Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies

This work studies the symmetries, the associated momentum map, and relative equilibria of a mechanical system consisting of a small axisymmetric magnetic body-dipole in an also axisymmetric external magnetic field that additionally exhibits a mirror symmetry; we call this system the ``orbitron". We study the nonlinear stability of a branch of equatorial quasiorbital relative equilibria using the energy-momentum method and we provide sufficient conditions for their $\mathbb{T}^2$--stability that complete partial stability relations already existing in the literature. These stability prescriptions are explicitly written down in terms of the some of the field parameters, which can be used in the design of stable solutions. We propose new linear methods to determine instability regions in the context of relative equilibria that we use to conclude the sharpness of some of the nonlinear stability conditions obtained.

The configuration space of the orbitron is SE (3) = SO(3) × R 3 The orbitron is a simple mechanical system Phase space is the cotangent bundle T * SE (3) of its configuration space SE (3) endowed with the canonical symplectic structure ω obtained as minus the differential of the corresponding Liouville one form with T (Π, p) := 1 2 where M is the mass of the axisymmetric magnetic body, the reference inertia tensor I ref = diag(I 1 , I 1 , I 3 ), x = (x, y , z) ∈ R 3 , µ is the magnetic moment of the axisymmetric rigid body/dipole, and B(x) is the strength of the magnetic field created by two magnetic poles/"charges" ±q placed at the points (0, 0, h) and (0, 0, −h), h > 0, that is, with D(x) + = x 2 + y 2 + (z − h) 2 , D(x) − = x 2 + y 2 + (z + h) 2 , and µ 0 the magnetic permeability of vacuum.

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron The standard and generalized orbitron Definition A small axisymmetric magnetized rigid body subjected to a external magnetic field of the form specified in (4) is called a standard orbitron.
The external magnetic field B in (4) has the following symmetry properties, namely: (i) Equivariance with respect to rotations R Z θ S around the OZ axis: (ii) Behavior with respect to the mirror transformation (x, y , z) −→ (x, y , −z) according to the prescription .

Definition
A small axisymmetric magnetized rigid body subjected to the influence of an arbitrary magnetic field in the magnetostatic approximation in a domain free of other magnetic sources that satisfies these symmetry properties is called a generalized orbitron.

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron

Equations of motion
The equations of motion of the orbitron are determined by Hamilton's equations:Ȧ The symbol I −1 ref Π stands for the antisymmetric matrix associated to the vector I −1 ref Π ∈ R 3 via the Lie algebra isomorphism : R 3 , × −→ (so(3), [·, ·]) and D for the differential.

Toral symmetry and associated momentum map
The axial symmetry of the magnetic rigid body + the rotational spatial symmetry of the external magnetic field w.r.t. rotations around OZ = toral symmetry. The action on SE (3): The cotangent lift Φ is a canonical symmetry given by Φ : , that has an invariant momentum map associated J : The invariance of the Hamiltonian w.r.t. the cotangent toral action By Noether's Theorem [AM78], the level sets of J are preserved by the associated Hamiltonian dynamics, i.e., if Ft is the flow of the vector field X h then J • Ft = J for any t.
Lyudmila Grigoryeva, Juan-Pablo Ortega, and Stanislav Zub Stability of Hamiltonian relative equilibria in symmetric magnetica

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron

Relative equilibria: setup and background
Consider a vector field X ∈ X(M) on a manifold M that is equivariant with respect to action of a Lie group G on it. We say that the point m ∈ M is a relative equilibrium with velocity ξ ∈ g and the infinitesimal generator ξ M associated to it if Or, equivalently, where Ft is the flow associated to X at m ∈ M, exp is the Lie group exponential map exp : g → G (the one-parameter Lie subgroup of G generated by ξ ∈ g). Consider now a symmetric Hamiltonian system (M, ω, h, G , J : M → g * ) and assume that the momentum map J is coadjoint equivariant; it can be shown [AM78] that the point m ∈ M is a relative equilibrium of the Hamiltonian vector field X h with velocity ξ ∈ g if and only if where J ξ := J, ξ . The combination h − J ξ is usually referred to as the augmented Hamiltonian.
If the relative equilibrium m ∈ M is such that J(m) = µ ∈ g * and we denote its isotropy subgroup with respect to the G action by G m , the law of conservation of the isotropy [OR04] and Noether's Theorem imply [OR99, is the normalizer group of G m in G µ (note that G m ⊂ G µ necessarily due to the equivariance of the momentum map). Finally, notice that the velocity of a relative equilibrium with nontrivial isotropy is not uniquely defined; indeed, it is clear in (12) that if ξ ∈ g is a velocity for the relative equilibrium m, then so is ξ+η for any η ∈ Lie (G m ).
Lyudmila Grigoryeva, Juan-Pablo Ortega, and Stanislav Zub Stability of Hamiltonian relative equilibria in symmetric magnetica

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron Relative equilibria equations of the orbitron Proposition Consider the orbitron system whose Hamiltonian function is given by (1) and let z = ((A, x) , (Π, p)) ∈ T * SE (3). Then: (i) The point z is a relative equilibrium of the orbitron with velocity (ξ1, ξ2) ∈ R 2 with respect to the introduced toral symmetry if and only if the following identities are satisfied: Lyudmila Grigoryeva, Juan-Pablo Ortega, and Stanislav Zub Stability of Hamiltonian relative equilibria in symmetric magnetica

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron x0) , (Π0, p0)) is a relative equilibrium of the standard orbitron with velocity (ξ1, ξ2), where ξ2 is an arbitrary real number and ξ1 is either arbitrary when x0 = 0 or when x0 = 0 (the existence is only guaranteed when µq < 0).
(iii) In the case of the generalized orbitron: Bz (x, y , z) = f (x 2 + y 2 , z) for some f ∈ C ∞ (R 2 ), and the spatial velocity ξ1 of the relative equilibria with x0 = 0 is where (exists only when µf 1 < 0).

The energy-momentum method
Definition Let X ∈ X(M) be a G -equivariant vector field on the G -manifold M and let G be a subgroup of G . A relative equilibrium m ∈ M of X , is called G -stable, or stable modulo G , if for any G -invariant open neighborhood V of the orbit G · m, there is an open neighborhood U ⊂ V of m, such that if F t is the flow of the vector field X and u ∈ U, then F t (u) ∈ V for all t ≥ 0.
Lyudmila Grigoryeva, Juan-Pablo Ortega, and Stanislav Zub Stability of Hamiltonian relative equilibria in symmetric magnetica

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron The energy-momentum method Nonlinear stability of the orbitron relative equilibria Linear stability/instability analysis of relative equilibria The energy-momentum method provides a sufficient condition for the Gµ-stability of a given relative equilibrium [Pat92, OR99, PRW04, MRO11].

Theorem (Energy-momentum method)
Let (M, ω, h) be a symplectic Hamiltonian system with a symmetry given by the Lie group G acting properly on M with an associated coadjoint equivariant momentum map J : M → g * . Let m ∈ M be a relative equilibrium such that J(m) = µ ∈ g * and assume that the coadjoint isotropy subgroup Gµ is compact. Let ξ ∈ LieN Gµ (Gm) be a velocity of the relative equilibrium. If the quadratic form is definite for some (and hence for any) subspace W such that then m is a Gµ-stable relative equilibrium. If dim W = 0, then m is always a Gµ-stable relative equilibrium. The quadratic form d 2 (h − J ξ )(m)| W ×W , will be called the stability form of the relative equilibrium m and W a stability space.

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron The energy-momentum method Nonlinear stability of the orbitron relative equilibria Linear stability/instability analysis of relative equilibria Nonlinear stability of the orbitron relative equilibria Theorem Consider the relative equilibria introduced in Proposition 3. Then: (i) The regular relative equilibria of the standard orbitron in part (ii) of Proposition 3, that is, those for which x0 = 0, are T 2 -stable whenever the following three inequalities are satisfied: where r 2 = x0 2 , ξ 0 1 = ± − 3hµqµ0 2πMD(x0) 5/2 1/2 , and µq < 0. The singular relative equilibria (x0 = 0) are always formally unstable, in the sense that the stability form (20) exhibits a nontrivial signature.

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron The energy-momentum method Nonlinear stability of the orbitron relative equilibria Linear stability/instability analysis of relative equilibria Theorem (Continued) (ii) The regular relative equilibria of the generalized orbitron in part (iii) of Proposition 3 are T 2 -stable whenever the following conditions hold: where r 2 = x0 2 , f ∈ C ∞ (R 2 ) is the function such that Bz (x, y , z) = f (r 2 , z), , and ξ 0 1 = ± −

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron The energy-momentum method Nonlinear stability of the orbitron relative equilibria Linear stability/instability analysis of relative equilibria

Theorem (Continued)
The singular branch (x0 = 0) is T 2 -stable if the following conditions are satisfied: where Π0 = I3 (ξ1 − ξ2) and we use the same notation as above for f0, f 1 , and f 2 , replacing v = r 2 by v = 0. When µf0 < 0 and f0 f 1 < 2 M I1, the conditions (29) and (30) can be replaced by the following single ξ1-independent optimal condition: This optimal condition is achieved by using the spatial velocities ξ1 = ± (−µf0/I1) 1/2 ; the positive (resp. negative) sign for the velocity corresponds to positive (resp. negative) values of Π0.

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The orbitron Relative equilibria of the orbitron Stability analysis of the relative equilibria of the orbitron The energy-momentum method Nonlinear stability of the orbitron relative equilibria Linear stability/instability analysis of relative equilibria Conditions (27)-(30) can be used in the design of magnetic fields capable of confining magnetic rigid bodies that do not exhibit spatial rotation. This is the working principle of devices such as magnetic contactless flywheels or levitrons. In the case of flywheels, up until now only actively controlled versions have been developed; as to the levitron, the potentials that have been considered so far [DE99, Dul04, KM06] do not allow to conclude nonlinear stability using the methods put at work in Theorem 7 and only the spectral stability of the corresponding linearized systems has been considered. We plan to explore in detail these systems in a future publication.
at m ∈ S coincides with the linear Hamiltonian vector field X Q on (W , ω W ) that has as Hamiltonian vector field the stability form (v) Suppose that the two tangent spaces Tm (Gµ · m) and Tm (G · m) coincide. Then TmM = W ⊕ W ω .
Additionally, let h ξ := h − J ξ ∈ C ∞ (M) be the augmented Hamiltonian and let X h ξ ∈ X(TmM) be the linearization of the Hamiltonian vector field X h ξ at m. Then where i W : W → TmM is the inclusion, P W : TmM −→ W is the projection according to (32), and X h ξ is the linearization of X h ξ at m.
(vi) If the linear vector field X Q is spectrally unstable in the sense that it exhibits eigenvalues with a nontrivial real part, then the relative equilibrium m ∈ M of X h is nonlinearly K -unstable, for any subgroup K ⊂ G .