Momentum mapping

Figure 13. Divisions of the energy-angular momentum map for the isomerizing HCN/HNC system, for which the cusps, Oj., in Figs. 10-12 lie at infinity. Taken from Ref [10] with permission of the American Physical Society, Copyright 2004.

Figure 21. Boundaries of the energy-momentum map for the resonant 1 1 2 oscillator, with a central singular thread. Taken from Ref. [13] with permission of the American Institute of Physics, Copyright 2004.

Fig. 1.2. Mass-resolved momentum imaging (MRMI) technique applied to the N2+ ion produced from N2 in intense laser fields ( 3.5PW/cm2). The two-dimensional momentum map is constructed from a set of time-of-flight mass spectra recorded at different angles a between the laser polarization direction and the TOF tube...

The two wings observed in the three-dimensional map (Fig. 1.6) are transformed into a pair of curved arms in the two-dimensional momentum map in the A >i2 - O12 plane, as shown in Fig. 1.7a, where Api2 represents the difference between the momentum values of the two S+ ions, i.e., Api2 = Pi - P2-As shown in the two-dimensional momentum map drawn on a logarithmic intensity scale, the observed curved arms are well-reproduced by the classical free-rotor model. 

The existence of the d integrals of motion, (/, h,..., fd), induces phase space structures which lead to further constraints on the trajectories in addition to the ones described above. In order to describe these structures and the resulting constraints it is useful to introduce the so-called momentum map, M [41, 42], which maps a point ( i, Pi, , Pd) in the phase space... 

As shown below, we have an important analogue of these assertions if the function / is replaced by the symplectic manifold momentum mapping induced hy a complete set of commuting integrals. In particular, one may examine one integral on a three-dimensional constant-energy surface of an integrable system. 

Bifurcation Diagram of Momentum Mapping for an Integrable System, The Surgery of General Position... 

Suppose t = sgradjET is a smooth system on a smooth symplectic manifold M, Suppose the system v is integrable, that is, there exist n independent (almost everywhere) smooth integrals /i,..., /n which are in involution. We will assume that fi = H, Let F — R be the momentum mapping corresponding to these... 

Using Lemma 2.2.5 for any such set a one can construct a symplectic manifold Za which has non-empty (of dimension 2n) intersections of Vk and Vk with Mj and Mj, respectively, and also has the momentum mapping which... 

The bar denotes here a topological closure in R . FVom Lemma 2.2.5 it follows that there exists such a symplectic manifold Z and such a momentum mapping... 

The results obtained above make it possible to describe visually the structure of singular fibres, that is, singular level surfaces of a second integral / (in the four-dimensional case) and singular fibres of the momentum mapping F (in an arbitrary multidimensional case). For the sake of definiteness we will dwell on the four-dimensional case. 

The Properties of Momentum Mapping of a System Integrable in the Noncommutative Sense. 

We will proceed to the proof of Theorem 3.3.2. At the same time, we investigate some important properties of momentum mapping, which are also of interest independent of Theorem 3.3.2. Let a maximal linear Lie algebra G of smooth functions be given on Consider the corresponding momentum mapping... 

Lemma 3.3.1. The momentum mapping F M G is ( invariant, where the group 0 = exp G acts on G in a coadjoint manner, and on the manifold M it acts as a group of symplectic transformations generated by the Lie algebra G of vector Gelds sgrad f,f G. 

PROOF This assertion follows from 0-invariance of the momentum mapping and from the definition of the Poisson bracket. We shall dwell on this fact in more detail. In fact, for any functions a and P on G, we have the identity... 

Momentum Mapping of Systems Integrable in the Noncommutative Sense by... 

Recall the definition of the momentum mapping with an account of identificar tion of G and G. The momentum mapping F M G brings a point X G M into correspondence with an element F(x) G G, such that (F(x),fi) = fi(x) for each t = 1,..., m. The element f is called semisimple if the operator ad is diago-nalizable or becomes such after complexification. 

A Lie algebra G is called compact if there exists a positive definite scalar product (, ) on G invariant under all inner automorphisms. Let a reductive Lie algebra G be a Lie algebra of functions with respect to the Poisson bracket on a symplectic manifold (A7,o ). The semisimplicity condition for the image of the momentum mapping F M G is automatically fulfilled by virtue of Theorem 3.3.6 for... 

The momentum mapping F M G carries the above-said action of the group 0 on Af into a coadjoint action on G, and therefore the image F Mh) of the surface Af/ is invariant with respect to Ad. ... 

Theorem 4.2.4 is applicable to the case of left-invariant Hamiltonian flows on Lie groups. In particular, Theorem 4.2.4 implies the following assertion. Let F T M — G be momentum mapping. 

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