2.4 First integrals and manifolds

In the distant past, when mathematical tradition demanded the solution of differ­ential equations by analytical integration, the somewhat strange concept of "first integrals" was coined. What was then understood by an integral is nowadays called solution.

The definition of the first integral used here has been adopted from Arnol'd (1980). Let F be a vector field, and the individual components F\, F? ■ • ■ Fn differentiate functions.

 

 

denotes a row vector and, correspondingly,

 

where the notation adopts the following conventions:

Definition: a function I(x) is called the first integral of the differential equation

 

 

(2.4.1)

if its so-called Lie derivative Lp (Olver, 1986) along the vector field F vanishes

(2.4.2)a column vector. Equation (2.4.2) implies the following characteristics of the first integral: on the one hand, the function I(x\,x2 ■ ■ ■ xn) remains constant along each given trajectory; on the other hand, each trajectory lies on a hypersurface in the phase space. The hypersurface is defined by I(x) = C, where C is a constant (see fig. 2.4.1 for a three-dimensional phase space). Each trajectory defined by the initial condition thus runs on a smooth hypersurface respectively forms it.

Fig. 2.4.1

Trajectory on the hypersurface I(xi,x2,xs) = C in the three-dimensional phase space

The hypersurface defined by I{x) = C is designated an (n - l)-dimensional man­ifold (see also the discussion at the end of this section). If all possible values are assigned to the parameter C, this leads to a one-parameter family of manifolds which fills the phase space completely. If a first integral / is known, the equation I(x) = C for a given C can be solved with respect to x\. If this x\ is substituted into the remaining (n — 1) equations of equ. (2.3.1), the whole set of equations is reduced to (n - 1) equations. The more first integrals are known, the lower the number of equations and the lower the dimension of the manifold to which each trajectory adheres.

In mechanics, it is the conservation theorems which often supply first integrals; unfortunately, however, no systematic rule is known which yields an easier deriva­tion of first integrals. The Hamilton function which represents a first integral for conservative Hamilton systems is a stroke of luck. In addition, it can be shown that in the case of conservative forces, the Hamilton function - if it is not time- dependent - corresponds with the total energy, i.e. the sum of kinetic and potential energy (see also section 4.1).

 

It is necessary to associate many phenomena with geometrical models which cannot be described in a simple form. In the case of dynamical systems, the geometrical models form manifolds. Manifolds play an important role in the determination ofthe global behaviour of trajectories. For example, non-chaotic attractors which specify the asymptotic state of many dynamical systems lie on manifolds or mould them.

 

An n-dimensional manifold M is a topological space which has the characteristic that each point and its neighbourhood can be mapped one-to-one onto the n- dimensional Euclidean space E. It is thus possible to apply the coordinates of E as local coordinates on M (Brocker and Janich, 1990; fig. 2.4.3). Using a simple example for a manifold, the limit cycle (fig. 2.4.2), the point by point one-to-one representation onto a line of the Euclidean space is demonstrated (fig. 2.4.3).

 

 

Fig. 2.4.3

One-to-one mapping of the limit cycle onto a line point by point

 

Fig. 2.4.4 Overlapping limit cycle in local coordinates and its mapping onto a line

 

Geometries which possibly do not admit an overall analytical representation are reduced to points and their associated surroundings. Each individual surrounding space can be described by local coordinates. By joining overlapping surrounding spaces, the total manifold and the corresponding mapping can be described ana­lytically. Particularly if we observe the limit cycle in fig. 2.4.4, we confirm that each point can be described by the coordinate ip in the interval from 0 to 2n. Hence, the limit cycle is a one-dimensional manifold. Since the one-to-one repre­sentation defines a line over the same interval from 0 to the ^-coordinate can be used directly as a local coordinate on the manifold, independent of the phase space coordinates which describe the limit cycle.

 

Fig. 2.4.5

Two-dimensional torus in the three-dimensional phase space

 

<P2

 

Fig. 2.4.6 The two-dimensional manifold torus in the local coordinates ipi and ip2 mapped one-to-one onto a square

 

A further example of a manifold in three dimensions is the two-dimensional torus (fig. 2.4.5). For such a torus, each surface element can be mapped one-to-one onto an element in the plane. The assembly of all the overlapping surface elements forms once again a torus. In a corresponding manner, the surface elements on the associated plane can be joined together. Their assembly yields the square in fig. 2.4.6. With scissors and glue and by rolling and bending, this square can be turned back into the torus; in this way, it becomes clear that each single point of the torus is defined by the coordinates if i and A "screw line" on the torus corresponds to a general straight line in the mapping plane. Figure 2.4.6 demonstrates both the coordinate lines tpi = const and tp2 = const.

Since both the limit cycle and the torus are hypersurfaces in the phase space and a velocity vector F exists at each point, they form differentiate manifolds.