant, that is ϕX (gx) = gϕX (x) for any g ∈ E(3). More generally, it is often assumed in classical mechanics that unknown external forces constrain the motion to a submanifold Q ⊂ R3n. In that case, the symmetries of the system are given by a Lie group G that acts on Q. For example, the motion of a rigid body may be described by a curve in the Euclidean group. If (R(t), b(t)) is such a curve, where R(t) is an element of the orthogonal group and b(t) is a translation, the motion of any particle of the rigid body is given by the curve xi(t) = R(t)Xi + b(t), where Xi is the position of the particle in a fixed reference body. The Euclidean group acts by left multiplication on the configuration space. If the rigid body has some rotational symmetry that is given by a subgroup K ⊂ SO(3) and preserves the reference body, there is an additional K-action by right multiplication. The equations of motion for the dynamical system on the manifold have a particularly simple form in the Hamiltonian formalism. A curve q : R ⊃ I → Q ∗ corresponds to a curve (q(t), p(t)) in the cotangent bundle, where p(t) ∈ Tq(t)Q is an expression of q(t) and q˙(t). The equations of motion are determined by the Hamiltonian function h ∈ C∞(T ∗Q, R). The Hamiltonian function h defines the Hamiltonian vector field Xh by dh(p) = ωp(Xh(p), ·) for any p ∈ T ∗Q, where ω denotes the canonical symplectic form of T ∗Q.A G-action on Q induces a G-action on T ∗Q, which lifts the G-action on Q and leaves ω invariant. The Hamiltonian system on T ∗Q has G-symmetry if the Hamiltonian function h is G-invariant. Then, the Hamiltonian vector field is G-equivariant and hence the G-action commutes with the flow of Xh. Theoretical mechanics is the original motivation for studying Hamiltonian dynamical systems, which is nowadays a research field in its own right. A 5 6 CONTENTS Hamiltonian system with symmetry consists of a symplectic manifold P with a Lie group G acting by symplectomorphisms on P and a G-invariant Hamiltonian function. One of the main objects of interest in the study of Hamiltonian systems with symmetry are trajectories that are contained in a single group orbit. A point of P whose trajectory has this property is a relative equilibrium. The G-action on P defines a map ξ 7→ ξP from the Lie algebra g of G into the space of vector fields on P such that ∪ξ∈gξP (p) coincides with the tangent space TpGp. If p ∈ P is a relative equilibrium, there is an element ξ ∈ gP such that Xh(p) = ξP (p). ξ is a generator of the relative equilibrium p. For free actions, the generator is uniquely defined, but in general, it is only unique modulo the Lie algebra gp of the isotropy subgroup Gp. The question of determining the relative equilibria of a given system has a long tradition in the study of mechanical systems. The relative equilibria of a mechanical system are often configurations of constant shape that rotate about a fixed principal axis. In particular, this holds for the n-body problem, which was described in the beginning. The investigation of relative equilibria goes back to Riemann and even fur- ther. In [Rie61], Riemann finds all solutions of constant shape for a model of ellipsoidal liquid drops, which was proposed by Dirichlet ([Dir60]). Moreover, he shows that all of these motions of constant shape consist of a rotation about one of the principal axis of the ellipsoid and an oval motion of the fluid particles within the ellipsoid, which corresponds to a rotation of a spherical reference body. A modern treatment of the same problem may be found in [RdS99]. Here, ∗ the problem is described as a Hamiltonian dynamical system on T GL+, the cotangent space of the group of 3 × 3-matrices with positive determinant. Outline In chapter 1, we introduce the basic concepts and notions of the theory of Hamiltonian systems with symmetry. In particular, we define the momentum map, which is a basic tool for the most modern approaches to determine the relative equilibria of a Hamiltonian system: Noether’s famous theorem of classical mechanics states that symmetries cor- respond to conserved quantities. For general Hamiltonian systems with symme- try, this is true at least locally. In this thesis, we will always suppose that these conserved quantities are given by globally defined functions on the symplectic space P . Putting these conserved quantities together in an appropriate way yields the momentum map J : P → g∗. The map Jξ := hJ(·), ξi is a Hamilto- nian function of the vector field ξP . Thus, the relative equilibria with generator ξ ξ coincide with the critical points of the augmented Hamiltonian hξ := (h − J ). In many relevant cases, the momentum map is equivariant with respect to the coadjoint action on g∗. For example if G is compact, an equivariant momentum map may be obtained from any given momentum map by averaging over G. For simplicity, this thesis focuses on actions of compact groups. The more general case is only formulated if the statement does not become more compli- cated. Otherwise, we refer to the literature. In particular, we will assume the equivariance of the momentum map with respect to the coadjoint action. CONTENTS 7 The equivariance of the momentum map implies in general that J(p) = J(gp) holds if and only if g ∈ Gµ for µ = J(p). Hence if p is a relative equilibrium, p is also a relative equilibrium with respect to the Gµ-action. If ξ is a generator of p, we obtain ξ ∈ gµ or equivalently coadξ µ = 0. This commutation relation occurs in different forms in most of the ap- proaches to analyse the structure of Hamiltonian relative equilibria presented in this thesis. For instance, it is equivalent to the first of two equations that occur in [MR-O13] and [MR-O15] and characterize relative equilibria. These equations are derived in chapter 2, in which the theory of the local structure of the system near a given G-orbit is treated. They are stated in the coordi- nates of the Marle-Guillemin-Sternberg normal form, which is a local model of the symplectic space P near a given G-orbit. The Marle-Guillemin-Sternberg normal form is often considered as the symplectic version of the Slice Theorem. For simplicity, let us assume that G is compact. Then the normal form the- orem states that for p ∈ P , there is a tubular neighbourhood U of Gp and a G-equivariant symplectomorphism to a G-invariant neighbourhood of [e, 0, 0] in the space ∗ Y = G ×Gp (m × N), where N is a symplectic Gp-representation and m is a Gp-invariant complement of gp within gµ for µ = J(p). The symplectic form on Y is determined by the symplectic form on N and the group G.