BANACH CENTER PUBLICATIONS, VOLUME 33 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES
WARSZAWA 1996
SINGULARITIES OF WAVE FRONTS AT THE BOUNDARY BETWEEN TWO MEDIA
O L E G M Y A S N I C H E N K O Moscow Aviation Institute
Volokolamskoe shosse 4, 125871, Moscow, Russia E-mail: mjasnich@k804.mainet.msk.su
1. Introduction. The subject of this talk is the study of Legendre singularities which arise from the problem of wave front refraction at a common boundary of two media.
Example. Let a wave front propagate in a plane divided by a line in two media with the propagation velocities v
1and v
2. It is well known that the formula of Snellius holds:
sin ϕ
1/ sin ϕ
2= v
1/v
2, where ϕ
1and ϕ
2are the incidence angles of incoming and refracted rays. If v
1< v
2, then there exists a value ¯ ϕ
1such that the refracted ray is tangent to the boundary. This point of the boundary is called the point of complete reflection. It is not difficult to see that at the point of complete reflection an envelope of the refracted rays appears and the refracted front becomes singular.
In terms of contact geometry this situation may be formulated as follows:
Let M be a smooth manifold of dimension n+1 , J
1(M, R) be the contact manifold of 1-jets of functions on M . For any hypersurface H ⊂ J
1(M, R) and an integral submanifold
∂L ⊂ H of dimension n (for the contact hyperplanes distribution (see [2], Chapter 3)), we call a Legendre submanifold L such that ∂L ⊂ L ⊂ H the solution of the Cauchy problem for H with the initial value ∂L. If L is a 1-graph of some function on M , then this function is the solution of the corresponding first order partial differential equation ([1]).
The front of a Legendre manifold L is the image of L under the natural projection of forgetting derivatives, π : J
1(M, R) → M × R. A family of hypersurfaces F (x, q, t) = 0, x ∈ X, q ∈ M, t ∈ R, where X is a space of additional parameters, defining the Legendre submanifold L = {(t, q, p) ∈ J
1(M, R) | ∃x, F = F
x0= 0, p = F
q0} with the front {(t, q) ∈ R × M | ∃x, F = F
x0= 0} is called a generating family of hypersurfaces of L ([2]).
1991 Mathematics Subject Classification: Primary 35R05; Secondary 70H20.
Research partially supported by ISF grant MSD000 and by RFFI grant 94-01-00255.
The paper is in final form and no version of it will be published elsewhere.
[235]
Let ∂M ⊂ M be a smooth hypersurface dividing M in two domains M
1and M
2; H
1, H
2⊂ J
1(M, R) be smooth hypersurfaces of first order partial differential equations.
We are interested in the discontinuous equation H = (H
1∩J
1(M
1, R))∪(H
2∩J
1(M
2, R)).
Let us consider the Cauchy problem for H
1with an initial value ∂L
0⊂ J
1(M
1, R).
Let L
1be the solution of this problem. Let ∂J
1(M, R) denote the preimage of ∂M × R under π : J
1(M, R) → M × R. The intersection ∂L
1= L
1∩ ∂J
1(M, R) defines the initial value for H
2as follows ([3]):
1. Let ρ : ∂J
1(M, R) → J
1(∂M, R) be the natural projection along the characteristics of ∂J
1(M, R) and ¯ ∂L = ρ(∂L
1). In general (L
1is transversal to ∂J
1(M, R)), ¯ ∂L is a Legendre submanifold of J
1(∂M, R).
2. Let ∂L
2denote the preimage of ¯ ∂L in H
2∩ ∂J
1(M, R) under ρ. We consider this integral (for the contact hyperplanes distribution) variety (smooth if ¯ ∂L is transversal to the critical values set of ρ restricted to H
2∩ ∂J
1(M, R)) as the initial value for H
2.
Definition. The solution L
2of the Cauchy problem for H
2with initial value ∂L
2is called refracted.
We construct the solution of the Cauchy problem for discontinuous H with the initial value ∂L
0as L = (L
1∩ J
1(M
1, R)) ∪ (L
2∩ J
1(M
2, R)) .
R e m a r k. The construction of ∂L
2naturally arises in theoretical mechanics: tra- jectories of a Hamiltonian system minimize the action functional. Studying Hamil- tonian systems with discontinuous hamiltonians h : T
∗M → R, h(q, p) = h
1(q, p) if q ∈ M
1, h(q, p) = h
2(q, p) if q ∈ M
2, where h
iare smooth, and minimizing the corre- sponding action functional one gets the construction described above.
At a point x ∈ ∂J
1(M, R)∩H
2let the restriction of ρ to H
2∩∂J
1(M, R) be equivalent to the A
i-singularity (i.e. to the projection of a hypersurface {(z, λ) | z
i+1+λ
1z
i−1+. . .+
λ
i= 0} in the total space of a fibration (z, λ) 7→ λ to the base space {λ}), ρ(x) ∈ ¯ ∂L and y = π
∂◦ ρ(x) where π
∂: J
1(∂M, R) → ∂M × R is the derivatives forgetting projection.
Definition. The point y is called an A
i-point of complete reflection.
Example. An optical wave front propagating in R
ndivided by a smooth ∂R
nin two isotropic and homogeneous media, in the natural coordinates (t, q, p) on J
1(R
n, R) (here q are some coordinates on R
n, t is a coordinate on R, p = t
0q) is the front of an H’s solution for H
i= {(t, q, p) | v
i2(p
21+ . . . + p
2n) = 1}, i = 1, 2, where v
iare the propagation velocities. One can see that the notion of complete reflection point defined above coincides with that of geometrical optics.
In the following, saying that an object is in general position (or simply generic) we mean that it belongs to an open and everywhere dense set in the Whitney topology.
2. Results
2.1. Wave front refraction
Theorem 1. If the front of ¯ ∂L at an A
i-point of complete reflection is smooth, then
for a generic refracted solution its front in some neighbourhood of this complete reflection
point is given by a generating family of hypersurfaces
yx
i+1+ y
2ϕ(x, y, q
3, . . . , q
n+1) + yx
i−1q
i+1+ . . . + yxq
3+ yq
2+ xq
0+ q
1= 0 for some i such that i < n + 1, where ϕ is some smooth function, and (q
0, . . . , q
n+1) are coordinates on M × R such that ∂M × R = {q
0= 0}.
Corollary 1. If the front of ¯ ∂L at an A
1-point of complete reflection is smooth, then for a generic refracted solution its front in some neighbourhood of this complete reflection point is given by the generating family of hypersurfaces
yx
2+ y
m−1+ y
m−2q
m−1+ . . . + yq
2+ xq
1+ q
0= 0 for some m such that 2 < m < n + 3.
Corollary 2. Let ϕ(0) 6= 0, then for a generic refracted solution its front in some neighbourhood of the considered A
i-point of complete reflection, in some coordinates (q
0, . . . , q
n+1) on M × R, is given by a generating family of hypersurfaces
x
2i+2+ x
2iq
i+ . . . + x
i+1q
2+ x
iϕ
1(q) + . . . + x
2ϕ
i−1(q) + xq
0+ q
1= 0, where i < n + 1 and ϕ
jare some smooth functions such that the rank of
∂(q∂(ϕ1,...,ϕi−1)i+1,...,qn+1)
|
q=0is equal to zero.
In what follows we assume H
2= {h(t, q, p) = 0}, the function h is smooth, quadratic and convex in p. This condition guarantees that only A
1-points of complete reflection may occur. Note that the Hamilton-Jacobi equations of classical mechanics’ natural systems belong to this class.
Theorem 2. If the restriction of the projection π
∂to ¯ ∂L at the point in J
1(∂M, R) over an A
1-point of complete reflection is equivalent to the A
n−k+1-singularity, then for a generic refracted solution in some neighbourhood of this point of complete reflection, in suitable coordinates (q
0, . . . , q
n+1) on M × R, its front is given by one of the following generating families:
1. n = 1: F (x, y, q) = x
2y + y
2ϕ(x, y) + yq
1+ xq
0+ q
2= 0,
2. n > 1: F (x, y, z, q) = z
n+2−k+ y(x
2+ z
m) + y
2ϕ(x, y, z, q
2, . . . , q
n−1) + z
n−kq
k+1+ . . .+zq
n+y(z
m−1q
m+. . .+zq
2+q
1)+xq
0+q
n+1= 0, where 0 < m ≤ k ≤ n, m < 2+n−k, ϕ is some smooth function.
This theorem provides typical singularities which may occur at points of complete reflection in J
1(M, R) for dim M = 2, 3, 4. Namely:
Corollary 3. Generic generating families 1. and 2. are V -equivalent (i.e. can be transferred to each other by a diffeomorphism of the form
(x, y, z) 7→ (x
0(x, y, z, q), y
0(x, y, z, q), z
0(x, y, z, q), q
0(q));
such families define Legendre equivalent germs) to the following generating families:
n = k = 1 : y
2+ x
4+ x
2q
1+ xq
2+ q
3= 0, n = k = 2 : z
2+ y
2+ x
4+ x
2q
1+ xq
2+ q
3= 0,
z
2+ y
3± yx
2+ y
2q
1+ yq
2+ xq
3+ q
4= 0,
n = 2, k = m = 1 : z
2+ y
2+ x
6+ x
4q
1+ x
3ϕ(q) + x
2q
2+ xq
3+ q
4= 0,
n = k = 3 : z
2+ y
2+ x
4+ x
2q
1+ xq
2+ q
3= 0, z
2+ y
3± yx
2+ y
2q
1+ yq
2+ xq
3+ q
4= 0, z
2+ y
4+ yx
2+ y
3q
1+ y
2q
2+ yq
3+ xq
4+ q
5= 0, n = 3, k = m = 2 : z
2+ x
3+ y
4+ y
2xq
1+ y
2q
2+ xyϕ(q) + yq
3+ xq
4+ q
5= 0,
n = 3, k = 2, m = 1 : z
2+ y
2+ x
6+ x
4q
1+ x
3q
2+ x
2q
3+ xq
4+ q
5= 0, n = 3, k = m = 1 : z
2+ y
2+ x
8+ x
6q
1+ x
5ϕ
1(q) + x
4q
2= 0,
+ x
3ϕ
2(q) + x
2q
3+ xq
4+ q
5= 0.
Here ϕ and ϕ
iare functional moduli.
2.2. Normal forms of smooth functions on Legendre submanifolds in the spaces of Legendre fibrations. For the proof of Theorem 2 we need to reduce to the normal form a generic smooth function on ¯ ∂L by a Legendre equivalence (i.e. a contactomorphism of the total space of a Legendre fibration preserving the fibers).
Let P T
∗R
n+1be the projectivization of the cotangent bundle of R
n+1, i : (L, x) ,→
(P T
∗R
n+1, y) be a Legendre submanifold germ, Φ : (P T
∗R
n+1, y) → (R, 0) be a smooth function germ, (q
1, . . . , q
n+1; p
1: . . . : p
n+1) be the standard coordinates on P T
∗R
n+1(q
1, . . . , q
n+1being some coordinates on R
n+1, and p
1: . . . : p
n+1the homogeneous coordinates on the fibers of the projectivized cotangent bundle induced from the canonical coordinates on the fibers of the cotangent bundle of R
n+1) in some neighbourhood of y.
Let the derivative of π ◦ i at x , where π : P T
∗R
n+1→ R
n+1is the cotangent bundle projectivization, have one-dimensional kernel.
Theorem 3. For a generic pair (i(L), Φ|
i(L)) its germ at the point i(x) such that (π ◦ i)
∗,xhas one-dimensional kernel and Φ(i(x)) = 0, is Legendre equivalent to the germ at (q = 0, p
1= . . . = p
n= 0) of the pair (L
0, Φ
0) defined by:
1. L
0= {(q; p) | q
n+1= f (q
k+1, . . . , q
n−1, p
n) + p
nq
n; p
i= ∂f /∂q
ifor i = 1, . . . , n − 1; q
n= −∂f /∂p
n; p
n+1= 1}, where f = p
n+2−kn+ p
n−knq
k+1+ . . . + p
2nq
n−1,
2. Φ
0(q
1, . . . , q
n−1, p
n) = p
mn+ q
mp
m−1n+ . . . + q
1for some k, m such that 0 < m ≤ k ≤ n, m < 2 + n − k.
3. Proofs
3.1. Preliminary results. Since we are interested in local situation, we may replace the Legendre fibration J
1(M, R) → M × R of derivatives forgetting by the Legendre fibra- tion P T
∗(M × R) → M × R (the projectivized cotangent bundle) because all Legendre fibrations of equal dimensions are locally isomorphic (see [2], Chapter 3).
Let us consider the action on ∂P T
∗(M ×R) of a Legendre equivalence of P T
∗(M ×R)
preserving ∂P T
∗(M × R). In the canonical coordinates (q; p) = (q
0, q
0; p
0: p
0) induced
from the coordinates q on M ×R such that ∂M ×R = {q
0= 0}, any Legendre equivalence
of P T
∗(M × R) preserving ∂P T
∗(M × R) is generated by a diffeomorphism of M × R
of the form
(q
0, q
0) 7→ (q
0ξ
0(q), ˜ q
0(q)), ξ
0(0) 6= 0.
Writing this Legendre equivalence explicitly one gets the following:
Lemma 1. In the chosen coordinates the restriction to ∂P T
∗(M × R) of any Legendre equivalence of P T
∗(M ×R) preserving ∂P T
∗(M ×R) is the superposition of the following two:
1. Legendre equivalence of P T
∗(∂M × R), 2. ˜ p
0= ξ
0(q
0)p
0+ . . . + ξ
n+1(q
0)p
n+1, ξ
0(0) 6= 0.
Conversely for any composition of 1. and 2. there exists a Legendre equivalence of P T
∗(M × R) preserving ∂P T
∗(M × R) and inducing this transformation.
Let H
2= {(q; p) | h(q, p) = 0}, with h smooth and quadratic in p. In the canonical coordinates (q
0, . . . , q
n+1; p
0: . . . : p
n+1) = (q
0, q
0; p
0: p
0) on P T
∗(M × R) such that
∂M × R = {q
0= 0} we have
h(0, q
0, p
0, p
0) = (ξ
0(q
0)p
0+ . . . + ξ
n+1(q
0)p
n+1)
2+ Φ(q
0, p
0) for some smooth ξ
i, ξ
0(0) 6= 0 and Φ : P T
∗(∂M × R) → R.
Applying Lemma 1 we get:
Lemma 2. The restriction of the canonical projection ρ to H
2is reducible to the form {(0, q
0; p
0: p
0) | p
20+ Φ(q
0, p
0) = 0} 7→ (q
0; p
0) by a Legendre equivalence preserving
∂P T
∗(M × R).
Notice once again that H
2is the zero set of some smooth and quadratic (in p) function.
3.2. Proof of Theorem 1. Let us choose a system of coordinates (q
0, . . . , q
n+1) in some neighbourhood of the considered A
i-point of complete reflection y = π
∂◦ ρ(x) such that
∂M × R = {q
0= 0} and π
∂( ¯ ∂L) = {q
0= q
1= 0}. This is possible because π
∂( ¯ ∂L) is smooth. These coordinates induce the canonical coordinates (q; p) (= (q
0, q
0; p
0: p
0) = (q
0, q
1, q
00; p
0: p
1: p
00)) in some neighbourhood of x. Under the assumption of the theorem at the point x the projection ρ : ∂P T
∗(M × R) ∩ H
2→ P T
∗(∂M × R) is equivalent to the A
i-singularity, hence ∂P T
∗(M × R) ∩ H
2= {(q; p) | q
0= 0, p
i+10+ f
0(q
0, p
0)p
i0+ . . . + f
i(q
0, p
0) = 0}.
We have ¯ ∂L = {(q; p) | q
0= q
1= p
0= p
00= 0}, hence ∂L
2= ρ
−1( ¯ ∂L) ∩ H
2is of the form
∂L
2= {(q; p) | p
i+10+ ϕ
0(q
00)p
i0+ . . . + ϕ
i(q
00) = q
0= q
1= p
00= 0, p
1= 1}, where ϕ
j(q
00) = f
j(0, q
2, . . . , q
n+1, 1, 0, . . . , 0). We can see that in general i ≤ n.
A diffeomorphism of M × R such that its inverse is of the form q
0= ˜ q
0, q
1= ˜ q
1−
˜
q
0ϕ
0(˜ q
0)/(i + 1), q
00= ˜ q
00, induces a Legendre equivalence of P T
∗(M × R) reducing ∂L
2(forgetting tilde): ∂L
2= {(q; p) | p
i+10+ ψ
1(q
00)p
i−10+ . . . + ψ
i(q
00) = q
0= q
1= p
00= 0, p
1= 1}.
In general the rank of
∂(q∂(ψ1,...,ψi)2,...,qn+1)
is maximal , say det
∂(q∂(ψ1,...,ψi)2,...,qi+1)
|
q=06= 0. A Legendre
equivalence of P T
∗(M × R) induced by ˜ q
0= q
0, ˜ q
1= q
1, ˜ q
2= ψ
i(q
00), . . . , ˜ q
i+1=
ψ
1(q
00), ˜ q
i+2= q
i+2, ˜ q
n+1= q
n+1preserves ∂P T
∗(M × R) and brings ∂L
2to the form
∂L
2= {(q; p) | p
i+10+ q
i+1p
i−10+ . . . + q
2= q
0= q
1= p
00= 0, p
1= 1}.
Assume that L
2is a Legendre submanifold containing ∂L
2. Locally it is given by some generating function S(p
J, q
I)(I ∪ J = {0, 2, . . . , n + 1}, 0, 2 ∈ J ) : L
2= {(q; p) | q
1= S, q
J= −S
p0J
, p
I= S
q0I
, p
1= 1}, and we know the values of S and its derivatives at p
2= 0:
S(p
0, 0, q
3, . . . , q
n+1) = S
p00(p
0, 0, q
3, . . . , q
n+1) = 0, S
q0i(p
0, 0, q
3, . . . , q
n+1) = 0, i = 3, . . . , n + 1, S
p02(p
0, 0, q
3, . . . , q
n+1) = p
i+10+ q
i+1p
i−10+ . . . + q
3p
0.
Hence S = p
i+10p
2+ p
22ϕ(p
0, p
2, q
3, . . . , q
n+1) + p
i−10p
2q
i+1+ . . . + p
0p
2q
3, where ϕ is some smooth function. Hence L
2can be given by the generating family from the statement of the theorem.
3.3. Proof of Theorem 2. The proof of the case n = 1 is exactly the same as in Theorem 1 (here i = 1 and in the generic case a point of complete reflection does not belong to the caustic in the first medium, hence the front of ¯ ∂L is smooth at this point).
Let n > 1. In the generic case ¯ ∂L is smooth. If the projection of ¯ ∂L to ∂M × R is equivalent to the A
n−k+1-singularity, then from Theorem 3 and Lemma 2 we get that
∂L
2= ρ
−1( ¯ ∂L) ∩ H
2is of the form
q
n+1= p
n+2−kn+ p
n−knq
k+1+ . . . + p
2nq
n−1+ p
nq
n. q
0= 0.
q
1= −p
20− p
mn− p
m−1nq
m− . . . − p
nq
2.
q
n= −(n + 2 − k)p
n+1−kn− (n − k)p
n−k−1nq
k+1− . . . − 2p
nq
n−1. p
1= . . . = p
k= 0.
p
k+1= p
n−kn, . . . , p
n−1= p
2n. p
n+1= 1.
Hence for any Legendre manifold L
2such that
∂L
2= L
2∩ ∂P T
∗(M × R), a generating function S is of the form
S(p
0, p
1, p
n, q
2, . . . , q
n−1) = p
n+2−kn+ p
n−knq
k+1+ . . .
+ p
2nq
n−1+ p
1(p
20+ p
mn+ p
m−1nq
m+ . . . + p
nq
2) + p
21ϕ(p
0, p
1, p
n, q
2, . . . , q
n−1).
This proves the theorem.
3.4. Proof of Theorem 3. The technique used here was developed in [5].
First of all we can reduce the germ of a generic Legendre submanifold i(L) ⊂ P T
∗R
n+1at a point i(x), where (π ◦ i)
∗has one-dimensional kernel, by a Legendre equivalence
of P T
∗R
n+1to the germ at the point q = p
1= . . . = p
n= 0 of the Legendre
submanifold L
0⊂ P T
∗R
n+1given by the generating function f (q
k+1, . . . , q
n−1, p
n) =
p
n+2−kn+ p
n−knq
k+1+ . . . + p
2nq
n−1, for some k < n, as in the theorem (see [2], Chapter 3).
Now we should reduce a generic smooth function Φ|
L0= Φ(q
1, . . . , q
n−1, p
n) (functions q
1, . . . , q
n−1, p
nform a coordinate system on L
0) such that Φ(0) = 0 to the normal form by a Legendre equivalence of P T
∗R
n+1preserving L
0.
Let us consider a subsidiary fibration κ : R
n+2→ R
n+1, κ : (x, q) 7→ q, and the Legendre submanifold N ⊂ P T
∗R
n+2having the smooth front {(x, q) ∈ R
n+2| F (x, q) = f (q
k+1, . . . , q
n−1, x)+xq
n+q
n+1= 0}. In P T
∗R
n+2consider an hypersurface P A formed by hyperplanes in spaces tangent to R
n+2, which contain lines tangent to fibers of κ. Then L
0= κ
∗π(N ∩ P A), where κ
∗π : P A → P T
∗R
n+1is the fibration induced from κ by π, κ
∗π : (x, q; 0 : p) 7→ (q; p) (see [2], Chapter 3). The function Φ on L
0can be lifted by κ
∗π to N ∩ P A and can be extended to a function (also denoted by Φ) on N . The manifold N is diffeomorphic to its front {F (x, q) = x
n−k+2+ x
n−kq
k+1+ . . . + q
n+1= 0}.
This diffeomorphism sends Φ on N to Φ on the front of N , {F = 0}. In the chosen coordinates it can be expressed much more simply: Φ(q
1, . . . , q
n−1, p
n) on L
0goes to Φ(q
1, . . . , q
n−1, x) on {F = 0} ⊂ R
n+2. In these terms the problem of reducing a function Φ(q
1, . . . , q
n−1, p
n) to the normal form by a Legendre equivalence of P T
∗R
n+1preserving L
0is equivalent to reducing the function Φ(q
1, . . . , q
n−1, x) to the normal form on the hypersurface {F (x, q) = 0} ⊂ R
n+2(i.e. on the generating family of hypersurfaces for L
0) by a diffeomorphism of R
n+2preserving {F = 0} and respecting κ. Indeed, preserving κ’s fibers we get a generating family defining a Legendre submanifold which is Legendre equivalent to L
0([2], Chapter 3). Preserving {F = 0} we get that in fact this new Legendre submanifold of P T
∗R
n+1coincides with L
0.
Definition. A diffeomorphism g : R
n+2→ R
n+2preserving {F = 0} and respecting κ is called admissible.
Now we are going to describe vector fields which are the velocities of one-parameter families of analytic admissible diffeomorphisms. Let E
n+2and E
n+1denote the rings of analytic functions’ germs at the points 0 ∈ R
n+2and 0 ∈ R
n+1, and let V denote the real vector space of germs at 0 of vector fields on R
n+2with pointwise addition and multiplication by scalars. In V we consider a subspace ˜ V = V
1⊕ V
2, where V
1is the E
n+2-module generated by v
0= F
∂x∂, V
2is the E
n+1-module generated by v
i, where v
i=
∂q∂i
for i = 1, . . . , k and v
i= v
ix(x, q)
∂x∂+ v
qik+1(q)
∂q∂k+1
+ . . . + v
qin+1(q)
∂q∂n+1
for i = k + 1, . . . , n + 1 are the solutions of the equations
v
i• F = F F
q0i(= F x
n+1−i), i = k + 1, . . . , n + 1.
Here the symbol • stands for the differentiation along the corresponding vector field.
Lemma 3. The space ˜ V is the tangent space to the space of germs at the origin of analytic admissible diffeomorphisms.
P r o o f. Let us consider a one-parameter family of admissible diffeomorphisms g
t: g
t(x, q) = (h
t(x, q), ϕ
1,t(q), . . . , ϕ
n+1,t(q)),
(1) H
t(g
t(x, q))F (g
t(x, q)) = F (x, q),
where H
tis some one-parameter family of analytic functions such that H
t(0) 6= 0.
Differentiating (1) with respect to t, for any t
0we get the equation (2) v
xF
x0+ v
qk+1F
q0k+1
+ . . . + v
qn+1F
q0n+1
= M F, where v
x, M ∈ E
n+2, v
qi∈ E
n+1, i = k + 1, . . . , n + 1, and
v = v
x∂
∂x + v
qk+1∂
∂q
k+1+ . . . + v
qn+1∂
∂q
n+1= d dt
t=t0g
t.
The function F is an R-versal deformation of the germ F (x, 0), hence it is R-infinite- simally versal, hence for M there exist functions v
x0∈ E
n+2; v
0qi∈ E
n+1, i = k+1, . . . , n+1, such that
(3) v
0xF
x0+ v
0qk+1
F
q0k+1
+ . . . + v
0qn+1
F
q0n+1
= M.
After substitution of (3) in (2), keeping in mind the uniqueness of decompositions (2) and (3) for analytic functions, we get that v = v
x ∂∂x
+ v
qk+1 ∂∂qk+1
+ . . . + v
qn+1 ∂∂qn+1
= v
x0v
0+ v
q0k+1
v
k+1+ . . . + v
q0n+1v
n+1. The vector fields v
1, . . . , v
kcorrespond to changes of q
1, . . . , q
k. The lemma is proved.
Corollary 4. 1. The v
iare polynomial vector fields.
2. The v
xiare regular with respect to x of order n + 2 − i for i = k + 1, . . . , n + 1 and of order n + 2 − k for i = 0.
3. v
i(0) = 0 for i = 0, k + 1, . . . , n + 1.
Consider a function Φ(q
1, . . . , q
n−1, x), with Φ(0) = 0. Suppose that
∂Φ
∂x (0) = . . . = ∂
m−1Φ
∂x
m−1(0) = 0 6= ∂
mΦ
∂x
m(0).
For a generic pair (L
0, Φ) we have 0 < k, m < k. If m ≥ 2 + n − k, then, since we are interested in the restriction of Φ to {F = 0}, we can divide Φ by F and get m < 2 + n − k.
For a generic Φ, changing only q
1, . . . , q
k, we can bring it to the form Φ = q
1+ q
2x + . . . + q
mx
m−1+ x
m+ f, f ∈ m
m+1n+2,
where m
n+2is the maximal ideal in C
n+2∞, the ring of smooth functions’ germs at 0 ∈ R
n+2.
Let us consider
Φ
0= q
1+ q
2x + . . . + q
mx
m−1+ x
m.
Lemma 4. There exists an admissible diffeomorphism g such that g(0) = 0 and , for the corresponding germs, Φ ◦ g = Φ
0.
P r o o f. Connect Φ and Φ
0by Φ
t= Φ
0+ tf . We are going to prove the existence of a family g
t: R
n+2→ R
n+2of admissible diffeomorphisms such that
(4) Φ
t◦ g
t= Φ
0.
After differentiating (4) with respect to t we get (5) ˜ v
xt(x, q) ∂Φ
t∂x + ˜ v
q1t(q) ∂Φ
t∂q
1+ . . . + ˜ v
qn+1t(q) ∂Φ
t∂q
n+1= −f,
where
˜
v
xt(x, q) = f
00(t, x, q)v
x0(x, q) + f
0k+1(t, q)v
xk+1(x, q) + . . . + f
0n+1(t, q)v
n+1x(x, q),
˜
v
qit(q) = f
i(t, q), i = 1, . . . , k,
˜
v
qit(q) = f
0k+1(t, q)v
k+1qi(q) + . . . + f
0n+1(t, q)v
n+1qi(q), i = k + 1, . . . , n + 1, where f
iand f
0jare some unknown functions. Let us put f
m+1= . . . = f
k= 0. Writing (5) in detail we have
(6) (mx
m−1+ (m − 1)q
mx
m−2+ . . . + q
2+ tf
x0)
× (x
n+2−k+ x
n−kq
k+1+ . . . + q
n+1)f
00(t, x, q) + (mx
m−1+ (m − 1)q
mx
m−2+ . . . + q
2+ tf
x0)
× (x
n+1−k/(n + 2 − k) + . . .)f
0k+1(t, q)
+ . . . + (mx
m−1+ (m − 1)q
mx
m−2+ . . . + q
2+ tf
x0)
× (x/(n + 2 − k))f
0n+1(t, q) + (x
m−1+ tf
q0m)f
m(t, q) + . . . + (1 + tf
q01)f
1(t, q)
+ tf
q0k+1(f
0k+1(t, q)v
k+1qk+1(q) + . . . + f
0n+1(t, q)v
n+1qk+1(q)) + . . . + tf
q0n+1
(f
0k+1(t, q)v
k+1qn+1
(q) + . . . + f
0n+1(t, q)v
qn+1n+1
(q)) = −f (x, q).
Proposition 1. For any t ∈ R there exist neighbourhoods V of t and W of 0 ∈ R
n+2such that (6) has a solution smooth in V × W .
P r o o f. In the ring C
n+2∞of infinitely smooth functions’ germs at 0 ∈ R
n+2we consider the ideal A = F
∂Φ∂xtC
n+2∞. Let B denote the quotient module C
n+2∞/A. For any t ∈ R the function F
∂Φ∂xtis regular with respect to x of order n + m + 1 − k. From Malgrange’s division theorem it follows that B is a finitely generated C
n+1∞-module (C
n+1∞stands for the ring of germs at 0 ∈ R
n+1of infinitely smooth functions of the variables q). The functions 1, x, . . . , x
n+m−kcan be chosen as generators. In B we consider the submodule C generated by the images in B of the functions
e
1= 1 + tf
q01, . . . , e
m= x
m−1+ tf
q0m,
e
m+1= (mx
m−1+ (m − 1)q
mx
m−2+ . . . + q
2+ tf
x0) x n + 2 − k + tf
q0k+1
v
qn+1k+1
(q) + . . . + tf
q0n+1
v
n+1qn+1
(q), . . . , e
m+n+1−k= (mx
m−1+ (m − 1)q
mx
m−2+ . . . + q
2+ tf
x0)
x
n+1−kn + 2 − k + . . .
+ tf
q0k+1
v
qk+1k+1
(q) + . . . + tf
q0n+1
v
k+1qn+1