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tionals considered in the preceding chapters. We begin by giving an invariant,
coordinate-free calculation of the formula (4.8) for the second derivative of a
functional under п¬Ѓxed-boundary variations. This formula has an interpretation
in terms of conformal structures induced on integral manifolds of the Euler-
Lagrange system, which we will describe. The role played by conformal geome-
try here is not to be confused with that in the previous chapter, although both
situations seem to reп¬‚ect the increasing importance of variational equations in
conformal geometry.
The usual integration by parts that one uses to establish local minimality
of a solution to the Euler-Lagrange equations cannot generally be done in an
invariant manner, and we discuss a condition under which this diп¬ѓculty can
be overcome. We considered in В§2.5 the example of prescribed mean curvature
hypersurfaces in Euclidean space; we will give an invariant calculation of the
second variation formula and the integration by parts for this example. We con-
clude by discussing various classical conditions under which an integral manifold
of an Euler-Lagrange system is locally minimizing, using the PoincarВґ-Cartan
e
form to express and prove some of these results in a coordinate-free manner.

4.1.1 A Formula for the Second Variation
We start by reconsidering the situation of В§1.2, in which we calculated the
п¬Ѓrst variation of a Lagrangian О› on a contact manifold. This amounted to
taking the п¬Ѓrst derivative, at some п¬Ѓxed time, of the values of the functional
FО› on a 1-parameter family of Legendre submanifolds. Our goal is to extend
the calculation to give the second variation of О›, or equivalently, the second
derivative of FО› on a 1-parameter family at a Legendre submanifold for which
the п¬Ѓrst variation vanishes. The result appears in formula (4.5) below, and in a
more geometric form in (4.8). This process is formally analogous to computing

133
134 CHAPTER 4. ADDITIONAL TOPICS

the Hessian matrix of a smooth function f : Rn в†’ R at a critical point, which
is typically done with the goal of identifying local extrema.
Let (M, I) be a contact manifold, with contact form Оё в€€ О“(I), and La-
grangian О› в€€ в„¦n (M ) normalized so that the PoincarВґ-Cartan form is given by
e
n
О  = dО› = Оё в€§ ОЁ. Fix a compact manifold N with boundary в€‚N , and a smooth
map
F : N Г— [0, 1] в†’ M
which is a Legendre submanifold Ft for each п¬Ѓxed t в€€ [0, 1] and is independent
of t on в€‚N Г— [0, 1]. Two observations will be important:

вЂў F в€— Оё = G dt for some function G on N Г— [0, 1], depending on the choice of
generator Оё в€€ О“(I). This holds because each Ft is a Legendre submanifold,
meaning that Ftв€— Оё = 0.

вЂў For every form О± в€€ в„¦в€— (M ), and every boundary point p в€€ в€‚N , we have

F в€—О± (p, t) = 0.
в€‚
в€‚t

This is equivalent to the п¬Ѓxed-boundary condition; at each p в€€ в€‚N , we
в€‚
have Fв€— ( в€‚t ) = 0.

We previously calculated the п¬Ѓrst variation as (see В§1.2.2)

d
Ftв€— О› G В· Ftв€— ОЁ,
=
dt Nt Nt

where О  = dО› = Оё в€§ ОЁ is the PoincarВґ-Cartan form for О›. This holds for each
e
t в€€ [0, 1].
We now assume that F0 is stationary for О›; that is, F is a Legendre vari-
ation of an integral manifold F0 : N в†’ M of the Euler-Lagrange system
EО› = {Оё, dОё, ОЁ}. This is the situation in which we want to calculate the second
derivative:

d2
Ftв€— О›
Оґ 2 (FО›)N0 (g) =
dt2 t=0 Nt
d
G Ftв€—ОЁ
=
dt t=0 Nt

L в€‚ (G F в€—ОЁ)
=
в€‚t
N0

g L в€‚ (F в€— ОЁ),
=
в€‚t
N0

в€—
where g = G|t=0, and the last step uses the fact that F0 ОЁ = 0.
To better understand the Lie derivative L в€‚ (F в€— ОЁ), we use the results ob-
в€‚t
tained via the equivalence method in В§2.4. This means that we are restricting
4.1. THE SECOND VARIATION 135

our attention to the case n в‰Ґ 3, with a PoincarВґ-Cartan form that is neo-
e
classical and deп¬Ѓnite. In this situation, we have a G-structure B в†’ M , where
G вЉ‚ GL(2n + 1, R) has Lie algebra
пЈ±пЈ« пЈј
пЈ¶
пЈІ (n в€’ 2)r 0 0 пЈЅ
j
g= пЈ­ пЈё : aj + ai = sij в€’ sji = sii = 0 .
i i i
в€’2rОґj + aj
0 0
пЈі пЈѕ
j j
nrОґi в€’ ai
di sij
(4.1)

The sections of B в†’ M are local coframings (Оё, П‰ i , ПЂi) of M for which:

вЂў Оё generates the contact line bundle I,

вЂў the PoincarВґ-Cartan form is О  = в€’Оё в€§ ПЂi в€§ П‰(i),
e

вЂў there exists a g-valued 1-form
пЈ« пЈ¶
(n в€’ 2)ПЃ 0 0
П•=пЈ­ пЈё,
в€’2ПЃОґj + О±i
i
0 0 (4.2)
j
j j
nПЃОґi в€’ О±i
Оґi Пѓij

satisfying a structure equation
пЈ« пЈ¶ пЈ« пЈ¶пЈ« пЈ¶
в€’ПЂi в€§ П‰i
Оё Оё
d пЈ­ П‰i пЈё = в€’П• в€§ пЈ­ П‰j пЈё + пЈ­ пЈё
в„¦i (4.3)
ПЂi ПЂj 0

where

в„¦i = T ijkПЂj в€§ П‰k в€’ (Sj П‰j + U ij ПЂj ) в€§ Оё,
i
(4.4)
j
with T ijk = T jik = T kji, T iik = 0; U ij = U ji; Sj = Si , Si = 0. The
i i

pseudo-connection form П• may be chosen so that also (cf. (2.48))

(n в€’ 2)dПЃ = в€’Оґi в€§ П‰i в€’ Sj ПЂi в€§ П‰j + ( nв€’2 U ij Пѓij в€’ ti ПЂi) в€§ Оё
i
2n

for some functions ti.

For any point p в€€ N , we consider a neighborhood U вЉ‚ M of F0 (p) on which we
can п¬Ѓx onesuch coframing (Оё, П‰ i , ПЂi) with pseudo-connection П•. All of the forms
and functions may be pulled back to W = F в€’1(U ) вЉ‚ N Г— [0, 1], which is the
setting for the calculation of L в€‚ (F в€— ОЁ). From now on, we drop all F в€—s.
в€‚t
We have ОЁ = в€’ПЂi в€§ П‰(i) , and we now have the structure equations needed
to diп¬Ђerentiate ОЁ, but it will simplify matters if we further adapt the forms
(Оё, П‰i , ПЂi) on W in a way that does not alter the structure equations. Note п¬Ѓrst
that restricted to each Wt = W в€© Nt , we have П‰i = 0, so (П‰1 , . . . , П‰n, dt)
forms a coframing on W . We can therefore write ПЂi = sij П‰j + gi dt for some sij ,
gi , and because each Wt is Legendre, we must have sij = sji . The structure
136 CHAPTER 4. ADDITIONAL TOPICS

group of B в†’ M admits addition of a traceless, symmetric combination of the
П‰j to the ПЂi , so we replace
ПЂi ; ПЂ i в€’ s o П‰ j ,
ij
1i
where s0 = sij в€’ n Оґj skk is the traceless part. Now we have
ij

ПЂi = sП‰i + gi dt

for some functions s, gi on W , so that

ОЁ = в€’ПЂi в€§ П‰(i) = в€’ns П‰ в€’ gidt в€§ П‰(i) .

Note that because F0 is assumed integral for the Euler-Lagrange system, we
have s = 0 everywhere on W0 вЉ‚ W ; in particular, ПЂi |W0 = 0.
With our choice of ПЂi , recalling that along W0, Оё = g dt for some function g
on W0 , and keeping in mind that restricted to W0 we have ПЂi = 0 and s = 0,
we can calculate
в€‚ в€‚
Lв€‚ ОЁ в€’ в€‚t d(ПЂi в€§ П‰(i) ) в€’ d( в€‚t (ПЂi в€§ П‰(i) ))
=
в€‚t t=0

(Оґi в€§ Оё + (nПЃОґi в€’ О±j ) в€§ ПЂj ) в€§ П‰(i)
j
в€‚
= i
в€‚t

+(sП‰i + gi dt) в€§ dП‰(i)
П‰i )П‰(i) в€’ ПЂi в€§ ( в€‚t
в€‚ в€‚
в€’d (gi + s в€‚t П‰(i))
в€’(gОґi + dgi + nПЃgi в€’ gj О±j ) в€§ П‰(i) .
= i

This gives our desired formula:

d2
g(dgi + nПЃgi в€’ gj О±j + gОґi ) в€§ П‰(i) .
=в€’
О› (4.5)
i
dt2 Nt N0
t=0

Unfortunately, in its present form this is not very enlightening, and our next
task is to give a geometric interpretation of the formula.

4.1.2 Relative Conformal Geometry
It is natural to ask what kind of geometric structure is induced on an integral
manifold f : N в†’ M of an Euler-Lagrange system EО› . What we п¬Ѓnd is:

If О  = dО› is a deп¬Ѓnite, neo-classical PoincarВґ-Cartan form, then an
e
integral manifold N of its Euler-Lagrange system EО› has a natural
conformal structure, invariant under symmetries of (M, О , N ), even
though there may be no invariant conformal structure on the ambient
M.

This is a simple pointwise phenomenon, in the sense that any n-plane V n вЉ‚
Tp M 2n+1 on which П‰i = 0 has a canonical conformal inner-product deп¬Ѓned
as follows. Taking any section (Оё, П‰ i , ПЂi) of B в†’ M , we can restrict the induced
4.1. THE SECOND VARIATION 137

quadratic form (П‰i )2 on T M to V вЉ‚ Tp M , where it is positive deп¬Ѓnite, and
then the action of the structure group (4.1) on (П‰ i ) shows that up to scaling,
this quadratic form is independent of our choice of section. Alternatively, one
can show this inп¬Ѓnitesimally by using the structure equations to compute on B
the Lie derivative of (П‰i )2 along a vector п¬Ѓeld that is vertical for B в†’ M ;
this Lie derivative is itself multiple of (П‰i )2 . Note that we have not restricted
to the conformal branch of the equivalence problem, characterized by T ijk =
U ij = Sj = 0 and discussed in В§3.3.
i

In particular, any integral manifold f : N в†’ M for the Euler-Lagrange
system EО› inherits a canonical conformal structure [ds2]f . We now want to de-
velop the conformal structure equations for (N, [ds2 ]f ), in terms of the structure
equations on B в†’ M , and our procedure will work only for integral manifolds
of EО› . We п¬Ѓrst note that along our integral manifold N we can choose local
sections (Оё, П‰i , ПЂi) of BN which are adapted to N in the sense that

Tp N = {Оё, ПЂ1, В· В· В· , ПЂn}вЉҐ вЉ‚ Tp M,

for each p в€€ N . In fact, such sections deп¬Ѓne a reduction Bf в†’ N of the principal
bundle BN в†’ N , having Lie algebra deп¬Ѓned as in (4.1) by sij = 0.
Restricted to Bf , we have the same structure equations as on B, but with
Оё = ПЂi = 0, and dОё = dПЂi = 0. Now observe that two of our structure equations
restrict to give

dП‰i (2ПЃОґj в€’ О±i ) в€§ П‰j ,
i
= j

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