Classical and quantum fold catastrophe in the presence of axial symmetry

We introduce a family of Hamiltonians with two degrees of freedom, axial symmetry and complete integrability. The potential function depends on coordinates and one control parameter. A fold catastrophe typically occurs in such a family of potentials and its consequences on the global dynamics are investigated through the energy-momentum map which defines the singular fibration of the four-dimensional phase space. The two inequivalent local canonical forms of the catastrophe are presented: the first case corresponds to the appearance of a second sheet in the image of the energy-momentum map while the second case is associated with the breaking of an already existing second sheet. A special effort is placed on the description of the singularities. In particular, the existence of cuspidal tori is related to a second-order contact point between the energy level set and the reduced phase space. The quantum mechanical aspects of the changes induced by the fold catastrophe are investigated with the quantum eigenstates computed for an octic potential and are interpreted through the quantum-classical correspondence. We note that the singularity exposed in this paper is not an obstruction to a global definition of action-angle variables.

Figure 1

Typical open subsets of the image of the energy-momentum map for a two-degree-of-freedom completely integrable system when the first action is a global action. The lattice of dots represents the joint spectrum of commuting observables and the filled quadrilaterals describe the evolution of an elementary cell along one closed loop. (a) Open subset with only regular values. (b) Open subset with regular values and one singular value indicated by the circle. (c) Open subset with two sheets of regular values glued along a singular dashed curve.

Figure 2

Reduced phase spaces of a two-degree-of-freedom SO(2)-invariant Hamiltonian in the space of invariant polynomials $(X,K,Y)$. (a) Singular reduced phase space for $m=0$. (b) Smooth reduced phase space for $m=2$.

Figure 3

Fold catastrophe: the germ ($\lambda =0$, solid curve) and its unfolding for $\lambda =-0.1$ (dashed line) and 0.1 (dotted line).

Figure 4

Local effective potential $V_{\mathrm{eff}}^{\mathrm{loc},-}(\rho ;m;\lambda )$ for ${\rho }_0=2$ and a fixed set of five values of angular momentum $m$. The effective potential with $m=0$ is represented by a thick curve. The angular momentum increases [values: $m_1=0.5$, $m_2=0.8$, $m_{\mathrm{deg},\mathrm{I}}\left(0.1\right)\simeq 0.903$, $m_3=1.0$] when jumping from one potential curve to the one above. Filled squares indicate degenerate critical points. Control parameter $\lambda =\left(\mathrm{a}\right)-0.1$, (b) 0, and (c) 0.1. Positions of the minima and maxima of the effective potentials are indicated by the dashed curves.

Figure 5

Five different cases of intersections between the reduced phase spaces and the energy level sets. Left column: three-dimensional view in the $(X,K,Y)$ space of invariant polynomials. The surfaces drawn in continuous curves are the reduced phase spaces and the level sets of the energy are drawn with dashed lines. The intersections of these two surfaces are drawn as thick curves whose individual points lift up to ${\mathbb{S}}^1$ circles. Right column: projection of the reduced phase space (gray domain) and the projection of the energy level sets (dashed curves) in the $(X,Y)$ space of invariant polynomials. (a),(b) Regular intersection. (c),(d) First-order contact point, ${\mathbb{S}}^1$ intersection. (e),(f) First-order contact point, locally bitoruslike intersection. (g),(h) Second-order contact point, locally cuspidal-like intersection. (i),(j) Three regular intersections.

Figure 6

Projections of the reduced phase spaces (gray domains) and energy level sets (dashed curves) in the $(X,Y+X)$ space of invariant polynomials for the case I with ${\rho }_0=2$. The symbols of Fig. 5 are used to label the singular intersection points.

Figure 7

Images of the energy-momentum map in case I. The preimage of a point in the light gray areas is one torus while the preimage of a point in the dark gray area consists of two tori. Selected points in the image of the energy-momentum map are marked by a cross and the figures attached to them show the effective potential (dashed curve), the energy (horizontal dashed curve), and the conjugate momenta $p_{\rho }$ (solid curve). Control parameter $\lambda =\left(\mathrm{a}\right)-0.1$, (b) 0, and (c) 0.1.

Figure 8

Local effective potential $V_{\mathrm{eff}}^{\mathrm{loc},+}$ $(\rho ;m;\lambda )$ for ${\rho }_0=2$ and a fixed set of five values of angular momentum $m$. The effective potential with $m=0$ is represented by a thick curve. The angular momentum increases [values: $m_1=0.5$, $m_2=0.8$, $m_{\mathrm{deg},\mathrm{II}}\left(0.1\right)\simeq 0.886$, $m_3=1.0$] when jumping from one potential curve to the one above. Filled squares indicate degenerate critical points. Positions of the minima and maxima of the effective potentials are indicated by the dashed curves. Control parameter $\lambda =\left(\mathrm{a}\right)-0.1$, (b) 0, and (c) 0.1.

Figure 9

Projections of the reduced phase spaces (gray domains) and energy level sets (dashed curves) in the $(X,Y+X)$ space of invariant polynomials for the case II with ${\rho }_0=2$. The symbols of Fig. 5 are used to label the singular intersection points.

Figure 10

Images of the energy-momentum map in case II. The preimage of a point in the light gray areas is one torus while the preimage of a point in the dark gray areas consists of two tori. Selected points in the image of the energy-momentum map are marked by a cross and the figures attached to them show the effective potential (dashed curve), the energy (horizontal dashed curve), and the conjugate momenta $p_{\rho }$ (solid curve). Control parameter $\lambda =\left(\mathrm{a}\right)-0.1$, (b) 0, and (c) 0.1.

Figure 11

Images of the energy-momentum map obtained from the octic potential $V_{\mathrm{oct}}$ [Eq. (13)] for ${\rho }_1=1.0$ and ${\rho }_2=1.5$. The joint spectrum of commuting observables is represented by small dots and is determined with the Planck constant set to $\hslash =0.05$. The insets shows the octic potential $V_{\mathrm{oct}}$. (a) Control parameter $\mu =-0.1$. (b) Control parameter $\mu =0$. The dashed rectangle delimits the area that is enlarged in Fig. 12a. (c) Control parameter $\mu =0.1$. The dashed rectangle delimits the area that is enlarged in Fig. 12b.

Figure 12

(a) Enlargement of Fig. 11b in the vicinity of the singularity. (b) Enlargement of the two sheets seen in Fig. 11c.

Figure 13

Images of the energy-momentum map obtained from the octic potential $V_{\mathrm{oct}}$ [Eq. (13)] for ${\rho }_1=1.5$ and ${\rho }_2=1.0$. The joint spectrum of commuting observables is represented by small dots and is determined with the Planck constant set to $\hslash =0.05$. The insets shows the octic potential $V_{\mathrm{oct}}$. (a) Control parameter $\mu =-0.1$. The dashed curves delimit the area that is enlarged in Fig. 14. (b) Control parameter $\mu =0$. (c) Control parameter $\mu =0.1$.

Figure 14

(a) Enlargement of Fig. 13a. The quantum eigenstates have been determined with the Planck constant $\hslash =0.005$. The energy axis is shifted by a linear function $\Delta h$ of the angular momentum $m$: $\Delta h\left(m\right)=1.6m-9.06$. (b)–(k) Effective potential (dashed curve), energy (horizontal dashed line), and radial momentum $\pm p_{\rho }\left(\rho \right)$ (solid curves) are drawn for ten representative points in the image of the energy-momentum map. The small filled circles represent $\pm p_{\rho }\left(\rho \right)$ curves reduced to one point. The evolution of a torus from (b) to (k) is seen in the continuous curves. (b) and (k) have two regular tori: one is drawn as a continuous curve and the other one is drawn as a dotted curve.