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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 ¬xed-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 re¬‚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 di¬culty can
be overcome. We considered in §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 Poincar´-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 §1.2, in which we calculated the
¬rst variation of a Lagrangian Λ on a contact manifold. This amounted to
taking the ¬rst derivative, at some ¬xed time, of the values of the functional
FΛ on a 1-parameter family of Legendre submanifolds. Our goal is to extend
the calculation to give the second variation of Λ, or equivalently, the second
derivative of FΛ on a 1-parameter family at a Legendre submanifold for which
the ¬rst 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 ’ 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 θ ∈ “(I), and La-
grangian Λ ∈ „¦n (M ) normalized so that the Poincar´-Cartan form is given by
e
n
Π = dΛ = θ § Ψ. Fix a compact manifold N with boundary ‚N , and a smooth
map
F : N — [0, 1] ’ M
which is a Legendre submanifold Ft for each ¬xed t ∈ [0, 1] and is independent
of t on ‚N — [0, 1]. Two observations will be important:

• F — θ = G dt for some function G on N — [0, 1], depending on the choice of
generator θ ∈ “(I). This holds because each Ft is a Legendre submanifold,
meaning that Ft— θ = 0.

• For every form ± ∈ „¦— (M ), and every boundary point p ∈ ‚N , we have

F —± (p, t) = 0.

‚t

This is equivalent to the ¬xed-boundary condition; at each p ∈ ‚N , we

have F— ( ‚t ) = 0.

We previously calculated the ¬rst variation as (see §1.2.2)

d
Ft— Λ G · Ft— Ψ,
=
dt Nt Nt

where Π = dΛ = θ § Ψ is the Poincar´-Cartan form for Λ. This holds for each
e
t ∈ [0, 1].
We now assume that F0 is stationary for Λ; that is, F is a Legendre vari-
ation of an integral manifold F0 : N ’ M of the Euler-Lagrange system
EΛ = {θ, dθ, Ψ}. This is the situation in which we want to calculate the second
derivative:

d2
Ft— Λ
δ 2 (FΛ)N0 (g) =
dt2 t=0 Nt
d
G Ft—Ψ
=
dt t=0 Nt

L ‚ (G F —Ψ)
=
‚t
N0

g L ‚ (F — Ψ),
=
‚t
N0


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

our attention to the case n ≥ 3, with a Poincar´-Cartan form that is neo-
e
classical and de¬nite. In this situation, we have a G-structure B ’ M , where
G ‚ GL(2n + 1, R) has Lie algebra
±« 

 (n ’ 2)r 0 0 
j
g=   : aj + ai = sij ’ sji = sii = 0 .
i i i
’2rδj + aj
0 0
 
j j
nrδi ’ ai
di sij
(4.1)

The sections of B ’ M are local coframings (θ, ω i , πi) of M for which:

• θ generates the contact line bundle I,

• the Poincar´-Cartan form is Π = ’θ § πi § ω(i),
e

• there exists a g-valued 1-form
« 
(n ’ 2)ρ 0 0
•= ,
’2ρδj + ±i
i
0 0 (4.2)
j
j j
nρδi ’ ±i
δi σij

satisfying a structure equation
«  « « 
’πi § ωi
θ θ
d  ωi  = ’• §  ωj  +  
„¦i (4.3)
πi πj 0

where

„¦i = T ijkπj § ωk ’ (Sj ωj + U ij πj ) § θ,
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 • may be chosen so that also (cf. (2.48))

(n ’ 2)dρ = ’δi § ωi ’ Sj πi § ωj + ( n’2 U ij σij ’ ti πi) § θ
i
2n

for some functions ti.

For any point p ∈ N , we consider a neighborhood U ‚ M of F0 (p) on which we
can ¬x onesuch coframing (θ, ω i , πi) with pseudo-connection •. All of the forms
and functions may be pulled back to W = F ’1(U ) ‚ N — [0, 1], which is the
setting for the calculation of L ‚ (F — Ψ). From now on, we drop all F —s.
‚t
We have Ψ = ’πi § ω(i) , and we now have the structure equations needed
to di¬erentiate Ψ, but it will simplify matters if we further adapt the forms
(θ, ωi , πi) on W in a way that does not alter the structure equations. Note ¬rst
that restricted to each Wt = W © Nt , we have ωi = 0, so (ω1 , . . . , ωn, dt)
forms a coframing on W . We can therefore write πi = sij ωj + 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 ’ M admits addition of a traceless, symmetric combination of the
ωj to the πi , so we replace
πi ; π i ’ s o ω j ,
ij
1i
where s0 = sij ’ n δj skk is the traceless part. Now we have
ij


πi = sωi + gi dt

for some functions s, gi on W , so that

Ψ = ’πi § ω(i) = ’ns ω ’ gidt § ω(i) .

Note that because F0 is assumed integral for the Euler-Lagrange system, we
have s = 0 everywhere on W0 ‚ W ; in particular, πi |W0 = 0.
With our choice of πi , recalling that along W0, θ = g dt for some function g
on W0 , and keeping in mind that restricted to W0 we have πi = 0 and s = 0,
we can calculate
‚ ‚
L‚ Ψ ’ ‚t d(πi § ω(i) ) ’ d( ‚t (πi § ω(i) ))
=
‚t t=0

(δi § θ + (nρδi ’ ±j ) § πj ) § ω(i)
j

= i
‚t

+(sωi + gi dt) § dω(i)
ωi )ω(i) ’ πi § ( ‚t
‚ ‚
’d (gi + s ‚t ω(i))
’(gδi + dgi + nρgi ’ gj ±j ) § ω(i) .
= i

This gives our desired formula:

d2
g(dgi + nρgi ’ gj ±j + gδi ) § ω(i) .
=’
Λ (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 ’ M of an Euler-Lagrange system EΛ . What we ¬nd is:

If Π = dΛ is a de¬nite, neo-classical Poincar´-Cartan form, then an
e
integral manifold N of its Euler-Lagrange system EΛ has a natural
conformal structure, invariant under symmetries of (M, Π, 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 ‚
Tp M 2n+1 on which ωi = 0 has a canonical conformal inner-product de¬ned
as follows. Taking any section (θ, ω i , πi) of B ’ M , we can restrict the induced
4.1. THE SECOND VARIATION 137

quadratic form (ωi )2 on T M to V ‚ Tp M , where it is positive de¬nite, and
then the action of the structure group (4.1) on (ω i ) shows that up to scaling,
this quadratic form is independent of our choice of section. Alternatively, one
can show this in¬nitesimally by using the structure equations to compute on B
the Lie derivative of (ωi )2 along a vector ¬eld that is vertical for B ’ M ;
this Lie derivative is itself multiple of (ωi )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 §3.3.
i

In particular, any integral manifold f : N ’ M for the Euler-Lagrange
system EΛ 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 ’ M , and our procedure will work only for integral manifolds
of EΛ . We ¬rst note that along our integral manifold N we can choose local
sections (θ, ωi , πi) of BN which are adapted to N in the sense that

Tp N = {θ, π1, · · · , πn}⊥ ‚ Tp M,

for each p ∈ N . In fact, such sections de¬ne a reduction Bf ’ N of the principal
bundle BN ’ N , having Lie algebra de¬ned as in (4.1) by sij = 0.
Restricted to Bf , we have the same structure equations as on B, but with
θ = πi = 0, and dθ = dπi = 0. Now observe that two of our structure equations
restrict to give

dωi (2ρδj ’ ±i ) § ωj ,
i
= j

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