Matrix monodromy

As an illustration, the base vector x in Fig. 4 is fixed as x = (1,0) and u = D x rotates in response to changes in A0 and T around the cycle. Thus on completion of the cycle, u/ = D/ x, where the subscript i has been added to indicate the direction of x in the initial lattice. Similarly, the projection of the final vector x/ onto the original lattice is given by X/ = D,Uy, where D, is the tangent matrix before translation. The overall change in Bohr quantized lattice vectors is therefore given by x/ = Mx where the monodromy matrix is given by... 

The tacit assumption above is that the monodromy matrix is defined with respect to the primitive unit cell, with sides (5v, 8fe) = (0,1) and (1, 0), because the twist angle that determines the monodromy is given by A9 = — (Sv/Sfe)j.. However, situations can arise where other choices are more convenient. For example, the energy levels within a given Fermi resonance polyad are labeled by a counting number v = 0,1,... and an angular momentum that takes only even or only odd values. Thus the convenient elementary cell has sides (8v, 8L) = (0,2) and (1, 0), and the natural basis, say, y, is related to the primitive basis, x, by... 

Since the reduced Hamiltonian in this case has a single isolated critical point, the transformed monodromy matrix, in the y basis, becomes... 

This emphasizes that the monodromy matrix is defined only within a similarity transformation... 

It is important in defining the monodromy matrix, which quantifies changes in the unit cell in Figs. 4 and 5, to specify the lengths of the unit cell sides that define the basis. The monodromy theorem—that the monodromy index is equal to the number of pinch points on the pinched torus [40]—applies in a basis in which the cell sides represent unit changes in the relevant quantum number. 

The coupling of two angular momenta with equal magnimdes, as in the previous model of the perturbed H atom, is of special interest in leading to a monodromy matrix with index n = 2. 

Moreover, a constant section can only contain integer n and I values separated by Al, An) = m+ —m - - 1), which means that the horizontal and vertical lattice vectors in Fig. 22 are given by x = 0,m + 1) and (1,0), respectively. Hence an alternative basis may be dehned with a unit cell having sides Av,APj = (l,m+ 1), in which case the argument in Section IIC3 shows that the two-dimensional monodromy matrix for the fixed km section transforms to... 

Analysis of the prototypical resonant swing spring model [11-13] shows that Fermi resonance with conserved angular momentum is an intrinsically three-dimensional phenomenon. The form of the 3x3 monodromy matrix was given. 

As a result of the balanced treatment of position and momentum, it involves a combination of all elements of the monodromy matrix... 

This kind of the scalar product is implicit in our derivations of the path integral. Expanding (48) in a power series, multiplying with respect to this scalar product, and resumming, we get the monodromy matrix ... 

The square root of the monodromy matrix gives rise to the so-called braiding matrix responsible for a proper form of skein relations, yielding knot or link invariants. 

The differential equation for M in (7) is non-antonomous and involves evaluation of the jacobian of the forced-model equations at the current value of the trajectory jc(x0, p, t) for each time step so that it must be integrated simultaneously with the system equations. Upon convergence on a fixed point, the matrix M becomes the monodromy matrix whose eigenvalues are those of the jacobian of the stroboscopic map evaluated at the fixed point and are called the Floquet multipliers of the periodic solution. 

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