Non-canonical Hamiltonian Structure Research Articles

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

Analytic Hall magnetohydrodynamics toroidal equilibria via the energy-Casimir variational principle

Equilibrium equations for magnetically confined, axisymmetric plasmas are derived by means of the energy-Casimir variational principle in the context of Hall magnetohydrodynamics (MHD). This approach stems from the noncanonical Hamiltonian structure of Hall MHD, the simplest, quasineutral two-fluid model that incorporates contributions due to ion Hall drifts. The axisymmetric Casimir invariants are used, along with the Hamiltonian functional to apply the energy-Casimir variational principle for axisymmetric two-fluid plasmas with incompressible ion flows. This results in a system of equations of the Grad–Shafranov–Bernoulli (GSB) type with four free functions. Two families of analytic solutions to the GSB system are then calculated, based on specific choices for the free functions. These solutions are subsequently applied to Tokamak-relevant configurations using proper boundary shaping methods. The Hall MHD model predicts a departure of the ion velocity surfaces from the magnetic surfaces which are frozen in the electron fluid. This separation of the characteristic surfaces is corroborated by the analytic solutions calculated in this study. The equilibria constructed by these solutions exhibit favorable characteristics for plasma confinement, for example they possess closed and nested magnetic and flow surfaces with pressure profiles peaked at the plasma core. The relevance of these solutions to laboratory and astrophysical plasmas is finally discussed, with particular focus on systems that involve length scales on the order of the ion skin depth.

Vorticity equation on surfaces with arbitrary topology embedded in three-dimensional Euclidean space

We derive the vorticity equation for an incompressible fluid on a two-dimensional surface with an arbitrary topology, embedded in three-dimensional Euclidean space and arising from a first integral of the flow, by using a tailored Clebsch parameterization of the velocity field. In the inviscid limit, we identify conserved surface energy and enstrophy and obtain the corresponding noncanonical Hamiltonian structure. We then discuss the formulation of the diffusion operator on the surface by examining two alternatives. In the first case, we follow the standard approach for Navier–Stokes equations on a Riemannian manifold and calculate the diffusion operator by requiring that flows corresponding to Killing fields of the Riemannian metric are not subject to dissipation. For an embedded surface, this leads to a diffusion operator, including derivatives of the stream function across the surface. In the second case, using an analogy with the Poisson equation for the Newtonian gravitational potential in general relativity, we construct a diffusion operator taking into account the Ricci scalar curvature of the surface. The resulting vorticity equation is two-dimensional, and the corresponding diffusive equilibria minimize dissipation under the constraint of curvature energy.

Impact of electron temperature anisotropy on the collisionless tearing mode instability in the presence of a strong guide field

We derive and analyze a dispersion relation for the growth rate of collisionless tearing modes, driven by electron inertia and accounting for equilibrium electron temperature anisotropy in a strong guide field regime. For this purpose, a new gyrofluid model is derived and subsequently simplified to make the derivation of the dispersion relation treatable analytically. The main simplifying assumptions consist in assuming cold ions, neglecting electron finite Larmor radius effects, decoupling ion gyrocenter fluctuations, and considering β⊥e≪1, with β⊥e indicating the ratio between the perpendicular electron thermal pressure and the magnetic pressure exerted by the guide field. This simplified version of the gyrofluid model is shown to possess a noncanonical Hamiltonian structure. The dispersion relation is obtained by applying the theory of asymptotic matching and does not predict an enhancement of the growth rate as the ratio Θe between perpendicular and parallel equilibrium electron temperatures increases. This indicates a significant difference with respect to the case of absent or moderate guide field. For an equilibrium magnetic shear length of the order of the perpendicular sonic Larmor radius and at a fixed β⊥e, we obtain that the tearing mode in the strong guide field regime gets actually weakly damped, as Θe increases. In the isotropic limit Θe=1, the dispersion relation reduces to a previously known formula. The analytical predictions are tested against numerical simulations, showing a very good quantitative agreement. We also provide a detailed discussion of the range of validity of the derived dispersion relation and of the compatibility among the different adopted assumptions.

Geometric Particle-in-Cell Simulations of the Vlasov--Maxwell System in Curvilinear Coordinates

Numerical schemes that preserve the structure of the kinetic equations can provide stable simulation results over a long time. An electromagnetic particle-in-cell solver for the Vlasov--Maxwell equations that preserves at the discrete level the noncanonical Hamiltonian structure of the Vlasov--Maxwell equations has been presented in [Kraus et al., J. Plasma Phys., 83 (2017)]. While the original formulation has been obtained for Cartesian coordinates, we extend the formulation to curvilinear coordinates in this paper. For the discretization in time, we discuss several (semi-)implicit methods either based on a Hamiltonian splitting or a discrete gradient method combined with an antisymmetric splitting of the Poisson matrix and discuss their conservation properties and computational efficiency.

Structure-preserving vs. standard particle-in-cell methods: The case of an electron hybrid model

We applied two numerical methods both belonging to the class of finite element particle-in-cell methods to a four-dimensional (one dimension in real space and three dimensions in velocity space) hybrid plasma model for electrons in a stationary, neutralizing background of ions. Here, the term hybrid means that (energetic) electrons with velocities close to the phase velocities of the model's characteristic waves are treated kinetically, whereas electrons that are much slower than the phase velocity are treated with fluid equations. The two developed numerical schemes are based on standard finite elements on the one hand and on structure-preserving geometric finite elements on the other hand. We tested and compared the schemes in the linear and in the nonlinear stage. We show that the structure-preserving algorithm leads to better results in both stages. This can be related to the fact that the spatial discretization results in a large system of ordinary differential equations that exhibits a noncanonical Hamiltonian structure. To such systems special time integration schemes with good conservation properties can be applied.

Non-canonical Hamiltonian structure and Poisson bracket for two-dimensional hydrodynamics with free surface

We consider the Euler equations for the potential flow of an ideal incompressible fluid of infinite depth with a free surface in two-dimensional geometry. Both gravity and surface tension forces are taken into account. A time-dependent conformal mapping is used which maps the lower complex half-plane of the auxiliary complex variable $w$ into the fluid’s area, with the real line of $w$ mapped into the free fluid’s surface. We reformulate the exact Eulerian dynamics through a non-canonical non-local Hamiltonian structure for a pair of the Hamiltonian variables. These two variables are the imaginary part of the conformal map and the fluid’s velocity potential, both evaluated at the fluid’s free surface. The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. The new Hamiltonian structure is a generalization of the canonical Hamiltonian structure of Zakharov (J. Appl. Mech. Tech. Phys., vol. 9(2), 1968, pp. 190–194) which is valid only for solutions for which the natural surface parametrization is single-valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface. In contrast, the new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to the Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations. In that case we identify powerful reductions that allow one to find general classes of particular solutions.

НОВЫЕ ИНТЕГРАЛЫ ДВИЖЕНИЯ И НЕКАНОНИЧЕСКАЯ ГАМИЛЬТОНОВА СТРУКТУРА ДЛЯ ДВУМЕРНОЙ ГИДРОДИНАМИКИ СО СВОБОДНОЙ ПОВЕРХНОСТЬЮ

We consider a potential motion of ideal incompressible fluid with a free surface and infinite depth in two dimensions with gravity forces and surface tension. A time-dependent conformal mapping z(w, t) of the lower complex half-plane of the variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid’s surface. We study the dynamics of singularities of both z(w, t) and the complex fluid potential Π(w, t) in the upper complex half-plane of w. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian structure for a pair of the Hamiltonian variables (Dyachenko et al., submitted), the imaginary part of z(w, t) and the real part at Π(w, t) (both evaluated of fluid’s free surface). The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. New Hamiltonian structure is a generalization of the canonical Hamiltonian structure of (Zakharov, 1968) (valid only for solutions for which the natural surface parametrization is single valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface). In contrast, new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations as in (Lushnikov and Zubarev, 2018) with the powerful reductions which allowed to find general classes of particular solutions. In Eulerian case we show the existence of solutions with an arbitrary finite number N of complex poles in zw(w, t) and Πw(w, t) which are the derivatives of z(w, t) and Π(w, t) over w (Dyachenko et al., submitted). These solutions are not purely rational because they generally have branch points at other positions of the upper complex halfplane with generally the infinite number of sheets of the Riemann surface for z(w, t) and Π(w, t) (Lushnikov, 2016). The order of poles is arbitrary for zero surface tension while all orders are even for nonzero surface tension. We find that the residues of zw(w, t) at these N points are new, previously unknown constants of motion. These constants of motion commute with each other with respect to the Poisson bracket. There are more integrals of motion beyond these residues. If all poles are simple then the number of independent real integrals of motion is 4N for zero gravity and 4N-1 for nonzero gravity. For higher order poles the number of the integrals is increasing. These nontrivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics. Work of A. Dyachenko, P. Lushnikov and V. Zakharov was supported by state assignment «Dynamics of the complex materials».

Hidden symmetries, trivial conservation laws and Casimir invariants in geophysical fluid dynamics

From a manifestly invariant Lagrangian density based on Clebsch fields and suitable for geophysical fluid dynamics, non-trivial conservation laws and their associated symmetries are described in arbitrary coordinates via Noether’s first theorem. Potential vorticity conservation is however shown to be a trivial law of the second kind with no relevance to Noether’s first theorem. A canonical Hamiltonian formulation is obtained in which Dirac constraints explicitly include the possibly time-dependent metric tensor. It is shown that all Dirac constraints are primary and of the second class, which implies that no infinite-dimensional symmetry transformations of Clebsch fields exist and that Noether’s second theorem does not apply to the governing equations. Therefore, the considered Lagrangian density does not admit a symmetry associated with potential vorticity conservation via Noether’s two theorems. Finally, the corresponding non-canonical Hamiltonian structure with time-dependent strong constraints is derived using tensor components for arbitrary coordinates. The existence of Casimir invariants is linked to trivial conservation laws of the second kind and to symmetries that become hidden after a transformation away from canonical dynamical fields.

Hamiltonian structure and relative equilibria of an axisymmetric rigid body in a second degree and order gravity field: cylindrical and generalized hyperbolic equilibria

The full dynamics of an axisymmetric rigid body in a uniformly rotating second degree and order gravity field are investigated, where orbit and attitude motions of the body are coupled through the gravity. Compared with the classical orbital dynamics with the body considered as a point mass, the full dynamics is a higher-precision model in close proximity of the central body where the gravitational orbit–attitude coupling is significant, such as a spacecraft about a small asteroid or an irregular-shaped natural satellite about a planet. The full dynamics are modeled by using the non-canonical Hamiltonian structure, in terms of variables expressed in the frame fixed with the central body. A Poisson reduction is carried out by means of the axial symmetry of the body, and a reduced system with lower dimension, as well as its non-canonical Hamiltonian structure and equations of motion, is obtained through the reduction process. With the second-order potential, three types of relative equilibria are found to be possible: cylindrical equilibria, generalized hyperbolic equilibria, and conic equilibria, which are counterparts to cylindrical equilibria, hyperbolic equilibria, and conic equilibria of an axisymmetric rigid body in a spherical gravity field, respectively. The geometrical properties and existence of the cylindrical equilibria and generalized hyperbolic equilibria are investigated in detail. It has been found that compared with the classical results in a spherical gravity field, the relative equilibria in this study are more complicated and diverse. The most significant difference is that the non-spherical gravity field enables the existence of non-Lagrangian hyperbolic equilibria, called generalized hyperbolic, which cannot exist in a spherical gravity.

A reduced Landau-gyrofluid model for magnetic reconnection driven by electron inertia

An electromagnetic reduced gyrofluid model for collisionless plasmas, accounting for electron inertia, finite ion Larmor radius effects and Landau-fluid closures for the electron fluid is derived by means of an asymptotic expansion from a parent gyrofluid model. In the absence of terms accounting for Landau damping, the model is shown to possess a non-canonical Hamiltonian structure. The corresponding Casimir invariants are derived and use is made thereof, in order to obtain a set of normal field variables, in terms of which the Poisson bracket and the model equations take a remarkably simple form. The inclusion of perpendicular temperature fluctuations generalizes previous Hamiltonian reduced fluid models and, in particular, the presence of ion perpendicular gyrofluid temperature fluctuations reflects into the presence of two new Lagrangian invariants governing the ion dynamics. The model is applied, in the cold-ion limit, to investigate numerically a magnetic reconnection problem. The Landau damping terms are shown to reduce, by decreasing the electron temperature fluctuations, the linear reconnection rate and to delay the nonlinear island growth. The saturated island width, on the other hand, is independent of Landau damping. The fraction of magnetic energy converted into perpendicular kinetic energy also appears to be unaffected by the Landau damping terms, which, on the other hand, dissipate parallel kinetic energy as well as free energy due to density and electron temperature fluctuations.

Gyrofluid modeling and phenomenology of low-βe Alfvén wave turbulence

A two-field reduced gyrofluid model including electron inertia, ion finite Larmor radius corrections, and parallel magnetic field fluctuations is derived from the model of Brizard [Brizard, Phys. Fluids B 4, 1213 (1992)]. It assumes low βe, where βe indicates the ratio between the equilibrium electron pressure and the magnetic pressure exerted by a strong uniform magnetic guide field, but permits an arbitrary ion-to-electron equilibrium temperature ratio. It is shown to have a noncanonical Hamiltonian structure and provides a convenient framework for studying kinetic Alfvén wave turbulence, from magnetohydrodynamics to sub-de scales (where de holds for the electron skin depth). Magnetic energy spectra are phenomenologically determined within energy and generalized cross-helicity cascades in the perpendicular spectral plane. Arguments based on absolute statistical equilibria are used to predict the direction of the transfers, pointing out that, within the sub-ion range, the generalized cross-helicity could display an inverse cascade if injected at small scales, for example by reconnection processes.

Structure and structure-preserving algorithms for plasma physics

Hamiltonian and action principle (HAP) formulations of plasma physics are reviewed for the purpose of explaining structure preserving numerical algorithms. Geometric structures associated with and emergent from HAP formulations are discussed. These include conservative integration, which exactly conserves invariants, symplectic integration, which exactly preserves the Hamiltonian geometric structure, and other Hamiltonian integration techniques. Basic ideas of variational integration and Poisson integration, which can preserve the noncanonical Hamiltonian structure, are discussed. Metriplectic integration, which preserves the structure of conservative systems with both Hamiltonian and dissipative parts, is proposed. Two kinds of simulated annealing, a relaxation technique for obtaining equilibrium states, are reviewed: one that uses metriplectic dynamics, which maximizes an entropy at fixed energy, and the other that uses double bracket dynamics, which preserves Casimir invariants. Throughout, applications to plasma systems are emphasized. The paper culminates with a discussion of geometric electromagnetic particle-in-cell [Kraus et al., J. Plasma Phys. (to be published); e-print arXiv:1609.03053v1 [math.NA]], a particle in cell code that incorporates Hamiltonian and geometrical structure preserving properties.

Variational integrators for reduced magnetohydrodynamics

Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics equations with applications to both fusion and astrophysical plasmas, possessing a noncanonical Hamiltonian structure and consequently a number of conserved functionals. We propose a new discretisation strategy for these equations based on a discrete variational principle applied to a formal Lagrangian. The resulting integrator preserves important quantities like the total energy, magnetic helicity and cross helicity exactly (up to machine precision). As the integrator is free of numerical resistivity, spurious reconnection along current sheets is absent in the ideal case. If effects of electron inertia are added, reconnection of magnetic field lines is allowed, although the resulting model still possesses a noncanonical Hamiltonian structure. After reviewing the conservation laws of the model equations, the adopted variational principle with the related conservation laws is described both at the continuous and discrete level. We verify the favourable properties of the variational integrator in particular with respect to the preservation of the invariants of the models under consideration and compare with results from the literature and those of a pseudo-spectral code.

Body-fixed orbit-attitude hovering control over an asteroid using non-canonical Hamiltonian structure

The orbit-attitude hovering means that both the position and attitude of the spacecraft are kept to be stationary in the asteroid body-fixed frame. The orbit-attitude hovering is discussed in the framework of the gravitationally coupled orbit-attitude dynamics, also called the full dynamics, in which the spacecraft is modeled as a rigid body to take into account the gravitational orbit-attitude coupling naturally. A feedback hovering control law is proposed by using the non-canonical Hamiltonian structure of the problem, which is consisted of two potential shapings and one energy dissipation. The first potential shaping is to create an artificial equilibrium at the desired hovering position-attitude. Then, the second potential shaping modifies the potential further so that the artificial equilibrium is a minimum of the modified Hamiltonian on the invariant manifold. Finally, the energy dissipation leads the motion to converge asymptotically to the minimum of the modified Hamiltonian, i.e., the artificial equilibrium for hovering. The feasibility of the hovering control law is verified through numerical simulations. The proposed hovering control law has a simple form and can be implemented by the spacecraft autonomously with little computation. This feature can be attributed to the utilization of the Hamiltonian structure and natural dynamical behaviors of the system in the control law design.

Development of variational guiding center algorithms for parallel calculations in experimental magnetic equilibria

Structure-preserving algorithms obtained via discrete variational principles exhibit strong promise for the calculation of guiding center test particle trajectories. The non-canonical Hamiltonian structure of the guiding center equations forms a novel and challenging context for geometric integration. To demonstrate the practical relevance of these methods, a prototypical variational midpoint algorithm is applied to an experimental magnetic equilibrium. The stability characteristics, conservation properties and implementation requirements associated with the variational algorithms are addressed. Furthermore, computational run time is reduced for large numbers of particles by parallelizing the calculation to use general-purpose graphics processing unit hardware.

Hamiltonian structure of a drift-kinetic model and Hamiltonian closures for its two-moment fluid reductions

We address the problem of the existence of the Hamiltonian structure for an electrostatic drift-kinetic model and for the related fluid models describing the evolution of the first two moments of the distribution function with respect to the parallel velocity. The drift-kinetic model, which accounts for background density and temperature gradients as well as polarization effects, is shown to possess a noncanonical Hamiltonian structure. The corresponding Poisson bracket is expressed in terms of the fluid moments and it is found that the set of functionals of the zero order moment forms a sub-algebra, thus automatically leading to a class of one-moment Hamiltonian fluid models. In particular, in the limit of weak spatial variations of the background quantities, the Charney-Hasegawa-Mima equation, with its Hamiltonian structure, is recovered. For the set of functionals of the first two moments, which, unlike the case of the Vlasov equation, turns out not to form a sub-algebra, we look for closures that lead to a closed Poisson bracket restricted to this set of functionals. The constraint of the Jacobi identity turns out to select the adiabatic equation of state for an ideal gas with one-degree-of-freedom molecules, as the only admissible closure in this sense. When the so called δf ordering is applied to the model, on the other hand, a Poisson bracket is obtained if the second order moment is a linear combination of the first two moments of the total distribution function. By means of this procedure, three-dimensional Hamiltonian fluid models that couple a generalized Charney-Hasegawa-Mima equation with an evolution equation for the parallel velocity are derived. Among these, a model adopted by Meiss and Horton [Phys. Fluids 26, 990 (1983)] to describe drift waves coupled to ion-acoustic waves, is obtained and its Hamiltonian structure is provided explicitly.

A Hamiltonian fluid-kinetic model for a two-species non-neutral plasma

A model for describing the dynamics of a pure electron plasma in the presence of a population of massive charged particles is presented. The model couples the fluid dynamics of the pure electron plasma with the dynamics of the massive particle population, the latter being treated kinetically. The model is shown to possess a noncanonical Hamiltonian structure and to preserve invariants analogous to those of the two-dimensional (2D) Euler equation for an incompressible inviscid fluid, and of the Vlasov equation. The Hamiltonian structure of the model is used to derive a set of stability conditions for rotating coherent structures of the two-species system, in the case of negatively charged massive particles. According to these conditions, stability is attained if both the equilibrium distribution function of the kinetic species and the equilibrium density of the electron fluid are monotonically decreasing functions of the corresponding single-particle energies in the rotating frame. For radially confined equilibria near the axis, the stability condition corresponds to the existence of a finite interval of rotation frequencies for the reference frame, with the upper bound determined by the presence of the kinetic population.

On the nonlinear stability of relative equilibria of the full spacecraft dynamics around an asteroid

The full dynamics of a spacecraft around an asteroid, in which the gravitational orbit–attitude coupling is considered, has been shown to be of great value and interest. Nonlinear stability of the relative equilibria of the full dynamics of a rigid spacecraft around a uniformly rotating asteroid is studied with the method of geometric mechanics. The non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are given in the differential geometric method. A classical kind of relative equilibria of the spacecraft is determined from a global point of view, at which the mass center of the spacecraft is on a stationary orbit, and the attitude is constant with respect to the asteroid. The conditions of nonlinear stability of the relative equilibria are obtained with the energy-Casimir method through the semi-positive definiteness of the projected Hessian matrix of the variational Lagrangian. Finally, example asteroids with a wide range of parameters are considered, and the nonlinear stability criterion is calculated. However, it is found that the nonlinear stability condition cannot be satisfied by spacecraft with any mass distribution parameters. The nonlinear stability condition by us is only the sufficient condition, but not the necessary condition, for the nonlinear stability. It means that the energy-Casimir method cannot provide any information about nonlinear stability of the relative equilibria, and more powerful tools, which are the analogues of the Arnold’s theorem in the canonical Hamiltonian system with two degrees of freedom, are needed for a further investigation.

Equilibrium attitude and nonlinear attitude stability of a spacecraft on a stationary orbit around an asteroid

The classical problem of attitude stability in a central gravity field is generalized to that on a stationary orbit around a uniformly-rotating asteroid. This generalized problem is studied in the framework of geometric mechanics. Based on the natural symplectic structure, the non-canonical Hamiltonian structure of the problem is derived. The Poisson tensor, Casimir functions and equations of motion, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. The equilibrium of the equations of motion, i.e. the equilibrium attitude of the spacecraft, which corresponds to a stationary point of the Hamiltonian constrained by Casimir functions, is determined from a global point of view. Nonlinear stability conditions of the equilibrium attitude are obtained in a modified energy-Casimir method. The nonlinear attitude stability is then investigated versus three parameters of the asteroid, including the ratio of the mean radius to the orbital radius, the harmonic coefficients C20 and C22. We find that when the spacecraft is located on the intermediate-moment principal axis of the asteroid, the nonlinear stability domain can be totally different from the classical Lagrange region on a circular orbit in a central gravity field. With the Lie group framework uncovered by geometric mechanics, besides determination of the equilibrium from a global point of view and the energy-Casimir method for nonlinear stability, several other powerful techniques can be performed, such as the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.