Lecture
We see evolutionary processes everywhere, from the motion of
atoms to the dynamics of planets. Newton understood that these
processes are described by differential equations,
and that it is useful to solve these equations… It turned out that
even the qualitative behavior of a solution can be very
complicated. The situation simplifies dramatically if one considers only equations in general position. From the point of view of physics, only the latter are of interest.
The quotation above, taken from [23], speaks of differential equations, but the situation it describes applies to objects of the
most varied kind, including germs of functions, mappings, etc.
101
Constructing a surveyable classification of literally all germs of smooth functions or mappings is just as unrealizable as an attempt to
solve all differential equations. Moreover, for a classification to be meaningful and surveyable, the normal forms
must not be too complicated, and for this reason one is forced
to restrict oneself to singularities satisfying certain additional conditions (recall, for instance, the singularities Aₖ, D in Section 4).
The philosophy of general position states that (first and foremost) one should
study typical singularities. Informally speaking, these are singularities that are stable with respect to small perturbations of the corresponding object (function, mapping, differential
equation, etc.), i.e. they do not disappear and do not turn into singularities of other types (see the examples and discussion in Section 1.3). Thus,
when speaking of the C∞-classification of germs of mappings (denote their set by the letter M), a germ is typical if any small
perturbation of it is C∞-equivalent to the original germ.
Here, by a small perturbation is meant that the germ
of the «perturbed» mapping (and all its derivatives) differs little from the original one (and its corresponding derivatives). This notion
can be given a rigorous mathematical meaning if one turns the set M into a topological space (with a suitable topology)
and thereby defines in it the notions of an open set and
a neighborhood of a point. Then a germ f ∈ M is typical if there
exists a neighborhood Uf ⊂ M such that f ∼ g for any g ∈ Uf. From this
it is easy to derive the openness of the set of typical germs: for g ∈ Uf
take a neighborhood Ug ⊂ Uf; then for any h ∈ Ug, from f ∼ g
and g ∼ h, by transitivity it follows that g ∼ h.
However, we have not yet said the most important thing: exactly how the «suitable» topology on M is introduced. Clearly, not just any topology
will do: for instance, if one introduces the topology generated by the discrete metric (in which the distance between any two distinct points is taken to be 1), then every germ turns out to be
stable, which of course is entirely vacuous.
The topology suitable for our purpose is called the fine topology (also the Whitney topology). It is defined by means of
K-jets of mappings of all orders, which corresponds to the requirement of smallness
of the perturbation of all derivatives of the given mapping. The degree of closeness of the mappings f and g is determined by arbitrarily given positive numbers estimating the difference of jets of the coordinate
102
functions of the mappings f and g. We shall not introduce the Whitney topology here,
referring the reader to the specialized literature, for example [15, 43].
We say that we are considering the case of general position in the class M if we study the classification not for all elements of this
class, but for some open and everywhere dense (in the Whitney
topology) subset M' ⊂ M. We have already discussed above the meaning of the openness condition, while
the requirement of everywhere-density means that although the set M' does not coincide with the whole of M, nevertheless for every f ∈ M, in any of its neighborhoods Uf there is found an element g ∈ M'. One can say
that the subset M' well approximates all elements of the class M.
Accordingly, the expression a germ in general position means that
the germ f ∈ M under consideration belongs to some open,
everywhere dense subset M' ⊂ M, and no further conditions on f
are imposed.
When studying specific objects, one is usually guided by
codimension considerations: this is convenient because it allows one to pass from abstract topological properties to entirely concrete
conditions on the jets of mappings (most often the matter is restricted to
jets of finite order, and the conditions on their coefficients take
the form of equalities). By studying singularities in order of increasing codimension (from the smallest, 1, up to n, the dimension of the source space), we thereby order the set of stable
singularities and divide it into parts, so-called strata. The larger
the codimension of a singularity, the rarer the representatives of the corresponding stratum.
Critical points whose codimension exceeds n are unstable with respect to arbitrarily small perturbations of germs, and therefore
do not occur for mappings in general position. However, they
may occur for families of mappings depending on parameters, and be stable within the class of families: under small perturbations of
families, critical points of the given type do not disappear, but merely pass from one member of the family to another. We have already discussed
this question in Section 1.3.
The notion of general position must be handled with care:
it depends essentially on the choice of the class of objects under study.
Sometimes degenerations of very large (or
even infinite) codimension may be of interest, if they occur in applications. Thus, for example, in the theory of differential equations (and indeed,
in mathematics and physics generally) an extremely important role is played by the so-called Hamiltonian systems — systems of autonomous differential
103
equations of a special form, by means of which many problems of mechanics, the calculus of variations, control theory and others are described. At the same time, in the (infinite-dimensional) space of all autonomous systems of differential equations, the class of Hamiltonian
systems has infinite codimension.
In this course we shall encounter two examples of this kind. First, in Section 11 we shall consider germs of smooth mappings R² → R² (this is the class M) in general position, but also germs of the so-called «frontal mappings», whose class MF
has infinite codimension in M. The notion of «general position»
in the class MF is entirely different from that in M, which naturally
leads to an entirely different list of normal forms.
The second example is even more interesting; it relates to differential
equations of first order not solved with respect to the derivative, and to the envelopes of families of their solutions. This is the so-called «Clairaut paradox», which we analyze in detail in Section 14.
From what has been said one should conclude that both the question of the choice of objects of study, and the question of defining «general position», are quite subtle, and in seeking an answer to them one must take into account not
only codimension, but other concrete considerations as well.
The remarkable book by three authors begins with a quotation from the story «On the Road» by A.P. Chekhov. We would like
to close this section with the continuation of the same quotation:
The thing is that every science has a beginning, but has
no end at all, just like a repeating decimal. Zoology has discovered 95,000 species of insects, chemistry counts
60 simple substances. If in time ten more digits are added to these figures on the right, zoology and chemistry will
be just as far from their end as they are now…
Comments