Commensurate Harmonic Oscillators:
Classical Symmetries
Abstract
The symmetry properties of a classical dimensional harmonic oscillator with rational frequency ratios are studied from a global point of view. A commensurate oscillator possesses the same number of globally defined constants of motion as an isotropic oscillator. In both cases invariant phasespace functions form the algebra with respect to the Poisson bracket. In the isotropic case, the phasespace flows generated by the invariants can be integrated globally to a set of finite transformations isomorphic to the group . For a commensurate oscillator, however, the group of symmetry transformations is found to exist only on a reduced phase space, due to unavoidable singularities of the flow in the full phase space. It is therefore crucial to distinguish carefully between local and global definitions of symmetry transformations in phase space. This result solves the longstanding problem of which symmetry to associate with a commensurate harmonic oscillator.
PACS: 02.30.Ik; 45.50.j; 02.20.Sv; 02.20.Tw
1 Introduction
Harmonic oscillators are ubiquitous in physics. To lowest order, motion close to a stable equilibrium of a classical system is often described by a Hamiltonian of the form
(1) 
Here the (appropriately rescaled) canonical coordinates and momenta have Poisson brackets , . If the frequencies are all equal,
(2) 
the Hamiltonian (1) describes an isotropic dimensional oscillator. This system is invariant under a set of transformations isomorphic to the group : on the one hand, the quadratic form (1) in variables is obviously invariant under proper rotations —on the other hand, canonical transformations need to be symplectic, hence they are elements of . However, any transformation in which is both (special) orthogonal and symplectic, must be (special) unitary [1]: . The group is represented by phasespace functions which, as constants of motion, generate symmetry transformations of the Hamiltonian. In fact, the isotropic oscillator is “maximally superintegrable” since it possesses the maximal number of functionally independent constants of motion, exceeding by far the number of globally defined invariants required for integrability [2].
Suppose now that the frequency ratios are positive rational numbers,
(3) 
This property defines a commensurate harmonic oscillator, or oscillator, with . As shown below, it also possesses globally defined phasespace invariants, apart from the Hamiltonian. Their Poisson brackets form the Lie algebra , as for the isotropic oscillator. It is known that in both systems all orbits are closed. Nevertheless, some difference is to be expected, since all orbits of an isotropic oscillator have the same period, while a commensurate frequencies allow for closed orbits with different periods. This is easily seen by exciting only individual degrees of freedom with frequencies .
In the following, the topological and grouptheoretical impact of rational frequency ratios (different from one) will be made explicit. First, various papers dealing with commensurate oscillators are reviewed in Section 2, which is independent of the later developments. The technical part starts with Section 3, where, for simplicity, the class of twodimensional oscillators will be studied in detail. The generalization to , given in Section 4, is not straightforward. Finally, the overall picture is summarized and conclusions are drawn. A study of quantum mechanical oscillators, including the classical limit to connect with the present results, will be presented elsewhere [3].
2 Symmetries of Harmonic Oscillators
The equations of motion of harmonic oscillators can be solved analytically for arbitrary frequency ratios. In spite of this exceptional property many authors have wrestled with the symmetries of such systems, the question being how their symmetries depend on the (ir) rationality of the frequency ratios. Most contributions are fostered by the difficulty to distinguish between local and global properties of phase space. Twodimensional oscillators with rational or irrational frequency ratios are discussed almost exclusively. Surprising claims have been made in the attempt to generalize properties of the isotropic oscillator in dimensions.
Jauch and Hill [4] address the problem of “accidental degeneracy” of quantummechanical energy eigenvalues. The obvious invariance of the threedimensional harmonic oscillator (as well as the hydrogen atom) under the group of rotations in configuration space is not sufficient to explain the observed degeneracy of the energy levels. They conclude that additional constants of motion must exist which account for extra degeneracies in the quantum mechanical energy spectrum. In fact, hermitean operators can be specified which commute with the Hamiltonian of the isotropic harmonic oscillator in dimensions. Their commutation relations turn out to be those of the algebra . Therefore, the oscillator is said to have the symmetry—which then leads to the correct degree of degeneracies of energy levels.
Pauli [5] and Klein [6] have pointed out that there is a connection between degeneracies of energy levels and the existence of further constants of motion in the associated classical system. Therefore, the result also should be manifest in the corresponding classical isotropic oscillator. Upon ‘dequantizing’ the quantum invariants, one obtains indeed constants of motion which constitute the algebra with respect to the Poisson bracket. Hence, the classical isotropic oscillator possesses indeed constants of motion other than the angular momentum. Its components generate obvious geometrical symmetry transformations while the additional constants are said to generate dynamical symmetry transformations. They cannot be visualized in configuration space because they mix coordinates and momenta.
However, to exhibit a set of conserved phasespace functions which form a particular algebra is not sufficient in order to prove invariance of the physical system in a global sense, i.e. in the entire phase space. Jauch and Hill assert that the “system of orbits” of a classical oscillator be invariant under a group of transformations isomorphic to the threedimensional group of proper rotations . However, this claim cannot be justified by local considerations only. In other words: global invariance under a particular group of transformations does not follow from specifying phasespace functions forming the corresponding algebra.
McIntosh reviews accidental degeneracy in classical and quantum mechanics in [7]. He notes that the phase space of the isotropic harmonic oscillator in two dimensions foliates into hyperspheres, being surfaces of constant energy. A discussion of the canonical transformations generated by three constants of the motion quadratic in the coordinates and momenta makes follows. It becomes obvious that the group of symmetry transformations is the special unitary group in two dimensions, —not the group of proper threedimensional rotations, , as Jauch and Hill suggested.
Dulock and McIntosh [8] devote a paper to the twodimensional harmonic oscillator with arbitrary frequency ratio. Using classical variables which mimic quantum mechanical creation and annihilation operators, they write down three constants of motion with Poisson brackets isomorphic to the algebra relations. A Hopf mapping is performed in order to visualize “how the rotational symmetry of , which is the threedimensional rotation group, chances also to be the symmetry group of the harmonic oscillator.” [8]. Formally, this method can be applied to oscillators with arbitrary frequency ratio. However, one of the transformations, which is onetoone in the isotropic case, becomes a multiplevalued map. For rational frequency ratios there is a finite ambiguity, turning to infinite multiplevaluedness if the frequencies ratios are irrationally. In spite of this result, the authors claim that the set of symmetry transformations for all types of oscillators investigated is isomorphic to the group —irrespective of the multiplevaluedness. Once more, the possibility to write down formal expressions which constitute particular algebraic relations is taken as a proof of the existence of an associated group of transformations.
Maiella and Vitale [9] react to the claim that “every classical system should possess a ‘dynamical’ symmetry larger than the ‘geometrical’ one” [9]. Using actionangle variables, they provide three constants of motion for the twodimensional oscillator which form the algebra. However, for irrational frequency ratio the invariants are not singlevalued—hence they consider the “su(2) symmetry” to be of “formal value” only. It is claimed to acquire physical relevanve only for commensurate and, a fortiori, isotropic oscillators. At the same time, no argument is given which would forbid the existence of the group for the irrational oscillator. The authors do not investigate whether, in the commensurate case, the invariants generate indeed finite singlevalued phasespace transformations in .
Maiella [10] extends this discussion to the dimensional oscillator and emphasizes that only singlevalued constants of the motion generate actual symmetry transformations. Initially, the group of all contact transformations for a given dynamical system is considered. Any subgroup of transformations which generated by singlevalued constants of motion and leave the Hamiltonian invariant, is called an “invariance group.” The classical degree of degeneracy determines the number of its generators: each linear relation between the classical frequencies of the system with rational coefficients is accompanied by the appearance of a singlevalued constant of motion. Subsequently, phasespace functions are given in actionangle variables which realize the algebra for an isotropic oscillator and the algebra , for smaller degeneracy. However, it is again not proven explicitly that the generators actually give rise to globally welldefined transformations.
In the late ’s, successful application of group theoretical concepts in elementary particle physics renewed the interest in symmetries of classical Hamiltonian systems and stimulated more general approaches. The invariance of the threedimensional Kepler problem under the group of fourdimensional rotations, , was explicitly shown by Moser [11] in for the first time. Already in 1965 Bacry, Ruegg and Souriau [12] proved that there exists a set of global symmetry transformations for the Kepler problem being isomorphic to the group . The transformations presented, however, do not act on variables in phasespace. The transformations of phasespace manifolds are parameterized by the components of angular momentum and of the RungeLenz vector. Representing only five independent constants of motion, the time at which the particle passes the perihelion of the orbit is taken as sixth parameter.
Dulock and McIntosh [13] claim that the Kepler problem has not only the symmetry but . Two papers by Bacry, Ruegg and Souriau [12] and by Fradkin [14] generalize this statement: all classical central potential problems should possess the dynamic symmetries and . This surprising statement is subject to the same criticism as the following, even more general claim by Mukunda [15, 16]: all classical Hamiltonian systems with N degrees of freedom have and symmetries. If this statement were true, then there would exist just one and only one global phasespace structure for systems with degrees of freedom—the wellestablished distinction between regular and chaotic systems would have no meaning at all.
Mukunda argues on the basis of an a theorem by Eisenhart [17]. Consider, in a Hamiltonian system with degrees of freedom, independent functions of canonically conjugate variables (subjected to weak conditions). They can always be supplemented by phasespace functions such that pairs of canonically conjugate variables result which define a symplectic basis of phase space. Hence, starting with the Hamiltonian of the system under consideration one can find (i) a variable being canonically conjugate to the Hamiltonian and (ii) additional pairs of phasespace functions with Poisson brackets equal to one, all commuting with the first pair and therefore with the Hamiltonian. Consequently, this theorem is a blueprint to construct independent constants of motion in any Hamiltonian system with degrees of freedom. The particular form of the Hamiltonian does not even enter into the construction. Next, two different sets of phasespace functions are defined in terms of the functions of this particular basis. Their Poisson brackets realize the relations characteristic of the algebras and , respectively. In a footnote, the author restricts the applicability of the results: “We concern ourselves only with constructing realizations of Lie algebras, not of Lie groups. Even when we talk of invariance under the group, for example, we really intend invariance under the algebra” [15]. Consequently, “invariance under the algebra” is a local concept only, so that Mukunda’s construction has formal value only. Actually, the phasespace functions written down by Mukunda do not neatly map phase space onto itself: the functions become imaginary if the range of the canonical variables is not restricted artificially. The lesson to be learnt is obvious: in order to establish the invariance of a system under a group of phasespace transformations it is not sufficient to realize specific Poissonbracket relations with invariants.
A related position is put forward by Stehle and Han [18, 19]. To identify a particular algebra by constants of motion does not guarantee the presence of a “higher symmetry”—a singlevalued, or at most finitely manyvalued realization of the group must exist in phase space. To show this, they show that a system is classically degenerate if the HamiltonJacobi equation of a particular system is separable in a continuous family of coordinate systems. This property is observable. Compare the Fourierseries representation of one specific orbit described with respect to two different (continuously connected) coordinate systems. For consistency, the frequencies appearing in its Fourier decomposition must be rationally related, which corresponds to a classical degeneracy. It is important to note that the transformation from one coordinate system to the other be singlevalued, otherwise the argument does not hold. Any phasespace function and, consequently, any constant of motion generates a transformation of phasespace onto itself; alternatively, it can be viewed as the generator for a transition to another coordinate system such that the Hamiltonian remains invariant. Only singlevalued constants of motion generate global singlevalued transformations—infinitely manyvalued “constants of motion” represent formal expressions only, not necessarily related to the existence of classical degeneracy. Therefore, they do not establish a higher symmetry group of the system.
To sum up, the construction of an algebra from constants of motion is only the first step in the proof of the existence of a potential higher symmetry group. It needs to be supplemented by a global investigation of the generated transformations.
3 The twodimensional commensurate oscillator
This Section deals with the symmetry properties of a twodimensional commensurate harmonic or oscillator described by the Hamiltonian
(4) 
where the integers and have no common divisor. Two pairs of canonical variables, label points in phase space , the only nonvanishing Poisson brackets being given by
(5) 
It will be useful to introduce two other sets of canonical variables. First, combine each pair into a complex variable
(6) 
with nonvanishing brackets
(7) 
where denotes the complex conjugate of . Second, actionangle variables and , are determined through modulus and phase of . Their nozero brackets read
(8) 
These coordinates provide alternative forms of the Hamiltonian,
(9) 
Constants of motion and Lie algebras
Commensurate harmonic oscillators possess a large number of constants of motion. The Hamiltonian itself is an invariant as . Motion of the system with given energy is thus restricted to a threedimensional hypersurface, an ellipsoid in phase space . Further, the actions and , having zero Poisson brackets with the Hamiltonian and among themselves, render the oscillator integrable. For fixed values of the actions, Arnold’s theorem [2] states that the motion takes place on a twodimensional torus . In fact, the entire phase space is foliated by tori with radii and , respectively. According to (9) the Hamiltonian is a linear function of these invariants.
A third, functionally independent (complex) constant of the motion is given by the expression
(10) 
As mentioned in [4], both its real and complex part are invariant which implies that the phase of the function ,
(11) 
is a constant of the motion, too. Considered as a generator of transformations in phase space, it connects energetically degenerate pairs of tori. The existence of a third invariant is expected to reduce the dimensionality of the accessible manifold. Indeed, fixed the values of the three invariants , , and (or, equivalently, ) single out a onedimensional orbit on the torus if the two frequencies are rationally related. Generic orbits, , , retrace themselves after a characteristic time , with winding numbers for and for . However, if the frequency ratio of the motion on the tori were not rational, an orbit would cover the torus densely—the function would represent a formal constant of the motion only, without any physical impact on the motion of the system. An important difference to the isotropic oscillator is due to the fact that orbits of an oscillator may have different orbits with frequencies and , respectively. This allows to distinguish experimentally the two cases.
The phase space of an oscillator has a particular discrete symmetry. Combine the variables into a column: now the Hamiltonian is obviously invariant under finite rotations, , , or explicitly,
(12) 
These transformations map the phase space to itself. They form a cyclic group , the direct product of two cyclic groups with and elements, respectively. In [20], has been called ambiguity group.
The Poisson bracket of two invariants results in a third invariant. Therefore, the collection of all invariants is a Lie algebra. Typically, it will contain an infinite number of elements, all of which depend functionally on a smaller number of invariants. By an appropriate choice of the invariants, however, algebras with a finite number of elements can be found. The simplest example is given by the three invariants , , giving rise to the following brackets:
(13) 
The algebra contains three independent elements—it is not possible to find an algebra with fewer elements since the oscillator has three invariants. It also contains two elements with vanishing Poisson bracket which, in a system with two degrees of freedom, is the maximum number of ‘commuting’ functionally independent invariants.
There is an alternative set of four invariants ,
(14)  
(15)  
(16)  
(17) 
Only three of these invariants are functionally independent because
(18) 
This constraint is conveniently rephrased by saying that the ‘four vector’ is ‘null’ or ‘light like.’ The functions are particularly interesting since they form the basis of a Lie algebra isomorphic to ,
(19) 
which has as a subalgebra, generated by the components of . Eqs. (19) has been at the origin of many attempts to associate a group of symmetry transformations with the twodimensional oscillator.
Reduced phase space and space of invariants
Consider the complex variables
(20) 
which satisfy
(21) 
In spite of these relations, the variables do not define pairs of canonical coordinates of since the map is not a onetoone transformation. The variables are, however, canonical coordinates in the reduced phase space . The reduced space is obtained from identifying those points of which satisfy , . The definition of the variables (21) is motivated by the invariance of the constants of motion in (14) under the ambiguity group .
The invariants (14) take a simple form when expressed in terms of the reduced variables,
(22)  
(23)  
(24)  
(25) 
Using the twocomponent ‘Weyl spinor’ , the invariants (22) can be written
(26) 
where and the Pauli matrices generate the algebra . Consequently, the invariants, which span the space of invariants, , turn into sesquilinear expressions on the reduced phase space . Their structure is similar to those of the isotropic or oscillator: formally, the reduced phase space and the original one coincide, . In some sense, the nonbijective map ‘linearizes’ the invariants at the expense of accounting for a fraction of phase space only. It will be shown later that the concept of the reduced space is natural in the present context. It is the appropriate setting to derive global statements with respect to symmetry transformations.
Topological aspects
Turn now briefly to the topology of the spaces involved. Consider the nontrivial transformations introduced so far: first, the original phase space has been mapped to the reduced phase space,
(27) 
second, introducing the invariants maps the reduced variables to the space of invariants, ,
(28) 
which is an upper cone in since .
The reduced phase space has the structure of a wellknown fiber bundle. To see this, consider an orbit in phase space . Its image in the reduced space is given by . The maps form a group which leaves invariant the map , , since the phase drops out from the sesquilinear expressions given in Eq. (22). Therefore, is indeed a fiber bundle : the invariants form the base, each orbit is a fiber, and the map is the projection. The global structure of the bundle follows from the fact that the restriction of to the submanifold with points is isomorphic to the sphere —as is obvious from the quadratic form (4). Thus, the restriction of the map to defines the Hopf fibration of . To each orbit in corresponds a point of the sphere of radius and a circle in the tangent space at this point.
It is interesting to look at space of invariants and the transformations among them from a general perspective. To do so, consider the complex instead of the real Lie algebra which also leaves invariant the Hamiltonian in (22) invariant. This is the Lie algebra associated with the group , the universal covering of the Lorentz group. The Lorentz group induced by in is the transitivity group of the upper (half) cone.
The elements of can be written as , where and are two real parameters, and each is a traceless complex matrix,
(29) 
The matrices belong to the group . Thus, they generate rotations and infinitesimal transformations can be written in terms of a Poisson bracket:
(30) 
The subsets represent Lorentz boosts mapping a point according to
(31) 
On the invariants, the transformation
(32) 
is induced. Hence, the sphere of radius is mapped to a sphere of radius with
(33) 
This is an infinitesimal Lorentz transformation which maps the upper cone to itself as is obvious from and remaining positive. Contrary to (30), it is not possible to express the righthandsides of (33) by means of Poisson brackets. This can be understood from a quantum mechanical point of view. A classical theory can only manage Boltzmann statistics whereas in quantum (field) theory, due to the anticommutativity of Weyl spinors, it would be possible to find a commutator to express the derivatives .
4 Global invariant vector fields
Each phasespace function generates a flow in phase space , as well as in the reduced phase space , and in the space of invariants . The invariants generate flows which commute with the Hamiltonian vector field. To be more specific, consider any element of the Lie algebra . When acting on an observable through the Poisson bracket,
(34) 
it defines a vector field in . Its integral lines satisfy the differential equation
(35) 
The solution of this differential equation is a map which will be written in the form
(36) 
where
(37) 
with smooth phase space functions and . In a simplified notation, the solutions (36) are written as
(38) 
each unit vector associated with a point of the unit sphere .
The crucial question now is to investigate whether the flow (34) and hence the maps (38) are defined everywhere in the space under consideration. Only in this case, the algebra formed by the closed set of Poisson brackets among the invariants integrates to a group of symmetry transformations. More specifically, one needs to find out whether the invariants (14) of the oscillator generate a set of transformations isomorphic to the group (or ). This is only possible if the associated vector fields are welldefined everywhere in the space where they act. The fields will be studied separately for functions from the spaces , or .
Vector fields in the space of invariants
The simplest case to look at is the orbits generated by the first component of , which is a multiple of the Hamiltonian, . Not surprisingly, one has
(39) 
that is, all components of are invariant under the action of . Rotations about the 3axis, i.e. with an axis passing through the poles , are generated by the invariant ,
(40) 
Each possible orbit is generated by a linear combination of invariants ,
(41) 
where the matrix represents a rotation by an angle about an axis parallel to the vector . In other words, every point of the sphere is mapped to another point of the same sphere, the energy being conserved.
These results are conveniently summarized by a group theoretical statement. The set
(42) 
of maps acting in is a representation of the group . In other words, there is a subset of all phasespace functions, such that its elements transform according to the group . Mathematically, this group is the integrated form of the adjoint representation of the algebra (19). Consequently, one can attribute this group as a symmetry group to the reduced oscillator, for any frequency ratio. Note, however, that this symmetry does not act on points in phase space but on points of the space of invariants.
Vector fields in the reduced phase space
Again, the action of the generators , and will be studied, now with respect to the variables . It is straightforward to see that
(43) 
which is just the time evolution with . Similarly, the invariant generates a flow
(44) 
Comparison with (21) shows that the function is left invariant. Transformation (44) is a special case of the map
(45) 
No ambiguities arise when mapping points under , for whatever values of the parameter and the directions . Therefore, the set
(46) 
of maps faithfully represents the group in . Consequently, an oscillator admits as symmetry not only the threedimensional rotation group in but also the special unitary group in .
In this restricted sense, and only in this one, oscillators are seen to possess both and as symmetry groups. This statement agrees with the fact that the algebras and are isomorphic. The next section deals with the question which groups, if any, are represented on the original phase space .
Vector fields acting in phase space
It will be shown in this section that the vector fields associated with the invariants are not defined globally when they act on the variables which span phase space . Consequently, it is not possible to implement the group on phase space . More explicitly, it will be shown that the action on is nonlinear, and that it is inevitably singular for some parameters and initial points . Contrary to one’s intuition the flows can be defined only locally, and they cannot be extended to define a group of symmetry transformations.
To begin with, consider the flows generated by and , respectively. The resulting orbits are welldefined for all initial points: they are given by
(47) 
and by
(48) 
respectively. Eq. (47) describes the time evolution of the point , hence both the energy and the torus are left invariant. Since the values of the actions change according to , the flow in (48) also conserves the energy while mapping a torus to another one, .
Now consider fields which are generated by arbitrary linear combinations of the invariants, . Denote potential solutions of the differential equation
(49) 
by , with some initial point . Explicitly, the complex twocomponent field reads
(50) 
It is finite but illdefined on the hyperplanes and . There are points which, when transported by the flow , hit the planes or for some value of . The associated orbits will be called singular since they cannot be continued unambiguously across the planes. This is due to the terms in (50) which contain and ,
(51) 
while all other terms are zero on and . Here is a toy example to illustrate the underlying problem. Consider a onedimensional system with variable , satisfying . The flow generated by is illdefined at the origin,
(52) 
as its value depends on the way the point is approached. If a trajectory were reaching the origin, it would be impossible to continue it unambiguously beyond this point. It is important to realize that this singularity as well as the one encountered in the singular planes is not due to a choice of coordinates but an intrinsic property of the flow.
To visualize the entire set of singular orbits, look at their images in , that is, the orbits , . For given energy , the points of correspond to the north pole of the sphere , while those of are mapped to its south pole, . By , an orbit goes to a circle . Singular orbits thus correspond to circles going through either one or both poles of the sphere, while regular orbits hit neither of them: for almost all flows, associated with a given vector , there exist two “critical” circles passing through the north pole and the south pole, respectively. These circles coalesce into a single one passing through both poles if the axis of rotation is in the equatorial plane, . They degenerate to points located at the poles if . Two conclusions can be drawn from this picture:

for any given unit vector , the map has at least one singular orbit in ;

any point can be sent to a singular hyperplane by a map with an appropriately chosen vector .
In fact, the vectors can be chosen from two continuous sets: they only need to be in a plane (passing through the origin) which is perpendicular to either of the vectors , or explicitly,
(53) 
Regular orbits of are easily computed without solving the differential equation (49). One needs to determine modulus and phase of the variables as a function of . It is useful to write down the orbits in the reduced phase space and in the space of the invariants. According to (45), the reduced variables evolve linearly,
(54)  
(55) 
while the invariants evolve in as
(56) 
Using , the dependence of the moduli is simply
(57) 
For the evolution of the phases, plug Eqs. (54) into
(58) 
giving
(59) 
and a similar equation for . The two phases must be continuous whenever reaches the value . They will both have a value which is a multiple of when the parameter takes the value . This result seems to suggest that might be an fold covering of the subgroup , , of the special unitary group, . Due to existence of singular orbits, however, this is not possible. Further, it is wellknown that the only universal covering of is this group itself. Nevertheless, one might describe the situation as a ramified covering of since the maps combine according to a group product law.
To visualize the obstruction of a global action of the group differently, recall that a given map sends a torus to a torus such that holds. For some and it happens that one of the actions vanishes, , say. This means that the initial twodimensional torus () is mapped to a onedimensional torus, i.e. a circle , and, therefore, one of the angle variables has lost its meaning. Once this has happened, it is impossible to unambiguously continue the trajectory which has hit the singular plane, as the missing angle could take any value. The phenomenon is similar to the passage of a spherical wave through a focus.
It will be useful to give a name to the situation encountered here. A system with phase space will be said to have a faint symmetry if it admits a set of globally defined invariants which form an algebra while the group associated with it cannot be realized on but only on a smaller part of it. Thus, a twodimensional commensurate oscillator has a faint symmetry.
5 The dimensional commensurate oscillator
To describe a commensurate harmonic oscillator in dimension, the present notation is straightforward to adapt. Let the label run from 1 to : the Hamiltonian of the oscillator with reads
(60) 
the complex canonical variables are given by , while actions and angles are defined through . Thus there are three sets of pairs of canonical variables to choose from, with brackets
(61) 
It will be assumed that the positive integer numbers do not have an overall common divisor. For the discussion to follow, two cases will be distinguished: a commensurate oscillator is said to be canonical if no pair of numbers and , , admits a common divisor but one. This class will be studied first. The presence of common divisors among subsets of the frequencies gives rise to interesting additional complications which will be considered later on.
Constants of motion and Lie algebras
Inspired by Eq. (10), each function
(62) 
is seen to be an invariant for the commensurate oscillator, . These constants of motion depend on only real invariants, independent actions , and relative angles
(63) 
As in the twodimensional case, the range of the functions must be restricted to the interval because two values and , respectively, correspond to the same orbit. The angles satisfy linear relations,
(64) 
Therefore, there are no more than functionally independent constants of motion, the maximum number of possibly independent invariants. As independent invariants, one may choose, for example, the actions and relative angles .
The  dimensional surface of constant energy is an ellipsoid in phase space . It contains the dimensional torus of constant actions as a submanifold. Lines of constant actions and angles are the orbits of the motion, winding around a torus . Each orbit is a onedimensional closed loop given by
(65) 
where . One revolution is completed after a time , with the number taking a value such that the winding numbers of each subsystem are integer without overall common divisor. In the canonical case, is equal to . Here is an example for which illustrates the noncanonical case: let . The number would then take the value .
It is important to note that in a canonical (but not isotropic) oscillator (i.e., all ), there exist orbits with different periods. There are orbits corresponding to motion of a single oscillator only; there are orbits winding aro