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