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preserved under
¯
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-

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. 38
( 48 .)



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