¯

θ± = θ± , ωi = ωi , πi = d ± θ β + e ± ω j + πi ,

¯± ±

¯ iβ ij

with e± = e± , because such a change has no e¬ect on the expression for Π

ij ji

n+1

modulo { J}. Finally, consider a change of the form

¯

θ± = θ± , ω i = ci θ β + ω i , ¯± ±

¯ πi = πi .

β

160 CHAPTER 4. ADDITIONAL TOPICS

We will prove the invariance of the symmetry condition in¬nitesimally, writing

instead of ωi the family

¯

ωi (µ) = µci θβ + ωi . (4.29)

β

This associates to each Hk a tensor Hk+1(µ) for each k ≥ 1, and we will show

d

that dµ |µ=0Hk+1(µ) ∈ Uk+1 . This just amounts to looking at the terms linear

in µ when (4.29) is substituted into (4.28). Noting that C = (ci ) ∈ W — — V , we

β

consider the commutative diagram

σ

k k k+1 k+1

W— — V — W— — V — W— — V ’ W— — V — W— — V

“ „k — 1 “ „k+1

k+1

W — k+1 V — W — — V k+2 k+2

—

W— —

’ V,

where σ is skew-symmetrization with the latter W — — V , and „k — 1 is an

extension of the earlier skew-symmetrization. The point is that given Hk — C

in the upper-left space of this diagram,

d

| H (µ) = σ(Hk — C).

dµ µ=0 k+1

So if we assume that (Hk ) ∈ Uk , then „k (Hk ) = 0, so σ(Hk — C) ∈ Uk+1 , which

is what we wanted to show.

This proves that the condition that the symmetry condition on Π = dΛ is

independent of the choice of adapted coframe. We can now state the following.

Theorem 4.1 Given a functional [Λ] ∈ “(M, n(J/I)), there is a unique ad-

n

J) such that Π = dΛ ∈ „¦n+1(M ) is symmetric.

missible lifting Λ ∈ “(M,

Proof. We inductively construct Λ = Λ0 + Λ1 + · · · + Λmin(n,s), with each

i

Λi ∈ { I} chosen to eliminate the fully skew-symmetric part of

def

Πi’1 = d(Λ0 + · · · + Λi’1).

Initially, Λ0 = F ω is the prescribed X-semibasic n-form modulo {I}. We know

from the existence of admissible liftings that there is some Λ1 ∈ {I} such that

def

Π1 = d(Λ0 + Λ1) ∈ {I}; and we know from Proposition 4.5 that Λ1 is uniquely

2

determined modulo { I}. Now let

2 n+1

ij ±

Π1 ≡ H±β πi § θβ § ω(j) (mod { I} + { J}).

If we add to Λ0 + Λ1 the I-quadratic term

ij ±

§ θβ § ω(ij),

1

Λ2 = 2!2 F±β θ

then the structure equation (4.24) shows that this alters the I-linear term Π1

only by

ij ij ij

H±β ; H±β + F±β .

4.2. EULER-LAGRANGE PDE SYSTEMS 161

ij ji ij ij

Because F±β = ’F±β = ’Fβ± , we see that F±β may be uniquely chosen so that

ij

the new H±β lies in U1 .

n

The inductive step is similar. Suppose we have Λ0 + · · ·+ Λl ∈ “( J) such

l’1

that Πl = d(Λ0 + · · · + Λl ) is symmetric modulo { I}. Then the term of

I-degree l is of the form

l+1

H±Aπi § θA § ω(I)

iI ±

1

Πl ≡ (mod {U1 } + · · · + {Ul’1 } + { I}).

l!2

iI

for some H±A. There is a unique skew-symmetric term

1

FA θA § ω(I)

I

Λl+1 = (l+1)!2

|I|=|A|=l+1

which may be added so that

l+1

Πl+1 ∈ {U1 } + · · · + {Ul } + { I}.

We can continue in this manner, up to l = min(n, s).

De¬nition 4.8 The unique Λ in the preceding theorem is called the Betounes

form for the functional [Λ].4 Its derivative Π = dΛ is the Poincar´-Cartan form

e

for [Λ].

The unique determination of Π, along with the invariance of the symmetry

condition under admissible coframe changes of M , implies that Π is globally

de¬ned and invariant under symmetries of the functional [Λ] and the multi-

contact structure (M, I, J).

It is instructive to see the ¬rst step of the preceding construction in coordi-

nates. If our initial Lagrangian is

Λ0 = L(x, z, p)dx,

then we have already seen in (4.26) that

Λ0 + Λ1 = L dx + θ± § ‚L

‚p± dx(i) .

i

The H1-term of d(Λ0 + Λ1 ) (see (4.27)) is

‚2L

dpβ § θ± § dx(i). (4.30)

j

‚p± ‚pβ

i j

Of course Lp± pβ = Lpβ p± , corresponding to the fact that H1 ∈ Sym2 (W — — V )

i j j i

2

automatically. The proof shows that we can add Λ2 ∈ { I} so that Π2 instead

includes

+ Lp± pβ )πi § θβ § ω(j) ∈ U1 = Sym2 (W — ) — Sym2V.

±

1

2 (Lp± pβ

ij j i

4 It was introduced in coordinates in [Bet84], and further discussed in [Bet87].

162 CHAPTER 4. ADDITIONAL TOPICS

In fact, this corresponds to the principal symbol of the Euler-Lagrange PDE

system (4.20), given by the symmetric s — s matrix

‚2L

H±β (ξ) = ξi ξj

‚p± ‚pβ

i j

ξ ∈ V —.

1

= (Lp± pβ + Lp± pβ )ξi ξj ,

2 ij j i

In light of this, it is not surprising to ¬nd that only the symmetric part of (4.30)

has invariant meaning.

Note also that if Lagrangians Λ, Λ di¬er by a divergence,

n’1

Λ ’ Λ = d», » ∈ “(M, J),

then the construction in the proof of Theorem 4.1 shows that the Poincar´-

e

Cartan forms are equal, though the Betounes forms may not be. A related but

more subtle property is the following.

n

Theorem 4.2 For a functional [Λ] ∈ “(M, (J/I)), the Poincar´-Cartan

e

form Π = 0 if and only if the Euler-Lagrange system is trivial, EΛ = I.

Proof. One direction is clear: if Π = 0, then the n-form generators Ψ± for EΛ

can be taken to be 0, so that EΛ = I. For the converse, we ¬rst consider the

I-linear term

2 n+1

ij ±

Π1 ≡ H±β πi § θβ § ω(j) (mod { I} + { J}).

ij ±

EΛ is generated by I and Ψβ = H±β πi § ω(j), 1 ¤ β ¤ s, and our assumption

EΛ = I then implies that these Ψβ = 0; that is,

ij ±

H1 = H±β πi § θβ § ω(j) = 0.

We will ¬rst show that this implies

H2 = H 3 = · · · = 0

n+1

as well, which will imply Π ∈ { J}. To see this, suppose Hl is the ¬rst

non-zero term, having I-degree l. Then we can consider

l n+1

0 ≡ dΠ (mod { I} + { J}),

and using the structure equations (4.24),

ijI β

±

H±βAπi § πj .

0=

|I|=|A|=l’1

Written out fully, this says that

i i ···i i i ···i

H±11 ±2···±l+1 = H±22 ±1···±l+1 .

2 l+1 1 l+1

4.2. EULER-LAGRANGE PDE SYSTEMS 163

iI

Also, H±A is fully skew-symmetric in I and A. But together, these imply in

that Hl is fully skew-symmetric in all upper and all lower indices, for

H±11i±i3 ······ H±22i±13 ······

i2 i 1i

=

2 ±3 ±3

’H±22i±i3 ······

i1

= 3 ±1

’H±13i±i3 ······

i2

= 2 ±1

H±13i±13 ······

i 2i

= ±2

H±21i±33 ······

i 1i

= ±2

’H±21i±i3 ······ .

i1

= 2 ±3

This proves full skew-symmetry in the upper indices, and the proof for lower in-

dices is similar. However, we constructed Π so that each Hk lies in the invariant

complement of the fully skew-symmetric tensors, so we must have Hk = 0.

Now we have shown that if the Euler-Lagrange equations of [Λ] are trivial,

n+1

then Π ∈ { J}. But that means that the Betounes form Λ is not merely

semibasic over the quotient space X, but actually basic. We can then compute

the (assumed trivial) ¬rst variation down in X instead of M , and ¬nd that for

any submanifold N ’ X, and any vector ¬eld v along N vanishing at ‚N ,

0= v dΛ.

N

But this implies that dΛ = 0, which is what we wanted to prove.

The preceding results indicate that Π is a good generalization of the classical

Poincar´-Cartan form for second-order, scalar Euler-Lagrange equations. We

e

note that for higher-order Lagrangian functionals on vector-valued functions of

one variable (i.e., functionals on curves), such a generalization is known, and

not di¬cult; but for functionals of order k ≥ 2 on vector-valued functions of

several variables, little is known.5

We want to brie¬‚y mention a possible generalization to the multi-contact

case of Noether™s theorem, which gives an isomorphism from a Lie algebra of

symmetries to a space of conservation laws. To avoid distracting global consid-