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 2
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

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