Rotation Forms and Local Hamiltonian Monodromy
The monodromy of torus bundles associated to completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach is that the residue-like formula can be shown to be local in a neighborhood of a singularity, hence allowing the definition of monodromy also in the case of non-compact fibers. This idea has been introduced in the literature under the name of scattering monodromy. We prove the coincidence of the two definitions with the monodromy of an appropriately chosen compactification.
A Liouville-Arnold integrable system is a map (called the map of integrals of motion or integral map) from a -dimensional symplectic manifold to such that the components , , of Poisson commute. Let denote a connected component of the set of regular values of and denote a connected component of the preimage . Assuming that the level sets of are compact, the Liouville-Arnold theorem Arnold1989 () states that is a -bundle over . If is not simply connected, then the -bundle over a simple closed path in may have non-trivial monodromy. Equivalently, there are no smooth action variables throughout Nekhoroshev1972 (); Duistermaat1980 (). In degree of freedom systems with a circle action, monodromy can be identified with an integer number. If the number of degrees of freedom is larger than , then could possibly have non-trivial second cohomology. In that case the Liouville-Arnold integrable system could have global action variables but have non-trivial Chern class or, equivalently, no corresponding global angle variables which together with the action variables give a symplectic chart Nekhoroshev1972 (); Duistermaat1980 ().
Non-trivial monodromy has been shown to exist in several integrable Hamiltonian systems such as the spherical pendulum Duistermaat1980 (); Cushman2015 (), the champagne bottle Bates1991 (), and the hydrogen atom in crossed electric and magnetic fields Cushman2000 (). In the mid-90’s it was realized that a common property of these systems was the existence of isolated, focus-focus, critical values in the image of . The presence of such focus-focus critical values causes a non-trivial fundamental group, , and it turns out that the corresponding bundle over a path in encircling the critical value has non-trivial monodromy Lerman1994 (); Matveev1996 (); Zung1997 (). This result, now referred to as the geometric monodromy theorem, has been further generalized to the non-Hamiltonian context Zung2002 (); Cushman2001 ().
In this paper we focus on degree of freedom systems where are smooth. The function is the Hamiltonian of a Hamiltonian vector field , while is the momentum of a Hamiltonian action whose infinitesimal action is . Establishing the non-triviality of monodromy along a closed path in such systems is often done through the study of the variation of the rotation number along . We give the definition of the rotation number in Section II, see Definition 4, where we discuss in detail how the non-trivial variation of the rotation number along is equivalent to the non-trivial monodromy of the bundle over . We only note here that the definition of the rotation number is based on a geometric construction but its computation is typically done through the evaluation of an (abelian) integral and the investigation of its dependence upon the values of the integrals of motion. Moreover, the variation of the rotation number has been used to describe fractional monodromy Efstathiou2007 (); Sugny2008 () and to define scattering monodromy Bates2007a ().
In the present work we relate the proofs of the non-triviality of monodromy based on the variation of the rotation number to a more geometric approach. In particular, we formalize an analytical computation of the rotation number through the notion of rotation 1-form (Definition 10), a closed 1-form whose integral over suitably defined orbit-segment of gives the rotation number up to a term which we prove to be unimportant for the variation. Moreover, we show that the variation is independent of the choice of the rotation 1-form provided that the latter satisfies a transversality condition (Definition 16).
It turns out that a rotation 1-form cannot be defined in the whole phase-space, but it must necessarily be singular on a subset, whose points we call poles. Such subset is essential for the non-triviality of monodromy. In all examples known to the authors, the set of poles is a 2-dimensional submanifold intersecting at a finite number of -orbits, cf. Section III. The main result in this article is the following theorem relating the analytic computation of the variation of the rotation number to the geometry of the set of poles of the rotation 1-form.
Consider a two-degree of freedom integrable Hamiltonian system , such that the fibers of are compact and connected. Consider a closed path in the set of regular values of and assume that there is a neighborhood of where is invariant under a Hamiltonian action generated by a momentum . Let be a rotation 1-form for the vector field , transversal to , and let be its polar locus, which we assume two-dimensional. Further, assume that transversally intersects at a finite number of values . Then the poles of the rotation 1-form in are a disjoint union of a finite number of -orbits , which we call polar orbits, and the monodromy number along , see Eq. (3), is given by
where is a loop in surrounding with appropriate orientation, see Figure 1.
Theorem 1 applies to any torus bundle, provided that a Hamiltonian circle action, leaving invariant, is defined in a neighborhood of the torus bundle. The Theorem reduces the problem of computing the variation of the rotation number to that of integrating the rotation -form along closed paths encircling the poles of . A method for constructing the rotation -form is given in Lemma 14. The integrals are real analogues of residues for the rotation -form around its set of poles. This is strongly reminiscent of the complex analytic approach of Ref. 20, where the variation of the rotation number is expressed as the integral around the pole(s) of a meromorphic -form.
The local form of an integrable Hamiltonian system in a neighborhood of a focus-focus critical point allows to apply Theorem 1 and obtain the following well known fact.
Let be a focus-focus critical point of and a simple closed path in the set of regular values of , such that is the only critical point in , where is the set bounded by . Then the monodromy number along is .
The main contribution of this paper does not lie in the computation of the monodromy, but in the systematic approach to monodromy through the variation of the rotation number and the expression of the latter as the integral of a rotation -form. More specifically, the monodromy number is given by the sum of the integral of the rotation -form along the cycles described in Theorem 1. Applying this approach to the case of focus-focus points yields as a consequence that the Hamiltonian monodromy relies only on the local structure of the foliation in a neighborhood of such points (cf. similar local approaches in Ref. 19 and Ref. 21). We preferred to present here the method in the easiest case of single focus-focus points and plan to apply it to more complicated cases, where additional difficulties appear, in a forthcoming work. In particular, we plan to deal with cases of non-isolated singularities such as the -resonance case. In such general cases, more complicated contribution given by Picard-Lefschetz formula can appear.
Moreover, understanding how monodromy is locally determined in the case of single focus-focus points permits a generalization of the notion of monodromy to completely integrable Hamiltonian systems having not necessarily compact fibers, avoiding the, frequently artificial, compactification of the fibers by adding suitable higher order terms to the Hamiltonian. We compare our local approach to monodromy based on the rotation -form to the notion of scattering monodromy introduced in Ref. 4. We show that the two concepts are similar, and we highlight the role played by the identification of incoming and outgoing asymptotic directions in scattering monodromy.
The plan of the paper is as follows. In Section II we give the definition of the rotation number and describe how the non-triviality of its variation is related to the non-triviality of monodromy. Then, we introduce rotation 1-forms and we show how they can be used to determine the variation of the rotation number. In Section III we give several examples of rotation 1-forms in specific examples of integrable Hamiltonian systems. In Section IV we study focus-focus singularities and show that the variation of the rotation number can be computed through the variation of an appropriate integral of a locally defined rotation -form. In Section V we define monodromy for non-compact fibrations and relate our results to scattering monodromy. We draw conclusions and give perspectives in Section VI.
Ii Monodromy and Rotation Number
As stated in the Introduction, in this work we restrict our attention to 2 degree of freedom integrable systems () under the very typical hypothesis that one of the integrals of motion is a function which is the momentum of a circle action , , with (where is possibly an open subset of the phase space). The other integral of motion is an -invariant function typically called Hamiltonian or energy of the system. For this reason, the map is often called energy-momentum map.
Consider a closed path in the set of regular values of and the -bundle . The monodromy of the -bundle is an automorphism of for any in the image of . Fixing a basis of , monodromy is then characterized by a matrix . On each fiber , , the existence of the -action gives a globally defined generator of . In a basis of the monodromy matrix has the form
The number is called the monodromy number and completely determines the topology of the -bundle . Therefore, the computation of monodromy boils down to the computation of the value of .
In this section we review the computation of monodromy through the variation of the rotation number. We first recall the definition of the rotation number and how it can be used to construct local action coordinates.
ii.1 Rotation Number and its Variation
In our setting, the momentum of the -action can be taken as an action coordinate for the system. A second action coordinate can be constructed in the following way. Consider a point in a regular fiber . Furthermore, let be the closed orbit of going through . The orbit of starting at will cross again at a point after a time , called first return time, giving a smooth function .
The rotation number is the minimal positive time it takes to flow with from to .
The rotation number is a function defined in and taking values in . With our definition, is smooth outside its zero level-set but is possibly discontinuous at . The set is typically a union of codimension-1 surfaces in and the function can possibly tend smoothly to zero from one side and smoothly to from the other.
Both, the first return time and the rotation number, are invariant under the flows of and , and hence are constant on the connected components of the level sets of . It follows that they are the pull-back via of functions defined on . With a little abuse of notation we will denote the rotation number and the first return time with the same name may they be defined in or in . The vector field defined as
can be shown to be Hamiltonian and -periodic Duistermaat1980 (); Cushman2015 (). It is hence associated to the second action coordinate wherever the function is smooth, that is, outside the set . Note that one can locally define a smooth action coordinate also at by adding, in a subset of the local neighborhood, an appropriate integer multiple of to so as to obtain a locally smooth function.
One of the most important singularities of the map , the focus-focus singularity, consists of an isolated point that is mapped by onto a point which is a puncture in . In this case the zero-set of locally consists of curves converging to , typically spiraling around Dullin2004a (). Considering a path that surrounds such singular value one can add the jumps of across such curves and obtain an integer multiple of .
Instead of the rotation number we could have used the rotation angle , a circle-valued function obtained by composing with the projection from to . The map can be shown to be smooth, while can have first-kind discontinuities with jump equal to .
The integer obtained by adding up the discontinuities of along , and dividing by , reveals the non-triviality of the -bundle over . It is connected to the non-existence of global action coordinates and we call it the variation of along . We formalize the notion of the variation of an -valued function along as follows.
Let be a function with a finite number of discontinuities , whose jumps across the discontinuities are respectively the real numbers
The variation of along is then defined as
Consider the function given by with parameterizing . Then is discontinuous at and the discontinuity jump is . Therefore .
Consider any step function . Then the discontinuity jumps must cancel so that , assuming that is continuous at . Therefore .
Note that , , also represents a periodic Hamiltonian vector field associated to the second action coordinate . Therefore, a variation of the rotation number by over implies a change of the corresponding action vector field by . Furthermore, since action vector fields generate a basis of the homology group we conclude that, going along , an initial cycle generated by is transported to the final cycle and therefore we have a non-trivial monodromy matrix. This comparison shows that the variation
measures the monodromy number and hence the non-triviality of the -bundle over .
Another way to obtain such integer is to consider the function that, being a map from a circle to itself, can possibly have a non-zero degree which is precisely the variation of along .
ii.2 Rotation 1-forms
In applications, the rotation number and its variation are typically computed by integrating a closed -form along the orbit . We formalize here this approach and clarify certain technical aspects of this computation.
A rotation 1-form is a -form , defined in an invariant subset of , such that is closed and . Points in the set are called poles and is called the polar set of .
The condition ensures that the integral of measures the natural time along the flow of when integrated along its orbits. Hence, one can use it to define a local angle coordinate along the orbits of . Moreover, the conditions in the definition imply that . The latter relation ensures that is an -invariant function, and therefore it descends to a function in the reduced space .
The polar set plays a central role in this work. For this reason we give the geometric intuition for the necessity of introducing and discuss its role and its properties. We first prove the following result partially characterizing .
Let be the generator of an action which is free outside fixed points and denote by the set of fixed points of the action. Consider the principal circle bundle defined by the flow of on . Then is such that the restriction of the circle bundle to defines a trivial principal circle bundle. Moreover, is a connection -form for the trivial circle bundle defined in .
Since the rotation -form satisfies and it is a connection -form for the principal circle bundle defined by the flow of on . Moreover, the condition implies that the curvature -form for the corresponding circle bundle is trivial and ensures the triviality of the bundle. Therefore, a rotation 1-form can be only defined on a set so that the restriction of the principal circle bundle to gives a trivial bundle. ∎
If is a fixed point of the action induced by then a rotation -form defined in a neighbourhood of must have a non-empty polar set with . Moreover, if the action is free in then must contain a two-dimensional manifold.
The rotation 1-form cannot be defined at since but whenever is defined. Therefore . Let now be a point in an -invariant open ball at which is defined. By invariance under the flow of , the form is defined in all points of the orbit through . Since we have that . If were trivial, then there would exist a disk in , bounded by the orbit , and then we would get the contradiction . Therefore, must be non-trivial and hence must contain a nonempty manifold passing through . The non-triviality of excludes simple possibilities of being contained in a manifold of dimension and . ∎
Note that Lemma 12 does not exclude the possibility that contains a manifold of dimension .
We now consider under what conditions a rotation -form can be defined and how it can be constructed. We start with the following result.
Suppose that the flow of defines a trivial principal circle bundle in . Then there exists a rotation -form without poles in .
Let be a smooth section for the bundle. Define an angle in as the time it takes for the flow of to move from the image of the section to a point . Then the -form can be shown to satisfy the requirements of Definition 10. The triviality of the principal bundle ensures that this -form is well-defined and has no poles. ∎
Then the idea for constructing a rotation -form is that given an action in we can obtain a trivial principal circle bundle by taking out a large enough set (which includes the points with non-trivial isotropy) so that in the remaining part we have a trivial principal circle bundle. Moreover, we have the following result.
Given as above, there always exists a set , finite union of submanifolds of of codimension at least , outside of which a rotation -form exists.
From the theory of CW complexes, one can always assume that admits a stratification of submanifolds of different codimension and a unique (contractible, open, and dense) cell of maximal dimension. The rotation -form can always be defined on the cell of maximal dimension since the corresponding principal circle bundle is trivial. ∎
Let be a fixed point of the circle action possibly defined in a neighborhood of and free except at . Then the circle action can be locally linearized as , where . In particular, the resulting principal circle bundle is isomorphic (up to orientation) to the Hopf fibration and is therefore non-trivial. This means that a rotation -form defined in a punctured neighborhood of a fixed point of the circle action must necessarily have a non-empty set of poles and the latter should have dimension at least . Assume that we take away the plane . A bundle section is given by
Then and , or
In what follows we assume that is a two-dimensional manifold, which is smooth outside fixed points of the action. This is a reasonable assumption given that the polar set of the rotation 1-form is a two-dimensional smooth manifold in all examples known to the authors, cf. Section III.
We make use of the following transversality notion.
A rotation -form with a two-dimensional manifold of poles is transversal to if has rank outside fixed points of the action.
Note that the rank of cannot equal since both and are invariant.
Consider a rotation 1-form transversal to , let , and assume that the circle action is free outside fixed points. Then is a smooth open one-dimensional manifold and for each the intersection consists of a finite number of orbits.
By our assumptions on , is a smooth two-dimensional manifold. Since the action is free on , the reduced is a one-dimensional manifold. By transversality to , is of rank , which implies that it reduces to a map of rank on . Therefore, the map is an immersion and it follows that its image is smooth one-dimensional.
For each , by transversality, is one-dimensional. From -invariance of and , is a union of orbits. There is a finite number of them by transversality of . ∎
Note that the manifold of poles is well-defined within the disk above which a circle action is well-defined. Of course, if as in many examples the action is global, then the manifold of poles is defined globally.
ii.3 Variation of the Rotation Number and Rotation 1-forms
Let , and let . As in the definition of the rotation number let be the orbit through and the segment of the orbit of on starting from and flowing until it meets at a point . We call the closed curve which is the result of joining with the curve . The latter is obtained by flowing along from for time , that is, until closing at .
The integration of along the paths and does not depend on the choice of the point in the fiber . Therefore, for we define
where is any point in . Note that is not defined and may not be extended by continuity whenever intersects .
The following facts hold:
Let be a closed path in the set of regular values which transversally intersects . Then
where is the rotation number and is given by Eq. (4).
Let . Then if and only if there exists a cycle , independent of the cycle defined by the -orbit, such that .
(a) Consider the representative of that goes from to along and then from to along the flow of for time . By construction, such a path is where is the time- orbit of from to . Then
where we used that , since . Therefore,
Parameterize by . The function is locally constant along since is a closed 1-form and the initial points for the construction of the cycles can be chosen so that these cycles form a cylinder. When meets a pole of then is not defined and the function has a discontinuity which, because of the local constancy, must be a jump discontinuity. Therefore, is a step function. The rotation number also only has jump discontinuities and these two facts, together with Eq. (5), imply that also has only jump discontinuities along . Therefore, using that all functions involved only have jump discontinuities, we obtain
Since is a step function we have , cf. Example 8. Therefore,
There may be fibers for such that . Thus and are not defined on these fibers. Nevertheless, is always defined (by construction) and, therefore, the difference extends to a well-defined function on such fibers.
Lemma 18 shows that is independent of the choice of the rotation 1-form and always equals . This means that we can choose in such a way so as to simplify the computation of the variation, even if it does not give the correct value for the rotation number on each fiber.
ii.4 Proof of the Main Theorem 1
To prove Theorem 1 consider a section of the bundle over and the -torus . Then consider the cylinder made up of orbit segments of as moves along , see Figure 3. Specifically, the orbit segment of starts at and ends at the first point where the orbit intersects . This construction defines the map sending to . In terms of homology classes in we have for some . Here is the homology class represented by any orbit of period .
Parameterize by with increasing along the traversing direction of and parameterize each orbit segment by with increasing along the flow of . Then is parameterized by and such choice fixes an orientation on .
Since is closed, Stokes’ theorem gives that equals the sum of the integral of along positively oriented cycles encircling the poles of on ,
The boundary of is , therefore
Moreover, the variation of along is given by
This concludes the proof of Theorem 1.
In this section we apply the concepts introduced in Section II to three specific examples: the champagne bottle, the spherical pendulum, and a system on the symplectic manifold .
iii.1 The Champagne Bottle
The champagne bottle consists of a particle in the plane which moves under the influence of a conservative force whose potential energy is
The phase space of this system is the cotangent bundle of , diffeomorphic to , with the canonical symplectic structure . The Hamiltonian function is .
This system admits the integral of motion . The function is the momentum of the oscillator that rotates clockwise in the -plane and counterclockwise in the -plane. Its infinitesimal action is the vector field .
This system admits the global rotation 1-form
whose polar set is the plane . The transversality condition of Definition 16 is easily verified. In fact, the intersection of with the critical fiber is only the critical point , and the energy-momentum map restricted to is the function which has rank 1 at all points of the plane except the critical point. The projection of in the energy-momentum domain is the positive semi-axis, see Figure 4.
To compute monodromy by applying Theorem 1, consider a closed path that encircles the origin in a counterclockwise direction and transversally crosses at a point . Then is locally parameterized by with . The polar orbit is given by and . A tubular neighborhood of , contained in , admits a chart where while lie in a small disk containing the origin in . Taking (and subsequently ) sufficiently small and using the transversality condition, the cylinder of orbits of defined in the proof of Theorem 1, Section II.4, intersects along a disk that can be parameterized by and is represented by . We have to determine the orientation of the chart with respect to the orientation used in the proof of Theorem 1. The latter is defined by where is an increasing parameter along and is time. Then we need to check the determinant
which we can evaluate at . Since , we find
Evaluating the last relation at , gives . This implies that a cycle , which is positively oriented on , is negatively oriented in the -plane and therefore
iii.2 The Spherical Pendulum
The spherical pendulum is a Hamiltonian system in the cotangent bundle of the sphere . This manifold can be symplectically embedded in the cotangent bundle of , that is diffeomorphic to with canonical coordinates , . In these coordinates the Hamiltonian of the system is the restriction to of the function . This Hamiltonian commutes with the function . This system admits the global rotation 1-form
In this case the poles of the rotation 1-form are the points satisfying the two equations , that form the two planes
The restriction of the energy-momentum map to the two planes is the function , which has rank 1 at all points except the poles (defined by ), which are singular points for the system. The image of this map, that is , consists of two rays, subsets of the -axis (see Figure 4).
To compute monodromy in this example we follow the same argument as for the champagne bottle, Section III.1. The main difference is that now intersects at two distinct points and contains three polar orbits. Consider a closed path that encircles the focus-focus value in a counterclockwise direction and transversally crosses at the points , . When crosses at it is locally parameterized by with . There are two polar orbits on , given by , , and , . Each polar orbit admits a tubular neighborhood , contained in . Each admits a chart , where . Taking sufficiently small and using the transversality condition, the cylinder of orbits of , intersects each of along a disk that can be parameterized by and is represented by . For the orientation we check the determinant
which we can evaluate at . We further have
which, evaluated at , gives . This implies that the cycles should be negatively oriented in the -plane and therefore
The polar orbit on is given by , , and , . Working as above, we find , therefore
Note, in particular, that the polar orbits and that belong to are bounded away from the focus-focus point as goes to zero and their contributions to the monodromy number cancel out. This means that only the polar orbit which approaches the focus-focus point as goes to zero contributes to monodromy. We show that monodromy is locally determined in Section IV.
Note that the argument we give here for the spherical pendulum, works in exactly the same way, for more general systems of the form on , provided that the path lies in the set of regular values of and it transversally intersects which is a subset of the -axis. In particular, this includes the case where the system does not have a focus-focus singularity but, instead, a more complicated arrangement of critical values forming an “island”, see Ref. 12, Chapter 4.
iii.3 The Hydrogen Atom in Crossed Fields
After a first reduction, the hydrogen atom in crossed electric and magnetic fields turns into a Hamiltonian system defined in that can be embedded into the manifold endowed with the Poisson structure coming from the Lie algebra (that is and where is the signature of the permutation , ). The phase space is a symplectic leaf of this space.
The Hamiltonian function for this system is , with a function of degree two or higher in the variables depending on the parameters (see Ref. 12, Chapter 3, for a detailed description). This system can be normalized so as to admit an invariance under the Poisson action of , that induces a simultaneous clockwise rotation in the two copies of about the and axes respectively. In this case, regardless of the choice of , a global rotation 1-form is
and its poles correspond to two submanifolds .
Theorem 1 applies. We illustrate the application considering a toy model with . The restriction of the function to is the function , which has rank 1 and projects onto two horizontal lines connecting each focus-focus critical value to the elliptic-elliptic critical value at the same height (see Figure 4). The addition of the term to or the choice of a different term would deform this picture but leave it qualitatively the same (as long as the quadratic part does not cause a bifurcation of the system).
To apply Theorem 1, consider a closed path that encircles both focus-focus values in a counterclockwise direction and transversally crosses at the points . Near each point the path is locally parameterized by with . The polar orbits on are given by and , . Tubular neighborhoods of , contained in , admit charts where . Taking sufficiently small and using the transversality condition, the cylinder of orbits of defined in the proof of Theorem 1, Section II.4, intersects along disks that can be parameterized by and are represented by . We have to determine the orientation of the charts with respect to the orientation used in the proof of Theorem 1. Then we need to check the determinant
which is being evaluated at . We further have
Therefore, in both cases and we get
Iv Local Monodromy
We now concentrate on a point that is a focus-focus singularity Zung1997 (); Bolsinov2004 (). Such singularities are isolated, rank-zero, singularities of . This implies that . Moreover, we assume that the fiber containing is a singly pinched torus, that is, the only critical point on this fiber is . Under these assumptions we show that the main theorem, Theorem 1, can be applied locally to determine the monodromy number near . In particular, we show that can be determined by restricting our attention to a non-saturated neighborhood of . We then make a specific choice of the rotation -form and we use it to compute that .
iv.1 Local Fibration and its Complement
Let us consider the fibration induced by in a neighborhood of a focus-focus point of a 2 degree of freedom Hamiltonian system. Let and denote by the reduction map of the Hamiltonian -action induced by the flow of . Finally, let denote the reduced energy-momentum map, satisfying