Lecture
In conclusion let us discuss one curious question connected with the existence of envelopes of families of solutions of implicit differential equations. At one time it gave rise to a dispute between the well-known French mathematicians Darboux and Catalan".
Consider an arbitrary family of curves in the (т, у)-plane,
given by the formula
Ф(т, у, с) = 0, (14.1)
with a smooth function Ф, where с is a parameter playing the role of the «number» of the curve in the family. It is not assumed a priori that these are the integral
curves of some differential equation. Let us find out under what conditions this family has an envelope
т=Ф(К, у=\(, ТЕГ. (14.2)
It is clear that a necessary condition for this is the following: there exists a function &(#), not constant on any interval,
for which the equalities hold,
(9,40, 0) =0, $.(2(, 4,8) =0. = (143)
Proof. Indeed, suppose that the curve
(14.2) is an envelope, but this condition does not hold. From the first it follows that for every { Е Г there is a real
«number» с for which Ф(х,у,с) = 0. Generally speaking, this «number»
1]еап Сазюоп ОагЬоих (1842-1917). Епяёпе-Сваез Сабайап (1814 - 1894).
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с may not be unique (the curves (14.1) may intersect); denote by с» all such «numbers».
From the second assumption it follows that for some +, Е Ги
for all corresponding со the equality Ф.(т., у», Со) 2 0 holds, where
т. = Ф(ЁЬ,) иц, = (Ё,), otherwise we could set &(1,) = съ with the
со for which the equality holds. By the implicit function theorem
the condition Ф.(т„, у», Со) = 0 means that in a neighborhood of the point (1„, У», Сао)
equation (14.1) is solvable for с, and the family of curves has
the form of level lines /(7, у) = с of some smooth function /. Consequently, by means of a suitable local diffeomorphism this family is transformed into a family of parallel lines, see fig. 1.1.
Since the property of being an envelope is preserved under a diffeomorphism,
the family of parallel lines must also have an envelope,
which is obviously false. |
In the case of a specific family this necessary condition usually
allows one to find envelopes or to prove their absence. Consider
the system of equations in three variables
Ф( т, у, с) — 0, Ф.(х, у, С) — 0, (14.4)
and eliminate с from it. This gives the equation О(т,у) = 0, which defines on the plane (т, у) the discriminant set (discriminant curve) of the family (14.1). In general position it
consists of one or several curves, and for each of them one can
easily check whether it is an envelope of the family (14.1) or not.
Below we give examples showing that the derived necessary condition is not sufficient.
EXAMPLE 14.1. Consider the family of curves
Ф=у+ 17 + 21с — ас? = 0, (14.5)
where с is the parameter of the family, «о is some fixed number. The equality Ф, = 0 is equivalent to х — ас = 0. Assuming that а = 0,
we get с = х/а. Substituting the latter expression into (14.5), we find that
the discriminant set is the curve
у (1+а` 7)? = 0. (14.6)
Now let us find the intersection points of the curves (14.5) and (14.6). This gives
the equation 12 +2тс— ос? = (1+ а 122, from which we find: х = —2ас.
Hence, for each value of the parameter с the curve (14.6) intersects
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the corresponding curve of the family (14.5), and at a single
point 1 = —2ас, which, as is easy to see, is a point of tangency.
Thus the curve (14.6) is an envelope of the family (14.5)
for any а = 0. It remains to consider the case а = 0. Then Ф. = 2х,
the discriminant set is given by the equations 1 = Оиу = 0, i.e., it
consists of a single point and cannot be an envelope.
EXAMPLE 14.2. Consider the family of curves
Ф =у+ 17 — 2аса + с? = 0, (14.7)
where с is the parameter of the family, @ is a fixed number. The equality
Ф. = 0 gives с = ах; substituting it into (14.7), we find that the discriminant
set is the curve
у = (а? - 117. (14.8)
If а = 0, the parabola (14.8) intersects each curve of the family
(14.7) at a single point т = с / а, which is a point of tangency.
Hence the curve (14.8) is an envelope. If а = 0, then
the parabola (14.8) coincides identically with one of the curves of the family
(14.7), and therefore it is not an envelope.
Consider the surface „Я, given by equation (14.1) in the three-dimensional space (5, у, с). The curve of the family (14.1) with parameter
с. «lifts» to the surface „Я in an obvious way: the result of the «lifting» is the curve obtained by cutting the surface
-# by the «horizontal» plane с = с,. All such curves are integral curves of the field of directions obtained by intersecting the field of tangent planes to „Я and the field of horizontal planes
4с = 0. Clearly, this field of directions can be given by means of
the vector field:
&=Ф, 9=-Ф, 6=0, (14.9)
whose integral curves become curves of the family (14.1)
after projecting л the surface „Я onto the plane (т, у) along the direction of the с axis.
Denote by 5 the criminant (the set of critical points)
of the projection mapping л. It is given by the system of equations
(14.4), and its projection л(5) is the discriminant set.
According to what was proved above, the envelope of the family (14.1) is entirely contained in its discriminant set.
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PROBLEM 14.1. Prove the following sufficient condition: if
at all points of the criminant the conditions
9(Ф,Ф.) Эк.) я0, Фе 0, (14.10)
hold, then the discriminant set is a regular curve without singularities and is an envelope of the family (14.1).
REMARK 14.1. A proof of the sufficient condition given above can be found in the book [20], in which envelopes of families
are studied in more detail and from a considerably more general point of view.
It contains many subtle and nontrivial observations about envelopes, which are omitted even in very good textbooks?.
REMARK 14.2. For families of curves given in parametric form
г=хХ(Ьс), у=У(Ьс) (14.11)
with smooth Х and У, there hold necessary and sufficient conditions,
analogous to those formulated above. For example, the equality Ф. = 0
is replaced by О 0 (с)
All these statements can also be found in the book [20].
The construction described above is reminiscent of the lifting of the implicit differential equation (13.1) to the surface „Я in the space „11,
including the fact that in both cases the envelope (if it exists) is contained in the discriminant set. However, between
these constructions there is a fundamental difference, caused by the different procedures for lifting curves.
In the case of a differential equation one uses the passage to
the 1-graph of the curve by formula (13.3), which we shall call
the «Legendre lifting», whereas for families (14.1) one uses the «parallel lifting» (т,у) =? (т,у,с), where the value of с is the same for
all points (т, у) lying on one curve of the family. These procedures generate different fields of planes in the corresponding
spaces: the «vertical» field of contact planes 4у = рат
in the space /! and the «horizontal» field of planes 4с = 0 in the space (1,1, с); see fig. 14.1.
See also the methodological note: Belyaeva T.B., Zalgaller V.A. On the exposition of
the theory of envelopes // Uspekhi Mat. Nauk. 1963. Vol. 18, issue 5. P. 147-149.
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EXAMPLE 14.3. Consider the equation у = у’, whose solutions
are given by the formula у = се", as a general-type equation (13.1). The corresponding surface in the space „11 is the plane р-у = 0.
The field of contact planes 4у = рат in /\ generates on it the vector field г = 1, у = р, р = р, whose integral curves after
projection give the family of solutions у = се”. On the other hand, after
the parallel lifting the curves of the family уу = се” form a nonlinear surface с = уе ® in the space (т, у, с).
Fig. 14.1 Left: Legendre lifting of curves in the space ./'. Right:
parallel lifting of curves of the family Ф(т, у, с) =0
This leads to a difference between the notions of «general position» in the class
of families of integral curves of implicit differential equations (13.1) and in the class of arbitrary families of the form (14.1) or (14.11).
Take a smooth function of general position ЁР = Ф of three variables and consider the surfaces given by the same equations Р(т,у,р) = Ои Ф(т,у,с) = 0 in the corresponding spaces, whose fields of directions are generated by the forms 4у = рат
and 4с = 0 respectively. At points of the criminant the tangent planes
to both surfaces are vertical, but in the intersection with the contact
planes 4у = рф they give vertical directions, while in the intersection with the horizontal planes с — 0 they do not.
The criminant of the equation Р(т, у, р) = 0 consists almost entirely of
points with a vertical direction of the lifted field (fig. 9.3); the only exception is the points at which the function
С — Е: + Е, vanishes (in the typical case such points are isolated on the criminant
). The discriminant curve of equation (13.1) is the geometric locus of cusps of the integral curves of this
equation (fig. 13.6, left). The case in which the discriminant curve
is an envelope (fig. 13.6, right) requires the equality
С’ = 0 at all points of the criminant, and is therefore not generic and
is destroyed by arbitrarily small perturbations of the function Р.
On the contrary, the criminant of the family Ф(т,у,с) = 0 consists of points with a
non-vertical direction of the lifted field; the only exceptions are points at which the field is simply undefined -— these are the critical points of the function Ф (see (14.9)). After projecting т the criminant turns into the envelope of the family Ф(т, у, с) = 0.
REMARK 14.3. An analogous situation holds for families given by formulas (14.11). The discriminant curve of a family in general position is an envelope, while cusps occur
rarely, only at isolated values of the parameter. For example, in the family (6.4) (example 6.1 from section 6) a cusp occurs only for one curve,
while all other curves are regular at all their points. The envelope
of this family is the axis у = 0 with one point removed — the origin, where precisely that single cusp is located.
EXAMPLE 14.4. In section 6.2.7 we considered implicit equations of a special type:
Е(р,гр-у) =0, р= ау/ ах, (14.12)
where Р(и,®) is an arbitrary smooth function. The nonsingular solutions of this
equation form a family of tangent lines to the discriminant curve, and for any lines the Legendre and parallel liftings obviously coincide. Hence the 1-graphs of nonsingular solutions are obtained as
cross-sections of the surface in the space „/\ by the planes р = с0п8.
It follows that the family of nonsingular solutions of equation (14.12)
is given by formula (14.1) with the function
Ф(т, у, с) = Е(с, те- у).
This property (the family of nonsingular solutions is obtained from the equation itself by replacing р with с) is sometimes called the Clairaut property.
REMARK 14.4. It might seem that only equations of the form (14.12) possess the Clairaut property, but this is not so. The Clairaut
property is possessed, for example, by the equation р? — ур + е? = 0, considered
by us in example 13.5. We found that this equation has two singular solutions — envelopes of the family of nonsingular solutions у = сет + 1/с,
с = 0. The latter can also be given by another formula. For example, if
one passes to a new constant б = 1/с, then after multiplying both sides of the
equality у = &+ е* /ё by & we get the formula 62 — уб+е* = 0, which
is obtained from the equation itself by replacing р with С.
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PROBLEM 14.2. We leave it to the reader to work out independently the reason for this phenomenon and to describe the entire class
of equations possessing the Clairaut property?.
In conclusion let us mention one historical curiosity connected
with implicit differential equations, described in [28].
In 1870 Darboux noticed that in general the discriminant curve of equation (13.1) is the geometric locus of cusps of its nonsingular
solutions, and not an envelope. Carnot objected that this statement contradicts a well-known fact: the discriminant curve of a typical family of curves is its envelope, and since the nonsingular solutions
of a differential equation constitute a particular case of a
general family of curves, in his opinion this should be typical for differential equations as well. The Darboux–Catalan controversy
drew the attention of many well-known mathematicians, in particular Alfred Clebsch, who examined the resulting «paradox» from the point
of view of the theory of connexes he had developed*.
REMARK 14.5. Clebsch's theory of connexes in fact leads to the construction described above of lifting an equation to a surface. In Clebsch's terminology, every differential equation has its own connex — the field of planes tangent to the surface of the given equation — and there is also, common to all equations,
the principal connex — the field of contact planes. Clebsch then considered the principal coincidence of the equation — the intersection of the principal
connex with the connex of the given equation, which in modern terminology corresponds to the lifted field of directions.
On November 23, 1872, Darboux gave a talk on this subject at a meeting of the Paris society of amateurs of knowledge (5061646 rПошайчие),
in which he gave an explanation of the «Catalan paradox» essentially coinciding with the reasoning given above. His talk was later
published as the article «On singular solutions of differential
equations of the first order»”.
3 Минтота О.5., Кейв Т.О. Мала опв ап сепегаЙваопя о СЛайтащ '5 едпаНопя //
Роы. Неюктаен. РаК., Ош. Веорга@., 5ег. Маф. Р1х. 1981. 716 734. Р.11 21,
Водо! Еиедись АНгед СЛебзсь (1833 1872), a German mathematician. Many of his
works were published posthumously. An exposition of Clebsch's theory of connexes can be found, for example, in the book «Höhere Geometrie» by F. Klein.
5 Рагвоих С’. Зиг 1ев зошоп8 мариЙгез Чез 6бацаНон8 Ч гепыеЦез ог4шашез Чи
ргепиег огаге /’ Ви. 8с1. пла. её авйгоп. 1873. \. 4. Р. 158-176,
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