i i j

±j = pj dx ’ pidx .

To describe the conservation laws, we ¬rst calculate for symmetry vector ¬elds

V ∈ gΠ the expression

•V = V Λ ∈ „¦n’1(P )

at points of J 1 (Rn, R+ ) ‚ P , and then restrict this (n ’ 1)-form to that sub-

manifold, where it will be a conserved integrand for the equation.

n+2

3.4. CONSERVATION LAWS FOR ∆u = Cu n’2 111

3.4.1 The Lie Algebra of In¬nitesimal Symmetries

We know that the Poincar´-Cartan forms

e

«

Πk = ρ § βI § ω(I)

|I|=k

on P are invariant under the simple, transitive left-action of the conformal group

SOo (n+1, 1). The in¬nitesimal generators of this action are the vector ¬elds on

P corresponding under the identi¬cation P ∼ SOo (n + 1, 1) to right-invariant

=

vector ¬elds. Our ¬rst task is to determine the right-invariant vector ¬elds in

terms of the basis

‚ ‚ ‚ ‚

, , ,

‚ρ ‚ωi ‚βi ‚±i j

of left-invariant vector ¬elds dual to the basis of left-invariant 1-forms used

previously; this is because the Maurer-Cartan equation in our setup only allows

us to compute in terms of left-invariant objects.

For an unknown vector ¬eld

‚ ‚ ‚ ‚

(Vji + Vij = 0)

+ V i i + Vi + Vji i

V =g (3.57)

‚ρ ‚ω ‚βi ‚±j

to be right-invariant is equivalent to the conditions

LV ρ = LV ωi = LV βi = LV ±i = 0; (3.58)

j

that is, the ¬‚ow of V should preserve all left-invariant 1-forms. We will solve the

system (3.58) of ¬rst-order di¬erential equations for V along the submanifold

J 1 (Rn, R+ ) ‚ P . Such V are not generally tangent to J 1 (Rn, R+ ), but the

calculation of conservation laws as V Λ is still valid, as J 1 (Rn , R+) is being

used only as a slice of the foliation P ’ M . The solution will give the coe¬cient

functions g, V i , Vi of (3.57) in terms of the coordinates (xi, u, pi) of J 1(Rn , R+ ).

We will not need the coe¬cients Vji , because they do not appear in •V = V Λ;

in fact, we compute g = V ρ only because it simpli¬es the rest of the solution.

First, we use the equation LV ρ = 0, which gives

0= d(V ρ) + V dρ

dg ’ 2 (Vi ωi ’ V i βi ).

1

=

We have the formulae (3.56) for the restrictions of ω i and βi to J 1 (Rn, R+ ), by

which the last condition becomes

1 ||p||2

2 2

Vi u n’2 dxi ’ V i u’ n’2 (dpi ’ pipj dxj + i

dg = 2 dx ) .

2

This suggests that we replace the unknowns Vi , V i in our PDE system (3.58)

with

2

def 1 i ’ 2 def 1 2 2

’ 2 V j u’ n’2 (’pj pi + δij ||p|| ).

vi = 1

V u n’2 , vi = Vi u n’2

2 2 2

112 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

Then we have the result

‚g ‚g ‚g

= ’vi ,

= vi , = 0. (3.59)

‚xi ‚pi ‚u

In particular, we now need to determine only the function g.

For this, we use the equation LV ωi = 0, which gives

ωi ) + V dωi

0= d(V

dV i ’ 2ρV i + ±i V j + 2gωi ’ Vji ωj .

= j

When we restrict to J 1 (Rn, R+ ) using (3.56) and use our new dependent vari-

ables vi , vi , this gives

dvi = (pi dxj ’ pj dxi)vj ’ (pj dxj )vi ’ g dxi + 1 Vji dxj . (3.60)

2

This says in particular that v i (x, u, p) is a function of the variables xi alone, so

along with (3.59) we ¬nd that

g(x, u, p) = f(x) + f i (x)pi,

for some functions f(x), f i (x). Substituting this back into (3.60), we have

df i = (pi f j ’ pj f i ’ 2 Vji + δj f)dxj .

1 i

This is a PDE system

‚f i

= pif j ’ pj f i ’ 2 Vji + δj f

i

1

j

‚x

for the unknowns f i (x), and it can be solved in the following elementary way.

We ¬rst let

hi = pif j ’ pj f i ’ 2 Vji = ’hj

1

j i

so that our equation is

‚f i

= hi + δj f.

i

(3.61)

j

j

‚x

Di¬erentiating this with respect to xk and equating mixed partials implies that

the expression

‚hi i ‚f k ‚f

k

’ δj k + δj i (3.62)

‚xj ‚x ‚x

is symmetric in j, k. It is also clearly skew-symmetric in i, k, and therefore

equals zero (as in (3.11)). Now we can equate mixed partials of hi to obtain

k

‚2f 2

‚2f ‚2f

j‚f

’ δk i l = δli j k ’ δlj i k .

i

δk

‚xj xl ‚x x ‚x x ‚x x

n+2

3.4. CONSERVATION LAWS FOR ∆u = Cu n’2 113

With the standing assumption n ≥ 3, this implies that all of these second partial

derivatives of f are zero, and we can ¬nally write

f(x) = r + 1 bk xk ,

2

for some constants r, bk . The reasons for our labelling of these and the following

constants of integration will be indicated below. Because the expressions (3.62)

vanish, we can integrate to obtain

hi = ’ 1 a i + 1 b j xi ’ 1 b i xj

j 2j 2 2

for some constants ai = ’aj , and then integrate (3.61) to ¬nd

j i

f i (x) = ’ 1 wi + (δj r ’ 1 ai )xj ’ 1 bi||x||2 +

i

b, x xi,

1

2j

2 4 2

bk xk and ||x||2 = (xk )2 . We summarize the

where we have written b, x =

discussion in the following.

Proposition 3.4 The coe¬cients of the vector ¬elds (3.57) on P preserving

the left-invariant 1-forms ρ, ω i along J 1 (Rn , R+) are of the form

b, x (1 + p, x ) + ’ 1 wi + δj r ’ 1 ai xj ’ 1 bi ||x||2 pi ,

i

1

g = r+ 2j

2 2 4

‚g

def 2

1 i ’ n’2

vi =’

= Vu ,

2 ‚pi

‚g

2

def 2 2

Vi u n’2 ’ 2 V j u’ n’2 (’pj pi + δij ||p|| ) =

1 1

vi = ,

2 2 ‚xi

where r, bi , wi , ai = ’aj are constants.

j i

It is easy to verify that such g, V i , Vi uniquely determine Vji = ’Vij such that

the vector ¬eld (3.57) preserves βi and ±i as well, but we will not need this fact.

j

Note that the number of constants in the Proposition equals the dimension of

the Lie algebra so(n + 1, 1), as expected.

The reader may be aware that one should not have to solve di¬erential

equations to determine right-invariant vector ¬elds in terms of left-invariant

vector ¬elds. In fact, an algebraic calculation will su¬ce, which in this case

would consist of writing an arbitrary Lie algebra element

«

2r bj 0

gL = w i ai bi

j

0 wj ’2r

interpreted as a left-invariant vector ¬eld, and conjugating by σ(x, u, p) ∈ P

regarded as a matrix with columns e0 (x, u, p), ej (x, u, p), en+1 (x, u, p) given by

(3.29, 3.30). The resulting so(n + 1, 1)-valued function on J 1(Rn , R+ ) then has

entries which are the coe¬cients of a right-invariant vector ¬eld V . The calcu-

lation is tedious, but of course the vector ¬elds so obtained are as in Proposi-

tion 3.4.

114 CHAPTER 3. CONFORMALLY INVARIANT SYSTEMS

3.4.2 Calculation of Conservation Laws

We can now use the formulae for the in¬nitesimal symmetries derived above to

calculate the conservation laws for Π, which are (n ’ 1)-forms on J 1 (Rn , R+)

that are closed when restricted to integral submanifolds of the Euler-Lagrange

system.

The Noether prescription is particularly simple in this case, because the

equations

LV Λ = 0 and dΛ = Π

mean that there are no compensating terms, and we can take for the conserved

integrand just

•V = V Λ.

This is straightforward in principle, but there are some delicate issues of signs

and constants. We ¬nd that for V as in Proposition 3.4,

Λ0 = V i ω(i),

V

and restricting to J 1(Rn , R+ ), using vi instead of V i , we obtain

2n

Λ0 )|J 1 (Rn ,R+ ) = 2u n’2 vi dx(i).

(V

The analogous computation for V Λ1 is a little more complicated and gives

Λ1)J 1 (Rn ,R+ ) = 2u2(’vj dpi § dx(ij) + (vi + n’2 i

v ||p||2)dx(i) ).

(V 2

On a transverse Legendre submanifold S of J 1(Rn , R+ ), we can use the condi-

tion ρ = 0 from (3.56) to write

’1 ‚u

2

pi = n’2 u ‚xi , (3.63)

and if we compute dpi and ||p||2 for such a submanifold, then we can substitute

and obtain

’uuxi xj vj + uuxj xj vi + uxi uxj vj + n’2 2

4

(V Λ1)|S = 2 u vi dx(i).

n’2

We summarize with the following.

Λ to the 1-jet graph of u(x1, . . . , xn)

Proposition 3.5 The restriction of V

equals

2n

2