The Poincare Integral Invariants

First we consider a system with two degrees of freedom (N = 2). Suppose we have two closed curves yi and y2 phase space, both of which encircle a tube of trajectories generated by Hamilton s equations of motion. These curves can be at two sequential times (tj, or they can be at two sequential mappings on a Poincare map These curves are associated with domains labeled (Dj, D2), which are the projections of the closed curves upon the coordinate planes (pj, qj. Because both the mapping and the time propagation are canonical transformations, the integral invariants (J-j,. 2) are preserved (constant) in either case. There are two of them, of the form  

We next presume that the system is bound within a single potential well. If we now take the curves (71,72) Poincare mapping images, such that 7, = 1/71 with the surface of section defined by q = i, 0, and since dq = 

Similarly, it can be shown that Eq. [55] is a statement of the preservation of phase space volume under propagation by Hamilton s equations of motion, that is, Liouville s theorem. It is important to note that the Poincare integral invariants are also preserved under a canonical transformation of any kind and not just the propagation of Hamilton s equations. 

In N degrees of freedom, a hierarchy of N integral invariants exists. For an arbitrary phase space surface S with symplectic projections consisting of 2 -dimensional volumes the th member of this hierarchy is of the form 

Let us define a surface of section for a bounded three-dimensional system such that q = q, pj 0. For N degrees of freedom, such a surface is of dimension IN - 1 (all points on it have q = cfi and H = ) here the surface of section is four-dimensional. The three integral invariants are 

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Figure 26 A schematic representation of the Poincare integral invariants for a system with three degrees of freedom. (A) The invariant is the sum of oriented areas projected onto the three possible (qi, pj) planes. (B) The invariant 2 is rhe sum of oriented volumes projected onto the three possible pj, q, pl) hyperplanes. Not shown (Liouville s theorem). Reprinted with permission from f. 119.

In a previous work [33] we suggest an effective approach to study of conditional symmetry of the nonlinear Dirac equation based on its Lie symmetry. We have observed that all the Poincare-invariant ansatzes for the Dirac field i(x) can be represented in the unified form by introducing several arbitrary elements (functions) ( ), ( ),..., ( ). As a result, we get an ansatz for the field /(x) that reduces the nonlinear Dirac equation to system of ordinary differential equations, provided functions ,( ) satisfy some compatible over-determined system of nonlinear partial differential equations. After integrating it, we have obtained a number of new ansatzes that cannot in principle be obtained within the framework of the classical Lie approach. 

Integrating the system of partial differential equations under study within the equivalence relations above, we obtain a set of ansatzes containing those equivalent to the Poincare-invariant ansatzes obtained in the previous section. That is why we concentrate on essentially new (non-Lie) ansatzes. It so happens that our approach gives rise to non-Lie ansatzes, provided the functions (x), 0 ( ) within the equivalence relations (89) have the form... 

The integral over the Gaussian curvature in Eq. [27] is a topological invariant.i "i85 For a closed orientable 2D surface (i.e., one without boundary), the Gauss—Bonnet theorem ties the value of this invariant to the genus g or the Euler—Poincare characteristic x of the surface ... 

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