# Stability of Hamiltonian equilibria

Post-publication activity

Curator: James Howard

An equilibrium $$z_0=(q_0,p_0) \in \mathbb{R}^{2n}$$ of an autonomous Hamiltonian flow is Lyapunov stable if all nearby orbits remain close to $$z_0$$ for all forward time, linearly stable if all orbits of the tangent flow are bounded, and spectrally stable if all eigenvalues of the tangent flow are pure imaginary. For example, consider the nonlinear pendulum of length $$l$$ and mass $$m\ ,$$ as shown in Figure 1 and described by the Hamiltonian $\tag{1} H(\theta,p_{\theta}) = \frac{1}{2m l^2} p_{\theta}^2 + mgl(1-\cos \theta),$

where $$p_{\theta}= m l^2\dot \theta$$ is the momentum canonically conjugate to the angle $$\theta \ .$$ Setting $$\partial H /\partial \theta = \partial H/\partial p_{\theta} = 0$$ gives the two equilibria, $$z_1 = (0,0)$$ and $$z_2 = (\pi,0) \ .$$ depicts the phase space $$(\theta,p_{\theta})$$ for this simple system, showing the equilibria $$z_1$$ and $$z_2\ .$$ As we shall see, the central equilibrium $$z_1$$ is not only linearly but also Lyapunov stable, while $$z_2$$ is always unstable.

## Introduction

Stability of motion is one of the oldest problems in mathematical physics, with important contributions dating back to the eighteenth century. In this article we discuss stability of equilibria of Hamiltonian flows. Stability of periodic orbits, by Floquet theory, and of quasiperiodic orbits, by the powerful Kolmogorov-Arnold-Moser Theory, are treated elsewhere in this encyclopedia.

Recall that a vector field of an autonomous Hamiltonian flow can be written as $\tag{2} \dot z = J \nabla H,$ where $$\nabla H$$ is the gradient of $$H$$ and $\tag{3} J = \left ( {\begin{array}{*{20}c} 0 & I_n \\ -I_n & 0\\ \end{array}} \right )$ is the Poisson matrix, with $$I_n$$ the $$n\times n$$ identity matrix. Typically we write $$z = (q,p)$$ with an $$n$$-dimensional configuration $$q$$ and corresponding canonical momenta $$p$$. A Hamiltonian system is called "natural" if $\tag{4} H(q,p) = T(q,p) + U(q)$ where $$T$$ is the "kinetic energy", and $$U$$ is the potential energy.

Since $$J$$ is nonsingular, a point $$z_0$$ is an equilibrium of (2) iff $$\nabla H(z_0) = 0$$. The simplest study of stability of $$z_0$$ proceeds by linearizing the equations, assuming $$\delta z = z - z_0$$ is infinitesimal, to obtain the variational equations, $\tag{5} \delta \dot z = L \delta z$ where the constant matrix $$L = J D^2 H(z_0)$$ is the linearization, with $$D^2 H$$ the Hessian matrix of second derivatives. Since $$D^2 H$$ is symmetric it follows that $$L$$ is a Hamiltonian matrix, i.e. $$L^T J + JL = 0$$. The solution of (5) is called the tangent flow, which, under the assumption of distinct eigenvalues, takes the form $\tag{6} \delta z(t) = \sum_{j=1}^{2n} c_j {\bf v}_j e^{\sigma_j t}$ where $$\sigma_j$$ are the eigenvalues, and $${\bf v}_j$$ the eigenvectors of $$L$$. The eigenvalues are the roots of the characteristic polynomial $\tag{7} P(\sigma) = \det(L -\sigma I).$ It is not difficult to show that the eigenvalues of a Hamiltonian matrix come in pairs $$\pm \sigma$$ (Arnold, 1980). Consequently, (6) has exponentially growing terms unless all of the eigenvalues of $$L$$ lie on the imaginary axis. Thus the problem of linear stability effectively reduces to finding eigenvalues and eigenvectors of the Hamiltonian matrix $$L$$.

## Historical Remarks

An historical account of progress in stability analysis for Hamiltonian systems reads like a Who's Who in Mathematics and Physics. For the computation of linear stability, and in particular spectral stability, important early contributions, dealing with properties of the zeros of polynomials, were made by Cauchy, Lagrange, Fourier, and Hermite, culminating in the great theorems of Sturm, which provided necessary and sufficient conditions that a given real polynomial have all its zeros in a specified real interval. The relevance of these seminal studies in the theory of equations to orbital stability was recognized and developed further by Airy and Clifford, and subsequently carried to completion by Maxwell and Routh. Maxwell, in his celebrated Adams Prize Essay of 1857 on the stability of the rings of Saturn obtained linear stability criteria in two and three degrees of freedom. Routh (1877) devised two different methods, the first based on Clifford's idea of forming the polynomial whose zeros are the twofold sums of the zeros of the characteristic polynomial, the second based on the Cauchy Index Theorem, coupled with Sturm's Theorem. Routh's investigations on the stability of governors evolved into the field now known as control theory. His second method is still widely used for dissipative systems, together with the equivalent method of Hurwitz.

Stability considerations for equilibria of Hamiltonian flows differ markedly from those encountered in dissipative systems, where the existence of attractors and repellors results in asymptotic stability. In contrast, stability of Hamiltonian systems is at best neutral rather than hyperbolic. For example, Routh-Hurwitz criterion cannot be used since it explicitly assumes that no eigenvalues lie on the imaginary axis. Other techniques, such as the powerful Lyapunov function method are widely used in dissipative systems and find occasional use in Hamiltonian systems (Rumyantsev and Sosnitskii, 1992). The familiar Lagrangian method (Goldstein, 1980) is applicable only when the kinetic energy is positive definite, excluding such important examples as the restricted three-body problem (Broucke, 1969).

Recently a new method was devised which overcomes all the above problems, yielding explicit linear stability bounds for Hamiltonian equilibria in arbitrary dimension. The method is based on two innovations: (i) introduction of a "reduced characteristic equation" of degree half that of the usual characteristic equation, (ii) use of Sturm's theorem to relate the eigenvalues of the reduced characteristic equation to spectral stability. More generally, nonlinear stability can be proven for the case (4) when $$T(q,p)$$ is a positive definite, quadratic function of $$p$$ and equilibrium occurs at a local minimum of the potential energy $$U(q)$$. In addition, Dirichlet's theorem of 1846 shows that the equilibrium is Lyapunov stable whenever the Hessian $$D^2H(z_0)$$ is positive definite (Krechetnikov and Marsden, 2007). Significant progress has also been made on converse stability, where conditions are obtained for instability of non-natural systems. See (Rumyantsev and Sosnitskii) for a recent review.

## Types of stability

There are many notions of stability of equilibria, but only three of these are of primary importance for Hamiltonian systems.

DEFINITION: An equilibrium $$z_0 \in \mathbb{R}^{2n}$$ is Lyapunov stable (nonlinearly stable) if for every neighborhood $$V$$ of $$z_0$$, there exists a neighborhood $$U\subseteq V$$ such that $$z(0) \in U \implies z(t) \in V$$ for all forward time.

DEFINITION: An equilibrium is linearly stable if all orbits of the tangent flow are bounded for all forward time.

Thus, nonlinear stability is a much stronger property than linear stability, as the sets $$U$$ and $$V$$ do not have to be infinitesimally small.

DEFINITION: An equilibrium is spectrally stable if all eigenvalues of its linearization are pure imaginary.

Note however, that spectral stability does not imply linear stability. Nevertheless, it can be shown that the boundaries of linear and spectral stability coincide. The precise relation is that an equilibrium is linearly stable if and only if it is spectrally stable and all its Jordan blocks are one-dimensional (Abraham and Marsden, 1978; Arnold 2006). Thus,

Nonlinear stability $$\implies$$ Linear stability $$\implies$$ Spectral stability,

but not vice versa. A famous counterexample is the Cherry Hamiltonian (Cherry, 1926) $\tag{8} H = -\frac{\omega_1}{2}(p_1^2+q_1^2) +\frac{\omega_2}{2} (p_2^2+q_2^2)+\frac{\alpha}{2}[2 q_1 p_1 p_2-q_2 (q_1^2-p_1^2)]$

where $$\omega_1$$ and $$\omega_2$$ are adjustable frequencies and $$\alpha$$ is a nonlinearity parameter. In spite of the linear stability of the origin ($$\sigma_{1,2} = \pm i\omega_1$$, and $$\sigma_{3,4} = \pm i\omega_2$$), for the case that $$\omega_2 = 2\omega_1$$, an explicit solution shows that the nonlinear terms lead to explosive growth. We shall return to this example below.

Other types of stability encountered in Hamiltonian flows are orbital stability, which describes the divergence of two neighboring orbits, regarded as point sets, and structural stability, which describes the sensitivity (or insensitivity) of the qualitative features of a flow to changes in parameters. See Stability for details.

A general method for analyzing stability in Hamiltonian systems is the energy momentum method which uses a combination of energy and momentum to provide a Lyapunov function. A special case of this is the energy Casimir method that uses Casimir functions, functions that commute with every other function under the Poisson bracket. These ideas were developed by Arnold (who originally applied them to fluids) and Marsden and his collaborators (see Marsden and Ratiu (1994) for references). The role that dissipation plays in either stabilizing or destabilizing equilibria is discussed in Bloch, Krishnaprasad, Marsden and Ratiu (1994, 1996). See also references therein including to the work of Chetaev.

Since $$L$$ is a real matrix, its eigenvalues also come in complex conjugate pairs. Hence, eigenvalues occur in the following configurations:

• imaginary pairs $$\pm i\omega,\, \omega\in \mathbb{R}^+$$
• real pairs $$\pm \sigma,\, \sigma \in \mathbb{R}^+$$
• complex quadruplets $$\pm a\pm ib,\, a,b\in \mathbb{R}^+$$
• $$\sigma=0 \ .$$

Moreover, $$\pm\sigma$$, and the complex conjugates $$\pm\sigma^*$$ all have the same multiplicity and Jordan block structure, while a zero eigenvalue has even multiplicity (Arnold, 2006)

Now consider a Hamiltonian $$H(z,\mu)$$ that depends smoothly on a set of parameters $$\mu$$ so that its eigenvalues also vary continuously with $$\mu$$. It follows that an equilibrium can loose spectral (and therefore linear) stability in only two ways:

• a pair of imaginary eigenvalues merge at 0 and split onto the real axis (saddle-node bifurcation)
• a pair of imaginary eigenvalues collide at a nonzero point and split off into the complex plane, forming a complex quadruplet (Krein bifurcation)

Of course, combinations of these configurations or eigenvalues of multiplicity greater than one are possible. The essential fact is that every equilibrium on the boundary of spectral stability must have a multiple eigenvalue which is either zero or nonzero imaginary. However, not every equilibrium satisfying one of these conditions is actually on the boundary of spectral stability! This is a consequence of the existence of additional invariants for Hamiltonian flows which may prevent an eigenvalue pair from leaving the imaginary axis. These are the Krein Signatures, which we now describe.

### Krein's theorem

When two pairs of eigenvalues $$\sigma=\pm i\omega$$ meet on the imaginary axis (Krein collision) they may either move out into the complex plane (Krein bifurcation) or simply pass through each other, remaining on the imaginary axis. Which eventually actually occurs depends on a special invariant peculiar to Hamiltonian flows, the Krein signature. For an isolated eigenvalue $$\sigma_k$$, this signature is $\tag{9} s_k = sgn (\xi_k^T JL \xi_k).$ where $$\xi_k$$ is an associated eigenvector (Arnold and Avez, 1968, Meiss, 2007).

If a pair of pure imaginary eigenvalues $$\sigma_1$$ and $$\sigma_2$$ collide and $$s_1=s_2$$, then the eigenvalues remain on the imaginary axis; if however, $$s_1 \ne s_2$$---the signature is mixed---and it is possible for the eigenvalues to leave the imaginary axis, forming a complex quadruplet. For more information on the mixed signature case see (Mackay, 1986). In general a pair of periodic orbits are formed and one speaks of a Hamiltonian-Hopf bifurcation. For example, suppose $$\alpha = 0$$ in the Cherry Hamiltonian (8), with both $$\omega_{1,2}$$ positive. It is then easily seen that the Krein signature is mixed, yet the two independent counter-rotating oscillators are completely unperturbed when $$\omega_1 - \omega_2$$ crosses zero. In the important special case of two degree of freedom natural flows, where the kinetic energy is positive definite, it is easy to see that that Krein bifurcations are impossible. Whether this extends to arbitrary dimension is an open question.

## Stability boundaries

Since its eigenvalues occur in $$\pm$$ pairs the characteristic polynomial (7) is even, $\tag{10} P_{2n}(\sigma) = \det (L - \sigma I) = \sigma^{2n} + A_1 \sigma^{2n-2} +\cdots + A_n$

where the $$A_k$$ may be expressed in terms of the elements of $$L$$ (Gantmacher, 1960). Introducing the new variable $$\tau = -\sigma^2$$ then gives the reduced characteristic polynomial, $\tag{11} Q_n(\tau) = (-1)^n P_{2n} = \tau^n - A_1 \tau^{n-1} + \cdots + (-1)^n A_n.$

Hence, a Hamiltonian equilibrium is spectrally stable iff all zeros of its reduced characteristic polynomial are positive. An equilibrium is on the boundary of spectral stability iff there is a zero at $$\tau=0$$ or a multiple zero at $$\tau>0$$ with mixed Krein signature. For example, for the pendulum (1), $L = \left( {\begin{array}{*{20}c} 0 & 1/ml^2 \\ -mgl\cos{\theta_0} & 0 \\ \end{array}} \right )\implies Q_1(\tau) = \tau - \det L=\tau - \omega_0^2 \cos{\theta_0}$

where $$\omega_0 = \sqrt{g/l}$$ is the frequency of small oscillations. For $$\theta_0 = 0$$ setting $$Q_1 = 0$$ gives $$\tau = \omega_0^2 \ ,$$ indicating spectral stability, while for $$\theta_0 = \pi$$ the result is $$\tau = -\omega_0^2 \ ,$$ indicating an unstable equilibrium.

### Two degrees of freedom

When $$n = 2$$, the characteristic equation (10) is quartic, but the reduced characteristic equation (11) is an easily-solved quadratic, $\tag{12} Q_2(\tau) = \tau^2 - A \tau + B = 0$ where (Gantmacher, 1959) $\tag{13} A = -\frac{1}{2} Tr (L^2) ,\qquad B = \det L.$

The zeroes of $$Q_2$$, $\tag{14} \tau = \frac{1}{2} A\pm \sqrt{\frac{1}{4}A^2 - B}$

are both non-negative iff $$A \ge 0$$ and $$0\le B\le A^2/4$$. The stability boundaries are thus

• zero root: $$B = 0$$, $$A\ge 0$$ (saddle-node)
• multiple root: $$B = \frac{1}{4} A^2$$ (Krein collision)

Figure 3 shows the stable region in the space of polynomial coefficients$$(A,B)$$. In general a saddle-node bifurcation occurs upon crossing the horizontal boundary $$B = 0$$, although a pitchfork bifurcation (MacKay, 1986) is possible when $$H$$ possesses certain spatial symmetries. If a locus of equilibria crosses the parabolic upper boundary a Krein bifurcation occurs. This is possible only if the Krein signatures of the merging eigenvalues are mixed. If $$H$$ is natural (positive definite kinetic energy) a Krein bifurcation cannot occur.

### Axisymmetric systems

Consider a natural Hamiltonian $$H = \frac{1}{2m} {\bf p}^2 + U({\bf r})$$, on the phase space $$z = ({\bf r}, {\bf p}) \in\mathbb{R}^6$$. Writing $${\bf r} =(\rho, z, \phi)$$, in cylindrical coordinates, suppose that $$H$$ is independent of the azimuth $$\phi$$. In this case, the angular momentum $$p_{\phi}$$ is conserved, and that part of the kinetic energy may be incorporated into an effective potential $\tag{15} U^e(\rho,z) = \frac{p_{\phi}^2}{2m\rho^2} + U(\rho,z)$ and the Hamiltonian can be viewed as having two-degrees of freedom, $\tag{16} H(\rho,z, p_{\rho}, p_z) = \frac{1}{2m} (p_{\rho}^2 + p_z^2) + U^e(\rho,z)$ with an implicit dependence on the constant angular momentum suppressed. A relative equilibrium of the original system is an equilibrium of this two-degree-of-freedom model and, when $$p_{\phi} \neq 0$$, corresponds to a circular orbit in 3D. Such points occur when $$p_{\rho} = p_z = 0$$ and $$\nabla U^e = 0$$. Stability of such an equilibria corresponds to orbital stability of the circular orbit.

#### Example: the Stark problem

An example of a relative equilibrium is the classical Stark problem (Howard, 1995a), in which a hydrogen atom is perturbed by a uniform electric field $${\mathbf F} = F \hat e_{z}$$ in the symmetry axis direction. The Hamiltonian becomes (16) with effective potential $U^e(\rho,z) = \frac{\mu} {2\rho^2} - \frac{1}{r} - z,$ $$r = \sqrt{\rho^2 + z^2}$$, $$\mu$$ the scaled value of the conserved $$\phi$$-component of angular momentum, and $$m = F = 1$$ in scaled atomic units.

This integrable system has relative equilibria at zero momentum and any critical point of the effective potential, $$\nabla U^e(\rho_0,z_0) = 0$$, giving $\rho_0^4 = \mu r_0^3,~~z_0 = r_0^3 \implies r_0(1-r_0^4)^2 = \mu .$ Since the kinetic energy is positive definite, stability is completely determined from the Hessian determinant, $$\Delta = \det D^2 U^e$$: it is not necessary to calculate the eigenvalues (this is implicit in the analysis of the Hessian). Figure 4 shows level sets of $$U^e$$ in scaled coordinates for three values of the angular momentum. Stable circular orbits occur when $$\Delta_0 > 0$$, corresponding to a local minimum of $$U^e$$ if $$U_{\rho\rho}^e >0$$ and a local maximum if $$U_{\rho\rho}^e <0$$. A saddle point, where $$\Delta_0 < 0$$, corresponds to an unstable circular orbit. Since $$H$$ is a natural two degree-of-freedom system, Krein bifurcations are impossible and stability can only be lost via a saddle-node or pitchfork bifurcation. Figure 4: Level sets of $$U^e$$ for the Stark Problem (a) $$\mu<\mu^* \ ,$$ (b) $$\mu = \mu^* \ ,$$ (c) $$\mu > \mu^* \ .$$

A critical point changes type when $$\Delta_0$$ passes through zero. It follows that $$\Delta_0$$ changes sign when $$\mu = \mu^* = 64\sqrt{3}/243 \ ,$$ at which point a stable-unstable pair of equilibria merge and disappear for $$\mu > \mu^*$$, as seen in Figure 4. Since $$H$$ is natural one obtains as an added bonus the Lyapunov stability of the elliptic points.

Other physically interesting, axisymmetric systems possessing relative equilibria include ion motion in Paul traps (Bluemel, 1995), planetary dust dynamics (Howard et. al., 1999), the problem of two fixed centers, and galactic dynamics.

#### Example: microwave ionization

As a example of a nonaxisymmetric system, consider a hydrogen atom perturbed by a circularly polarized microwave field with the electron orbit lying in the plane of polarization, which is described in a co-rotating frame by the autonomous Hamiltonian (Howard, 1992) $H = \frac {1}{2} p_{\rho}^2 + \frac {p_{\phi}^2}{2 \rho^2} -\omega p_{\phi}-\frac{1}{\rho} + F\rho \cos\phi .$

where $$p_{\phi} = \rho^2 (\dot\phi + \omega) \ ,$$ and $$\omega$$ and $$F$$ are the frequency and amplitude of the microwave field. Again we have a relative equilibrium, but with the additional complication of nonconstant $$p_{\phi}$$ and consequent indefinite kinetic energy. In this case stability does not correspond to the type of the critical point and a more sophisticated approach is needed. The equilibria are obtained from $$\nabla H = 0 \implies \phi_0 = (0,\pi),~p_{\phi}=\omega \rho_0^2 \ ,$$ where $\tag{17} \omega^2 \rho_0^3 - \sigma \rho_0^2 - 1 = 0$ Figure 5: Zero-velocity curves for microwave ionization problem with $$\mu = 800$$. There is a pole at the origin, a saddle at $$\rho = 8.962$$ (unstable orbit), and a maximum at $$\rho = -9.629$$ (stable orbit).

with $$\sigma = \cos\phi_0 =\pm 1$$. Working out the linearization gives a polynomial (12) with a coefficient $$B$$ that cannot change sign, while the discriminant $$\Delta$$ changes sign at $$\mu = \mu^* = 648$$, signalling a Krein bifurcation. In fact it is possible to work out the equilibrium locus $$B = B(A)$$ in closed form for this system, showing that $$\lim_{\mu\to 0} \Delta = 0$$ and $$\lim_{\mu\to \infty} B = 0$$. Thus, the equilibrium is linearly stable for $$\mu > \mu^*$$. It is not necessary to calculate the Krein signature or solve the cubic (17). Since this system does not have positive definite kinetic energy, the question of nonlinear stability requires KAM theory.

Although an effective potential does not exist for this problem, it is useful to employ the zero-velocity function as the locus of points where the kinetic energy vanishes, with scaled radius $$\rho \to \omega^2 \rho/F \ ,$$ $Z(\rho,\phi) = -\frac{1}{2} \rho^2 -\frac{\mu}{\rho} + \rho\cos\phi$

with dimensionless parameter $$\mu = \omega^4/F^3$$. Setting $$Z = E$$ then yields the Zero-Velocity Curves (ZVC), which form boundaries for trapped and untrapped motion. It is also easy to see that the critical points of $$Z$$ are identical to those of $$H \ ,$$ with the caveat that a stable orbits can occur at a maximum of $$Z$$. Figure 5, shows level sets of $$Z$$ for $$\mu = 800$$. An orbit with energy $$E$$ cannot cross a ZVC with the same energy.

## Stability in arbitrary dimension

When $$n = 3$$ it is possible to use Descartes' rule of signs on (11) to obtain stability bounds. When $$n > 3$$ however, it is preferable to employ Sturm's method (Dickson, 1939), which yields a full set of necessary and sufficient conditions for spectral stability.

### Sturm's theorem

This method (magic bullet) to characterize the zeroes of a polynomial $$Q(\tau)$$, widely used in the 19th century, consists in defining the Sturm sequence $$\{F_k(\tau)\}$$ by $$F_0(\tau) = Q(\tau)$$, $$F_1(\tau) = Q^{\prime}(\tau)$$. At each stage one divides $$F_{k-1}$$ by $$F_{k-2}$$ to get the quotient $$G_{k-1}$$ and remainder $$-F_k$$. Thus, one obtains $$G_{k-1}$$ and then $$F_k$$ for $$k\ge 2$$ by $\tag{18} F_{k-2}(\tau) = G_{k-1}(\tau) F_{k-1}(\tau) - F_k(\tau),\qquad deg F_k < deg F_{k-1}$ until at stage $$K$$, a constant $$F_{K}$$ is obtained. There is a multiple root iff $$F_{K} = 0$$. Let $$V(\tau)$$ be the total number of variations in sign in proceeding through the Sturm sequence at $$\tau$$ (ignoring zeroes). Then the number of distinct real roots in the interval $$(a,b]$$ is exactly $$V(a) - V(b)$$.

In the present case, to have stability, all the zeroes of $$Q(\tau)$$ are required to be non-negative real. By Sturm's theorem, this is true iff $$V(0) - V(\infty) = n$$, which can be achieved iff $$K = n$$. Note that the same conditions guarantee nonlinear stability for natural systems.

### Three degrees of freedom

The reduced characteristic equation (11) reads $\tag{19} \tau^3 - A\tau^2 + B\tau - C = 0$

where $\tag{20} A = -\frac{1}{2}\mbox{Tr} (L^2) ,\qquad 8B = [\mbox{Tr}(L^2)]^2 - 2 \mbox{Tr}(L^4) ,\qquad C = \det L.$

Working out the Sturm sequence for the cubic (19) shows that the motion is spectrally stable iff $A>0,\qquad B>0,\qquad C> 0,\qquad \Delta > 0.$

where $\Delta = 4(A^2-3B)(B^2-3AC) - (AB-9C)^3$

is the discriminant (Dickson, 1939).

The same conditions result upon applying Descartes' rule of signs to (20) to conclude that there are either 1 or 3 positive zeros and no negative zeros, provided that $$A, B, C$$ are all positive. Further requiring that the discriminant be non-negative excludes the possibility of complex roots. The 3D stable region is depicted in Figure 6. The stability boundaries are given by the plane $$C = 0$$ (saddle-node) and the quartic surface $$\Delta = 0$$ (Krein collision), with $$A,B > 0$$. Whether a Krein collision actually occurs when $$\Delta = 0$$ of course depends on the Krein signatures. If $$H$$ is natural, Krein collisions are impossible. For $$n \ge 4$$ Descartes' rule does not suffice but Sturm's theorem still applies. The quartic case is worked out in detail in (Howard and MacKay, 1987).

## Natural flows

While the above methods yield explicit linear stability bounds for arbitrary Hamiltonian equilibria in arbitrary dimension, they provide absolutely no information about nonlinear stability. Indeed, as the Cherry problem demonstrates, linear stability is no guarantee of nonlinear stability. However, there is an important subclass of Hamiltonians for which a sufficient condition for nonlinear stability may be obtained.

• Dirichlet's Theorem. Let $$z_0$$ be a locally quadratic equilibrium of the natural Hamiltonian, $$H = T(q,p) + U(q)\ ,$$ where $$T$$ is a positive definite, quadratic function of $$p$$. Then $$z_0$$ is Lyapunov stable.

In such cases linear stability is tantamount to nonlinear stability. For a proof see (MacKay, 1986). Examples of natural flows include the nonlinear pendulum and the Stark problem, both described above. A very important example of a non-natural flow is the restricted three-body problem (Broucke, 1969): the point is that though the $$n$$-body problem it is natural in the inertial coordinates, the restricted problem is formulated in a coordinate system that is rotating with the primary bodies, which introduces the Coriolis force.

The polynomial coefficients are especially simple for 2D natural flows: $A = \mbox{Tr}\, D^2 U , \qquad B = \det D^2U$ from which it can be shown that $$\Delta>0 \ ,$$ so that Krein bifurcations cannot occur. Whether this extends to higher dimension is not known. A stronger result applies to non-separable Hamiltonians (Krechetnikov and Marsden, 2007):

Lagrange-Dirichlet Theorem: Let the second variation of the Hamiltonian $$\delta^2 H$$ be definite at an equilibrium $$z_0 \ .$$ Then $$z_0$$ is stable.

For more information see (Arnold et al., 2006; Krechetnikov and Marsden, 2007).

## The elliptic Paul trap

As an example of a 3D natural flow consider two-ion motion in an elliptic Paul trap, whose averaged motion is described by the pseudopotential (Howard and Farrelly, 1994) $\tag{21} U(x,y,z) = \frac{1}{2} (\lambda_1^2 x^2 + y^2 + \lambda_2^2 z^2) + \frac{1}{r}$

where $$r = \sqrt{x^2 + y^2 + z^2}$$ is the inter-ion distance and $$\lambda_{1,2}$$ are dimensionless parameters depending on the trap geometry. This highly symmetric potential possesses three pairs of critical points, which form a set of Morse saddles (Poston and Stewart, 1980). It turns out that in general only one of the three pairs is stable. Figure 7 is a 3D contour plot of $$U(x,y,z)$$ for $$\lambda_1 = 0.4, \lambda_2 = 0.6$$, sliced along the $$z=0$$ plane, with the $$x$$-axis vertical. There is a pair of stable equilibria on the $$x$$-axis and an unstable pair along the $$y$$-axis. Figure 7: Perspective view of ellipsoidal potential. A pair of unstable Morse saddles on the $$y$$-axis can be seen in the foreground.