THE OBSTACLE PROBLEM AND DIRECT PRODUCTS OF UNFURLED SWALLOWTAILS

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1997

THE OBSTACLE PROBLEM AND DIRECT PRODUCTS OF UNFURLED SWALLOWTAILS

O L E G M Y A S N I C H E N K O

Faculty of Applied Mathematics, Moscow Aviaton Institute Volokolamskoe shosse 4, 125871, Moscow, Russia

E-mail: mjasnich@k804.mainet.msk.su

1. Introduction. The obstacle problem has been studied in papers by Arnold, Given- tal, Shcherbak and other authors. This is the problem of investigating Lagrangian varieties naturally arising in variational problems with one-sided constraints.

Example. Let us consider a Euclidean space M and an obstacle in it bounded by a smooth hypersurface Γ. The shortest path between two points u, v ∈ M going around the obstacle consists of an interval of a ray (oriented straight line) l

1

going through the point u and tangent to Γ, of an interval of a geodesic l

₂

on Γ tangent to l

₁

and of an interval of a ray l

3

which is tangent to l

2

and goes through the end point v. We call l

1

the inbound ray (or the inbound geodesic), l

₃

— the outbound.

Considering the shortest paths between u and points in some neighbourhood of v we get a family of inbound rays, a family of geodesics on Γ starting at points of tangency between the inbound rays and Γ and a family of outbound rays.

The family of geodesics on Γ starting at the points of tangency determines a La- grangian variety L

_Γ

⊂ T

^∗

Γ: L

_Γ

= {(q, p) ∈ T

^∗

Γ | q belongs to a geodesic from the family, p is tangent to this geodesic, kpk = 1}.

Definition. L

^Γ

is called the variety consisting of geodesics on Γ starting at points of tangency.

The variety L

Γ

lies in the hypersurface H

⁰⁰

of all unit (co)vectors on Γ. Let l

Γ

denote the image of L

Γ

in the symplectic space of the characteristics of the hypersurface H

⁰⁰

.

Definition. l

Γ

is called the variety of geodesics on Γ starting at points of tangency.

1991 Mathematics Subject Classification: Primary 58F05; Secondary 70H99.

This research is partially supported by ISF grant MSD300 and by RFFI grant 94-01-00255.

The paper is in final form and no version of it will be published elsewhere.

[123]

In the same way the outbound rays determine a Lagrangian variety L

M

⊂ H ⊂ T

^∗

M , where H is the hypersurface of all unit (co)vectors on M .

Definition. L

M

is called the variety consisting of outbound geodesics tangent to the family of geodesics on Γ given by l

Γ

.

Let l

_M

denote the image of L

_M

in the symplectic space of the characteristics of the hypersurface H.

Definition. l

M

is called the variety of outbound geodesics tangent to the family of geodesics on Γ given by l

Γ

.

Lagrangian varieties described above are really varieties but not manifolds. The sim- plest singularity is the unfurled (open) swallowtail.

Definition ([3]). The unfurled swallowtail τ

n

of dimension n is a Lagrangian subva- riety in the symplectic space P

2n

of polynomials of the form

x

²ⁿ⁺¹

/(2n + 1)! + Q

1

x

²ⁿ⁻¹

/(2n − 1)! + . . . + Q

n

x

ⁿ

/n! − P

n

x

ⁿ⁻¹

/(n − 1)! + . . . + (−1)

ⁿ

P

1

(Q, P — Darboux coordinates) formed by polynomials with a root of multiplicity exceed- ing n.

The k-dimensional suspension over τ

_n

is τ

_n

× R

^k

⊂ P

2n

× T

^∗

R

^k

. It is denoted by τ

n+k,k

.

The following was proved in [3].

Theorem 1. If l

^Γ

is smooth then generically l

M

is symplectomorphic to τ

n+k−1,n

(k + n = dim M ) in some neighbourhood of an outbound geodesic tangent to Γ of order k.

It turns out that the assumption “l

Γ

is smooth” generically is violated. Namely ([5]):

Theorem 2. Generically l

Γ

is symplectomorphic to τ

_m+k−1,m

(m+k = dim Γ) in some neighbourhood of a geodesic on Γ which starts at a point of order k tangency between Γ and an inbound geodesic.

So, we have a very natural question: what is l

M

if l

Γ

is singular? The following gives partial answer to this question.

1. Generically l

M

is formally (at least ) diffeomorphic to τ

1

×l

Γ

in some neighbourhood of an outbound geodesic tangent to Γ of order 2.

2. If l

_Γ

is locally diffeomorphic to l × R, where l is some analytic variety, then generi- cally l

M

is formally (at least ) diffeomorphic to τ

2

×l in some neighbourhood of an outbound geodesic tangent to Γ of order 3.

R e m a r k. It follows from Theorem 2 that generically the condition “l

Γ

is locally diffeomorphic to l × R” is fulfilled except when dim M = 4 and l

_Γ

is considered in some neighbourhood of an inbound ray tangent to Γ of order 3. Indeed, l

Γ

is locally symplectomorphic to τ

m+k−1,m

and m = 0 along isolated geodesics on Γ. Along such “the most degenerate” single geodesic the order of tangency between Γ and the outbound ray is equal to 1, at isolated points the order of tangency is equal to 2, and nowhere except of the tangency point between Γ and the incoming geodesic the order of tangency exceeds 2.

Combining these results with Theorem 2 we get the following:

Generically in the obstacle problem

1. l

_M

is formally diffeomorphic to τ

₁

× τ

_n,k

, n = dim M − 2, in some neighbourhood of an outbound geodesic tangent to Γ of order 2.

2. l

_M

is formally diffeomorphic to τ

₂

× τ

_n−1,k

, n = dim M − 2, in some neighbourhood of an outbound geodesic tangent to Γ of order 3 provided either dim M 6= 4 or the outbound ray does not coincide with the corresponding inbound ray.

2. The obstacle problem in terms of symplectic geometry. Let M be the configuration manifold of a Hamiltonian system with a Hamiltonian h, Γ = {q ∈ M | F (q) = 0} be a smooth hypersurface in M restricting an obstacle, T e

^∗

M be the total space of the cotangent bundle over M , ω be the standard symplectic form on T

^∗

M , π : T

^∗

M → M be the cotangent bundle projection, F = π

^∗

F . Let us consider the e hypersurfaces H = {x ∈ T

^∗

M | h(x) = 0} and T

_Γ^∗

M = {x ∈ T

^∗

M | F (x) = 0}. Let ρ : T

_Γ^∗

M → T

^∗

Γ be the natural projection along the characteristics of the hypersurface T

_Γ^∗

M , κ : H → N be the natural projection along the characteristics of the hypersurface H (locally N is a symplectic manifold).

Lagrangian varieties formed by extremals of the action functional of a Hamiltonian system with one-sided constraint can be described as follows ([1], [2]):

Let B ⊂ M be a submanifold (initial front), L

B

⊂ T

^∗

M be a Lagrangian submanifold of covectors at points of B vanishing on tangent to B spaces, L be the union of the characteristics of the hypersurface H going through the points of L

B

∩ H. For B in general position L is a Lagrangian manifold (at least locally). For B and Γ in general position the manifold L transversally intersects T

_Γ^∗

M , hence l = T

_Γ^∗

M ∩ L is smooth and not tangent to the characteristics of the hypersurface T

_Γ^∗

M , hence ρ(l) ⊂ T

^∗

Γ is a Lagrangian submanifold. Let us consider H

⁰

and H

⁰⁰

— the sets of critical points and critical values of the restriction ρ|

H∩T_Γ^∗M

. Denote the projection along the characteristics of the hypersurface H

⁰⁰

⊂ T

^∗

Γ by κ

⁰⁰

: H

⁰⁰

→ N

⁰⁰

, the union of the characteristics of H

⁰⁰

going through the points of ρ(l) ∩ H

⁰⁰

by L

Γ

, its image in N

⁰⁰

by l

Γ

. Generically L

Γ

is singular ([5]).

Denote the union of the characteristics of the hypersurface H going through the points of H ∩ρ

⁻¹

(L

_Γ

) by L

_M

. Finally we introduce a filtration of H ∩T

_Γ^∗

M = H

⁽⁰⁾

by the order of tangency between the characteristics of H and the hypersurface T

_Γ^∗

M : H

⁽⁰⁾

⊃ H

⁽¹⁾

⊃ . . ., where H

⁽ⁱ⁾

= {x ∈ T

^∗

M | F (x) = h(x) = hF, hi(x) = . . . = . . . hF, hi, . . . , h(x) = 0}

(i times the Poisson bracket h·, ·i). It is not difficult to see that H

⁰

= H

⁽¹⁾

.

Example. In the previous example concerning extremals on a Riemannian manifold with a boundary we have: H = {(q, p) ∈ T

^∗

M | kpk = 1}, H

⁽⁰⁾

is the set of all unit covectors at points of Γ, H

⁽¹⁾

is the set of all unit covectors tangent to Γ, H

⁽²⁾

\ H

⁽³⁾

— tangent to Γ of order 2 etc. The initial front B is the initial point u, the manifold L is given by a system of rays (geodesics) going through u: L = {(q, p) ∈ T

^∗

M | q belongs to a geodesic going through u, p is tangent to this geodesic, kpk = 1}. The varieties L

Γ

, l

Γ

, L

M

, l

M

are exactly the same as described above.

Keeping in mind this example, for any Hamiltonian, we will call L

Γ

the variety con-

sisting of geodesics on Γ starting at points of tangency; l

Γ

— the variety of geodesics on Γ starting at points of tangency; L

M

— the variety consisting of outbound geodesics tangent to the family of geodesics on Γ given by l

_Γ

and l

_M

— the variety of outbound geodesics tangent to the family of geodesics on Γ given by l

Γ

.

3. Results. In what follows we assume that the Hamiltonian h is quadratic and convex in momenta.

Theorem 3. Generically l

^M

is formally diffeomorphic to τ

1

×l

Γ

in some neighbourhood of a characteristic of the hypersurface H tangent to T

_Γ^∗

M of order 2 (i.e. going through a point of H

⁽²⁾

\ H

⁽³⁾

).

R e m a r k. The genericity conditions are the following:

1. The restriction of κ (the projection along the characteristics of the hypersurface H) to H

⁽⁰⁾

is locally equivalent to the A

2

-singularity.

2. l

_Γ

is locally diffeomorphic to an analytic variety.

Theorem 4. If l

Γ

is locally diffeomorphic to l × R for some analytic l then generi- cally l

M

is formally diffeomorphic to τ

2

× l in a neighbourhood of a characteristic of the hypersurface H tangent to T

_Γ^∗

M of order 3 (i.e. going through a point of H

⁽³⁾

\ H

⁽⁴⁾

).

R e m a r k s.

1. The genericity conditions are the following:

1.1. The restriction of κ to H

⁽⁰⁾

is locally equivalent to the A

3

-singularity.

1.2. Let x ∈ ρ

⁻¹

(L

_Γ

) ∩ (H

⁽³⁾

\ H

⁽⁴⁾

) be the point under consideration (more precisely, the characteristic considered in the theorem goes through this point). We need the fol- lowing: at the point ρ(x) = (y

₁

, y

₂

) ∈ l × R

²

(∼ = L

Γ

) the edge y

₁

× R

²

is transversal to ρ(H

⁽³⁾

) in H

⁰⁰

.

2. It is not difficult to see that the decomposition of l

Γ

into l × R (or L

Γ

into l × R

²

) generically is possible except when dim M = 4 and the considered geodesics of the hypersurface H belongs to L (issues from the initial front). The reasons are exactly the same as in the case of geometrical optics (see the introduction).

4. Proofs. The main result we use in the proofs is the following theorem (proved in [1]) which gives the symplectic classification of pairs (H, H

⁽⁰⁾

), where H is a hypersurface in a symplectic manifold, H

⁽⁰⁾

is a hypersurface in H.

Theorem 5. In some neighbourhood of a point where the restriction to H

⁽⁰⁾

of the natural projection along the characteristics of the hypersurface H is equivalent to the A

k

-singularity (i.e. in some coordinates may be written in the form {(x, t

1

, . . . , t

n

) | x

^k+1

+ x

^k−1

t

₁

+ . . . + t

_k

= 0} 7→ (t

₁

, . . . , t

_n

)) the pair (H, H

⁽⁰⁾

) is reducible to the form ({q

0

= 0}, {q

0

= F = 0}) by a formal (at least ) symplectomorphism. Here q, p are Darboux coordinates and

(a) F = p

²₀

+ p

₁

if k = 1, (b) F = p

³₀

+ p

1

p

0

+ q

1

if k = 2,

(c) F = p

⁴₀

+ p

1

p

²₀

+ q

2

p

0

+ p

2

if k = 3.

R e m a r k. The case (a) was firstly proved in [4] on the C

^∞

–level.

4.1. Proof of Theorem 3. For a quadratic and convex in momenta Hamiltonian h we have ρ

⁻¹

(H

⁰⁰

) ⊂ H

⁰

(= H

⁽¹⁾

) and the restriction of ρ to H ∩ T

_Γ^∗

M is equivalent to the A

1

-singularity. Hence the restriction of ρ to H

⁰

is a diffeomorphism and the hypersurface H

⁰⁰

⊂ T

^∗

Γ is smooth. Denote by l

H

the preimage of L

Γ

in H: l

H

= ρ

⁻¹

(L

Γ

) ∩ H.

The variety l

_H

is isotropic (because ρ

⁻¹

(L

_Γ

) is Lagrangian); L

_Γ

⊂ H

⁰⁰

, hence l

_H

= (ρ|

_H(1)

)

⁻¹

(L

Γ

), hence l

H

is diffeomorphic to L

Γ

.

Let x ∈ l

H

∩ (H

⁽²⁾

\ H

⁽³⁾

), from part (b) of Theorem 5 it follows that the pair (H, H

⁽⁰⁾

) = ({h = 0}, {h = 0} ∩ T

_Γ^∗

M ) can be reduced to the form ({q

0

= 0}, {q

0

= p

³₀

+ p

₁

p

₀

+ q

₁

= 0}) by a formal symplectomorphism. Hence in some neighbourhood of x we have:

H (= {q

0

= 0}) ⊃ H

⁽⁰⁾

(= {q

0

= p

³₀

+ p

1

p

0

+ q

1

= 0})

⊃ H

⁽¹⁾

(= {q

0

= 0, p

1

= −3p

²₀

, q

1

= 2p

³₀

}) ⊃ H

⁽²⁾

(= {q

0

= q

1

= p

0

= p

1

= 0}), the functions (p

0

, q

2

, . . . , q

n

, p

2

, . . . , p

n

) = (p

0

, q

⁰

, p

⁰

) are coordinate functions on H

⁽¹⁾

, ω|

_H(1)

= dp

2

∧ dq

2

+ . . . + dp

n

∧ dq

n

.

Now we are going to prove that l

H

∈ H

⁽¹⁾

can be given by a system of equations not dependent on p

₀

. The main ingredient of the proof is the fact that l

_H

is isotropic.

Proposition 1. At points y ∈ l

H

where l

_H

is smooth ∂/∂p

₀

∈ T

y

l

_H

.

P r o o f. Let us consider an auxiliary fibration of coordinate spaces: α : R

²ⁿ⁻¹

→ R

²ⁿ⁻²

, α : (p

0

, q

⁰

, p

⁰

) 7→ (q

⁰

, p

⁰

), and the symplectic structure ω

⁰

= dp

⁰

∧ dq

⁰

on R

²ⁿ⁻²

. Our l

_H

⊂ R

²ⁿ⁻¹

and (α

^∗

ω

⁰

)|

_lH

= (ω|

_H(1)

)|

_lH

= ω|

_lH

= 0 because l

_H

is isotropic. Assume that ∂/∂p

0

6∈ T

y

l

H

and l

H

is smooth at y. Projecting T

y

l

H

: α

_∗,y

: T

y

l

H

→ T

_α(y)

R

²ⁿ⁻²

we get an isotropic subspace of dimension n in the symplectic space T

_α(y)

R

²ⁿ⁻²

. This contradiction proves the proposition.

It follows from Proposition 1 that for any smooth function g on H

⁽¹⁾

, vanishing on l

_H

, ∂g/∂p

₀

= 0 at points where l

_H

is smooth. But the singular locus is a proper and closed subset of l

H

and ∂g/∂p

0

is continuous hence ∂g/∂p

0

|

l_H

= 0. Let us denote by C

^∞

(H

⁽¹⁾

) the ring of germs at x of infinitely smooth functions on H

⁽¹⁾

, by J (l

_H

) the ideal of germs of vanishing on l

H

functions. Let f = (f

1

, . . . , f

r

)

^T

, where f

i

are some smooth representatives of the ideal J (l

H

) generators. We get:

∂f /∂p

0

(p

0

, q

⁰

, p

⁰

) = ξ(p

0

, q

⁰

, p

⁰

)f (p

0

, q

⁰

, p

⁰

)

where ξ = kξ

i,j

k is an r × r-matrix, ξ

i,j

are some smooth functions. Let Ψ(p

0

, q

⁰

, p

⁰

) be a principal matrix solution of the system of ordinary differential equations dependent on the parameters q

⁰

, p

⁰

dy/dp

0

= ξ(p

0

, q

⁰

, p

⁰

)y.

Then f (p

0

, q

⁰

, p

⁰

) = Ψ(p

0

, q

⁰

, p

⁰

)ψ(q

⁰

, p

⁰

), where ψ = (ψ

1

, . . . , ψ

r

)

^T

, ψ

i

(q

⁰

, p

⁰

) are some

smooth functions. Hence l

H

= {(p

0

, q

⁰

, p

⁰

) | ψ

1

(q

⁰

, p

⁰

) = . . . = ψ

r

(q

⁰

, p

⁰

) = 0}. From

this and the fact that l

_H

(∼ = L

Γ

) is locally diffeomorphic to l

_Γ

× R it follows that the

variety {(q

⁰

, p

⁰

) | ψ(q

⁰

, p

⁰

) = 0} is locally diffeomorphic to l

Γ

. Projecting l

H

along the

characteristics of the hypersurface H we get:

l

M

= {(q

1

, q

⁰

, p

1

, p

⁰

) | 4p

³₁

+ 27q

²₁

= 0, ψ(q

⁰

, p

⁰

) = 0}.

Hence l

_M

is locally diffeomorphic to τ

₁

× l

Γ

.

4.2. Proof of Theorem 4. Let x ∈ l

H

∩ (H

⁽³⁾

\ H

⁽⁴⁾

), from part (c) of Theorem 5 it follows that the pair (H, H

⁽⁰⁾

) = ({h = 0}, {h = 0} ∩ T

_Γ^∗

M ) can be reduced to the form ({q

0

= 0}, {q

0

= p

⁴₀

+ p

1

p

²₀

+ q

2

p

0

+ p

2

= 0}) by a formal symplectomorphism. Hence in some neighbourhood of x we have

H (= {q

0

= 0}) ⊃ H

⁽⁰⁾

(= {q

0

= p

⁴₀

+ p

1

p

²₀

+ q

2

p

0

+ p

2

= 0})

⊃ H

⁽¹⁾

(= {q

0

= 0, q

2

= −4p

³₀

− 2p

1

p

0

, p

2

= 3p

⁴₀

+ p

1

p

²₀

})

⊃ H

⁽²⁾

(= {q

0

= 0, p

1

= −6p

²₀

, q

2

= 8p

³₀

, p

2

= −3p

⁴₀

}),

the functions (p

0

, q

1

, p

1

, q

3

, . . . , q

n

, p

3

, . . . , p

n

) = (p

0

, q

1

, p

1

, q

⁰⁰

, p

⁰⁰

) are coordinate func- tions on H

⁽¹⁾

, the restriction of the symplectic structure to H

⁽¹⁾

:

ω

⁰

= ω|

_H(1)

= dp

1

∧ d(q

1

+ 12p

⁵₀

/5 + 2p

1

p

³₀

/3) + dp

⁰⁰

∧ dq

⁰⁰

or, after the change Q

1

= q

1

+ 12p

⁵₀

/5 + 2p

1

p

³₀

/3, ω

⁰

= dp

1

∧ dQ

1

+ dp

⁰⁰

∧ dq

⁰⁰

. As in the proof of Theorem 3 one proves that l

_H

may be given by a system of equations independent of p

0

:

l

H

= {(p

0

, Q

1

, p

1

, q

⁰⁰

, p

⁰⁰

) ∈ H

⁽¹⁾

| ψ

1

(Q

1

, p

1

, q

⁰⁰

, p

⁰⁰

) = . . . = ψ

r

(Q

1

, p

1

, q

⁰⁰

, p

⁰⁰

) = 0}.

From the condition of the theorem L

Γ

∼ = l × R

²

, the edge y

1

× R

²

is transversal to ρ(H

⁽³⁾

) at the point ρ(x) = (y

1

, y

2

) ∈ l × R

²

it follows that the variety l

H

is locally diffeomorphic to l × R

²

, the edge x

1

× R

²

is transversal to the submanifold H

⁽³⁾

⊂ H

⁽¹⁾

, H

⁽³⁾

= {p

0

= p

1

= 0} at the point x = (x

1

, x

2

) ∈ l × R

²

.

Proposition 2.

T

x

(x

1

× R

²

) ⊂ span {∂/∂p

0

, (∂ψ

i

/∂p

1

)(x)∂/∂Q

1

+ (∂ψ

i

/∂p

⁰⁰

)(x)∂/∂q

⁰⁰

− (∂ψ

i

/∂Q

1

)(x)∂/∂p

1

− (∂ψ

i

/∂q

⁰⁰

)(x)∂/∂p

⁰⁰

, i = 1, . . . , r}

P r o o f. We can locally decompose l

_H

into ˜ l × R

^m

, where m is maximal. Obviously T

x

(x

1

× R

²

) ⊂ T

x

(˜ x

1

× R

^m

), where x = (˜ x

1

, ˜ x

2

) ∈ ˜ l × R

^m

. We claim that

T

x

(˜ x

1

× R

^m

) = span {∂/∂p

0

, (∂ψ

i

/∂p

1

)(x)∂/∂Q

1

+ (∂ψ

i

/∂p

⁰⁰

)(x)∂/∂q

⁰⁰

− (∂ψ

i

/∂Q

₁

)(x)∂/∂p

₁

− (∂ψ

i

/∂q

⁰⁰

)(x)∂/∂p

⁰⁰

, i = 1, . . . , r}.

This follows from the facts:

1. The image of l

H

under the projection α : R

²ⁿ⁻¹

→ R

²ⁿ⁻²

, α : (p

0

, Q

1

, p

1

, q

⁰⁰

, p

⁰⁰

) 7→ (Q

1

, p

1

, q

⁰⁰

, p

⁰⁰

), is Lagrangian (the symplectic structure is dp

1

∧ dQ

1

+ dp

⁰⁰

∧ dq

⁰⁰

).

2. T

_x

(˜ x

₁

× R

^m

) ∼ = R∂/∂p

0

⊕ T

_α(x)

α(˜ x

₁

× R

^m

).

3. The tangent space to α(˜ x

1

× R

^m

) is spanned by the Hamiltonian vector fields with Hamiltonians ψ

_i

(functions ψ

_i

are representatives of J (l

_H

) and, simultaneously, representatives of J (α(l

H

))).

It follows from Proposition 2 that there exists ψ

i

(say ψ

1

) such that the vector

(∂ψ

1

/∂p

1

)(x)∂/∂Q

1

+ (∂ψ

1

/∂p

⁰⁰

)(x)∂/∂q

⁰⁰

− (∂ψ

1

/∂Q

1

)(x)∂/∂p

1

− (∂ψ

1

/∂q

⁰⁰

)(x)∂/∂p

⁰⁰

is transversal to the hypersurface {p

1

= 0}, that is, ∂ψ

1

/∂Q

1

(x) 6= 0. Using the implicit function theorem we get:

l

_H

= {(p

₀

, Q

₁

, p

₁

, q

⁰⁰

, p

⁰⁰

) ∈ H

⁽¹⁾

| Q

1

= g

₁

(p

₁

, q

⁰⁰

, p

⁰⁰

), g

_i

(p

₁

, q

⁰⁰

, p

⁰⁰

) = ψ

_i

(g

₁

(p

₁

, q

⁰⁰

, p

⁰⁰

), p

₁

, q

⁰⁰

, p

⁰⁰

) = 0, i = 2, . . . , r}.

In the coordinate (p

1

, q

⁰⁰

, p

⁰⁰

)-space R

²ⁿ⁻³

we consider the variety l

⁰

= {(p

1

, q

⁰⁰

, p

⁰⁰

) | g

2

(p

1

, q

⁰⁰

, p

⁰⁰

) = . . . = g

r

(p

1

, q

⁰⁰

, p

⁰⁰

) = 0}. The variety l

H

can be constructed from l

⁰

by the embedding into the hypersurface {Q

1

= g

1

(p

1

, q

⁰⁰

, p

⁰⁰

)} and the multiplication by p

0

- axis. Hence l

⁰

is diffeomorphic to l × R in some neighbourhood of the considered point x

⁰

= (x

₁

, x

⁰₂

) ∈ l × R (= l × R

²

∩ {p

0

= 0}). From the conditions of the theorem it follows that the edge x

1

× R is transversal to the hypersurface {p

1

= 0}, hence it is transversal to {p

₁

= ε} for any sufficiently small ε. Thus the intersections l

⁰

∩ {p

₁

= 0} (∼ = l) and l

⁰

∩ {p

1

= ε} (∼ = l) are locally diffeomorphic: there exists a diffeomorphism G

ε

: {p

1

= 0} → {p

1

= ε} sending {(q

⁰⁰

, p

⁰⁰

) | g

i

(0, q

⁰⁰

, p

⁰⁰

) = 0, i = 2, . . . , r} onto {(q

⁰⁰

, p

⁰⁰

) | g

i

(ε, q

⁰⁰

, p

⁰⁰

) = 0, i = 2, . . . , r}. The diffeomorphism

G : H

⁽¹⁾

→ H

⁽¹⁾

, G : (p

0

, Q

1

, p

1

, q

⁰⁰

, p

⁰⁰

) 7→ (p

0

, Q

1

, p

1

, G

⁻¹_p1

(q

⁰⁰

, p

⁰⁰

)) brings the variety l

_H

to the form

l

H

= {(p

0

, Q

1

, p

1

, q

⁰⁰

, p

⁰⁰

) ∈ H

⁽¹⁾

| Q

1

= g

1

(p

1

, G

p₁

(q

⁰⁰

, p

⁰⁰

)) = φ(p

1

, q

⁰⁰

, p

⁰⁰

), g

i

(0, q

⁰⁰

, p

⁰⁰

) = 0, i = 2, . . . , r}.

Hence l

H

= {(q, p) ∈ T

^∗

M | q

0

= 0, q

2

= −4p

³₀

− 2p

1

p

0

, p

2

= 3p

⁴₀

+ p

1

p

²₀

, q

1

=

−12p

⁵₀

/5 − 2p

1

p

³₀

/3 + φ(p

1

, q

⁰⁰

, p

⁰⁰

), g

i

(0, q

⁰⁰

, p

⁰⁰

) = 0, i = 2, . . . , r}. Changing Q = q

1

− φ(p

1

, q

⁰⁰

, p

⁰⁰

) and projecting l

H

along the characteristics of the hypersurface H (forgetting p

₀

) we finally get:

l

M

= {(Q, q

⁰⁰

, p

1

, p

⁰⁰

) | g

2

(0, q

⁰⁰

, p

⁰⁰

) = . . . = g

r

(0, q

⁰⁰

, p

⁰⁰

) = 0, the polynomial in p

0

: p

⁵₀

/5 + p

1

p

³₀

/3 + q

2

p

²₀

/2 + p

2

p

0

+ Q/2 has a root of multiplicity ≥ 3}.

This proves the theorem.

References

[1] V. I. A r n o l

⁰

d, Lagrangian manifold singularities, asymptotic rays and open swallowtails, Funct. Anal. Appl. 15 (1981), 235–246.

[2] V. I. A r n o l

⁰

d, Singularities of Caustics and Wave Fronts, Mathematics and its Applica- tions (Soviet Series) 62, Kluwer Academic Publishers, Dordrecht, 1990.

[3] A. B. G i v e n t a l

⁰

, Singular Lagrangian varieties and their Lagrangian mappings, in: Sovre- mennye Problemy Matematiki. Noveishie Dostizheniya 33. Itogi Nauki i Tekhniki. Akad.

Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, 55–112 (Russian);

English transl. in J. Soviet Math. 52 (1990), 3246–3278.

[4] R. M e l r o s e, Equivalence of glancing hypersurfaces, Invent. Math. 37 (1976), 165–191.

[5] O. M. M y a s n i c h e n k o, Geodesics on the boundary in the obstacle problem and unfurled swallowtails, Funct. Anal. Appl. 29 (1995), 82–84.

[6] O. P. S h c h e r b a k, Wave fronts and reflection groups, Russian Math. Surveys 43 (1988),

149–194.