Revista Brasileira de Flsica, Vol. 16, n? 4 , 1986
A Comment on the Proof of Noether's Theorem is Smooth Manifolds
QUINTINO A.G. DE SOUZA
Instituto de Física, Universidade Estadual de Campinas, Caixa Postal 1170, Campinas, 13100,
SP, Brasil
and
WALDYR A. RODRIGUES JR.
Instituto de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, Campinas, 13100,
SP, Brasil
Recebido em 11 de março de 1986
Abstract
We g i v e a p r o o f o f N o e t h e r ' s theorem i n a smooth manifold which
does n o t use t h e h y p o t h e s i s t h a t t h e one parameter f a r n i l y o f d i f f e o m o r phisms o f M, ( h s , s 6 R ] = C , w h i c h leaves the Lagrangian i n v a r i a n t , i s
a group. We a l s o d i s c u s s t h e p h y s i c a l meaning o f t h e r e s t r i c t i o n t h a t C
i s an one- paraneter group o f d i f f e o m o r p h i s m s .
1. INTRODUCTION
Let
M
be a smooth* m a n i f o l d ,
a smooth f u n c t i o n . L e t h :
M+M
TM
t h e tangent b u n d l e o f M, L : T h ?
be a diffeornorphism. We say
that
the
p a i r ( M , L ) i s i n v a r i a n t under h I f
I n eq.
(1) h,
i
t h e rnap h .
W + T M i s t h e d e r i v a t i v e rnapping o f
I n v ( ~ )be t h e s e t o f a11 diffeomorphisms o f
M such
that
for
Let
each
h 6 l n v ( L ) e q . ( l ) h o l d s t r u e . L e t S c l n v ( ~ ) .We say t h a t t h e p a i r (MC)
i s i n v a r i a n t under t h e s e t o f d i f f e o m r p h i s m s
each element i n
S i f eq.(l) i s v a l i d for
S. Note t h a t whereas l n v ( L ) i s a group,
n o t i n g e n e r a l a group.
the
set S is
I n p a r t i c u l a r , l e t C:R+ Inv(L) b e a curve i n the
space o f diffeomorphisrns, such t h a t hs
= C(sl)
and h
1
= C(sZ).
This
82
c u r v e w i l l g e n e r a t e a one-parameter group o f diffeomorphisrns
uncler t h e
operation o f composition i f f
*
I n t h i s paper smooth rneans d i f f e r e n t i a b l e o f c l a s s
t h a t t h e statements made a r e v a l i d .
2, w i t h
k
such
Revista Brasiieira.de Fkica, Vol. 16, n? 4, 1986
( i ) hs i s d i f f e r e n t i a b l e
(ii)
h,(x)
= x
Wx E M
V s , , s, 6 W
( i i i ) hs
s1+S2
When ( i ) and ( i i ) a r e s a t i s f i e d b u t ( i i i ) i s not s a t i s f i e d
cal l the curve C E { h
s'
we
s E R ) a one- parameter fami l y of diffeomorphisms
o f M (OPFDM).
The u s u a l p r e s e r l t a t i o n o f N o e t h e r ' s theoreml"
admits t h a t the
c u r v e C c I n v ( L ) generates a one- parameter group o f d i f f e o m o r p h i s m s .
In
t h i s paper we g i v e i n s e c t i o n 2 a p r o o f o f N o e t h e r ' s theorem where o n l y
the hypothesis that
I n s e c t i o n 3 we d i s c u s s t h e
C i s an OPFDM i s used.
p h y s i c a l meaning o f t h e group h y p o t h e s i s , u s i n g an example.
We a r r i v e
a t t h e c o n c l u s i o n t h a t t h e more g e n e r a l h y p o t h e s i s t h a t C i s a one-parameter group o f d i f f e o m o r p h i s m s does n o t g i v e g e n u i n e l y new c o n s e r v a t i o n
laws.
2. NOETHER'S THEOREM
Theorem: I f t h e system ( M , L ) . i s i n v a r i a n t under a OPFDM, C =
!
h
s€ R
s'
then t h e Lagrangian system o f e q u a t i o n s c o r r e s p o n d i n g t o L has a f i r s t
integral.
Proof: ( A ) F i r s t assume M = R'". L e t xi : R~
-+
-
R be t h e c a n o n i c a l
and l e t 4 : R +
by t * $ ( t ) be a c u r v e . D e f i n e t h e
d
i-.i
{ri o @ = i
q ,
{q,41.
Write
g = q , i = I , . , .,nl
q . Suppose now t h a t t h e c u r v e 4 i s a s o l u t i o n o f t h e E u l e r -
dinate functions,
mapping
-
L = L
0
coor-
q-
=
-Lagrange e q u a t i o n s , w h i c h we w r i t e as
where $,/aq'
means
(aE/axz) o 4 , e t c .
As L i s supposed t o be i n v a r i a n t
under h
S
E C
the
curve
hs o $ : R + f i s a l s o a s o l u t i o n o f t h e Euler-Lagrange e q u a t i o n s
s E R. D e f i n e
a11
for
Revista Brasileira de Flsica, Vol. 16, nQ4, 1986
and
i= L
o
g.
We have, f o r f i x e d s,
Since h
S
E C, we can w r i t e
(4) impl ies
Eq.
that
Us i ng now eq. (3) i n t o eq.
(5)
we have
then
and we g e t
Eq. (7) must be v a l i d f o r a l l s E R.
s = O,
where
i.e.,
6
I n the p a r t i c u l a r case-
when hs = I d we o b t a i n
=
i
x o hs o 4 , i = I
,...,n.
This concludesthe f i r s t p a r t o f the p r o o f . We must now showthat
the same r e s u l t i s obtained when M i s a n a r b i t r a r y smooth m a n i f o l d .
( 0 ) Let then A =
{(Uj.$j)ijêJ ( J E N)
a maximal a t l a s o f M.
It
Revista Brasileira de Fisica, Vol. 16, n? 4, 1986
TU^,
f o l l o w s t h a t t h e s e t TA =
- 1 (u.) and
TUj = n
3 ,
R",
: U j + U'.c
3
7'$
: TU
i
j
(
E
q
{hs,
i n the a t l a s
f
x
j
i s the
tangent
map
the
of
:
+
s 6 R},
s € R ) where
A, i .e.,
R as t h e r e p r e s e n t a t i o n o f L
in
the
, = ~ ( q , + ) ~
E Jj , S i n c e L i r i n v a r i a n t under t h e
a c t i o n o f t h e OPFOM {hs,
of t h e OPFDH
Ut
where
i s an a t l a s f o r TM,
and n : T M + M i s t h e c a n o n i c a l p r o j e s t i o n .
We now d e f i n e
a t l a s TA, i e . ,
-+
?)IjW
hs
=
qk
E
hS
:
w i l l be i n v a r i a n t under
L?+$
0 hs O
-1
qj
the
acticn
i s the representation
(see f i g . 1 ) .
Indeed,
o f hs
we must
have
where Ths : TM
hs*(q)
+
T M i s t h e t a n g e n t mapping o f h
S'
í"hS(q,j)
=
.
Since by h y p o t h e s i s
However ,
and then
L(q,j) = L ( ~ k ~ ( q , $i)t follows that
( h s (q),
Revista Brasileira de Física, Vol. 16. n? 4, 1986
(6)
Fig.1
-
Haps used i n the proof o f Noether's Theorem.
Revista Brasileira de Física, Vol. 16, n? 4, 1986
Now t h e f a c t t h a t t h e f u n c t i o n
{Es,
: RZn
-+
R i s i n v a r i a n t under t h e OPFDH,
s E R ] impl i e s as a l r e a d y shown t h a t t h e q u a n t i t i e s
To complete t h e p r o o f we must show t h a t t h e
ob t a i ned
resul t
But t h i s
does n o t depend on t h e a t l a s used t o p a r a m e t r i z e t h e m a n i f o l d .
p o i n t i s indeed t r i v i a l , s i n c e we assumed A t o be a maximal a t l a s .
3. THE PHVSICAL MEANING OF THE GROUP PROPERTY FOR THE WFDM, (h, s
E
Ri
I t may seem s t r a n g e t o t h e reader t h a t we d i d n o t use
p o i n t o f the proof o f Noether's
{ h s , s E R ) must be a group.
t h e o r e m the f a c t
that
at
the
As i s now c l e a r a i 1 we need
any
OPFDM
in
p r o o f i s t h a t t h e f a m i l y o f d i f f e o m o r p h i s m s can be p a r a m e t r i z e d
the
by t h e
s e t o f r e a l numbers. We then g e t t h e f o l l o w i n g q u e s t i o n .
s E R) i s an oneWhat mot i v a t e s t h e u s u a l assurnpt i o n t h a t { h
s'
-parameter group o f d i f f e o m o r p h i s m s ? The answer, as w i l l be made c l e a r
by an elementary example,
i s t h a t we do n o t o b t a i n r e s u l t s
g e n e r a l i t y by imposing L t o be i n v a r i a n t under a OPFOM.
with
To
more
convince
o u r s e l v e s t h a t t h i s i s indeed t h e case c o n s i d e r t h e Lagrangian L:TR'+.F,
by
where
known,
m i s a r e a l (and p o s i t i v e ) parameter and
-i
.L. =
(z,~).
AS
L r e p r e s e n t s t h e Lagrangian o f a f r e e p a r t i c l e i n
two
s i o n s . A s e t o f n a t u r a l c o n s e r v a t i o n laws f o r t h i s system i s
is
d imenthe
e s t a b l i s h i n g t h e constancy o f t h e l i n e a r momenta i n t h e d i r e c t i o n
t h e c o o r d i n a t e axes
well
one
of
Revista Brasileira de Fisica, Vol. 16, n? 4, 1986
aL
px = 7=
ax
=-=
ai
p~
ry
= constant
~
= constant
Eqs. (12) a r e consequences o f t h e i n v a r i a n c e o f t h e Lagrangian
under t r a n s l a t i o n s a l o n g t h e x and y d i r e c t i o n s . More f o r m a l l y , t h e f a c l :
t h a t L i s i n v a r i a n t under t h e one- parameter group o f diffeomorphisms
i.e.,
aL áh"
--=
L(ha(x,y),
.
h;(x,9))
= ~(x,y),(i,;))
implies
that
m i s a c o n s t a n t o f m o t i o n . Analogously, s i n c e
quanti t y
the
t h e Lagrangian
a2 da
i s i n v a r i a n t under t h e one-parameter group o f diffeomorphisms
the quantity
-r
aI
dhB
=
6 ir
conserved d u r i n g t h e m t i o n o f t h e p a r t i c l e .
d~
Now, from N o e t h e r ' s theorem we can g e t t h e r e s u l t
that
the
p r o j e c t i o n o f t h e rnomentum v e c t o r i s c o n s t a n t a l o n g an a r b i t r a r y d i r e c t i o n i n t h e p l a n e . Indeed, i t i s enough t o observe t h a t
t h e Lagrangian
i s i n v a r i a n t under t h e one- parameter g r o u p o f diffeomorphisms
We t h e n g e t t h a t t h e q u a n t i t y ma
+
myb i s conserved. Now, l e t
us c o n s i d e r , o n l y an OPFDM i n s t e a d o f a group. Put
where f and g a r e a r b i t r a r y C '
Lagrangian, eq.
functions.
I t i s t r i v i a l t o show t h i t the
(1 1) i s i n v a r i a n t under t h e
transíations
eq. (14) and N o e t h e r ' s theorem g i v e s as conserved q u a n t i t y
d e f ined
by
Revista Brasileira de Física, Vol. 16, n? 4, 1986
i.e.,
t h e p r o j e c t i o n o f t h e l i n e a r momentum i s conserved i n t h e
t ion o f the vector
direc-
(g,$$.
Now, f o r each a = a, ( f i x e d ) t h e r e e x i s t s a
one- parameter
group o f d i f f e o m o r p h i s m s t h a t reproduce t h e above r e s u l t . Indeed, i t i s
9
d a
enough i n eq. (13) t o p u t a =
.
and b = 691
da a,. I t f o l !ows
a
do n o t o b t a i n r e a l nove1 t i e s a1 l o w i n g f o r t h e s e t { h
,
that
a E R
a
we
more
general s t r u c t u r e as an OPFDM.
We end t h i s paper w i t h two more comnents:
(a) The a d m i s s i o n o f a more g e n e r a l s t r u c t u r e than t h a t
o f an
OPFDM i s n o t p o s s i b l e i n more g e n e r a l p h y s i c a l aDpl i c a t i o n s .
We mus t
keep i n mind t h a t i n t h e mathematical i m p l e m e n t a t i o n
o f any
theory,
p h y s i c a l q u a n t i t i e s a r e more c o n v e n i e n t l y
by
described
geometrical
o b j e c t s d e f i n e d on t h e m a n i f o l d , w h i c h can be c h a r a c t e r i z e d
transformation properties i n r e l a t i o n t o the g r o u p
of
by
thei r
diffeonorfisms
o f the manifold.
(b) We must a l s o observe t h a t i n t h i s paper we work o n l y
wi t h
one- parameter f a m i l i e s and groups o f d i f f e o m o r f i s m s . The t r e a t m e n t , can
indeed, be g e n e r a l i z e d t o t h e case where we have an r - p a r a m e t e r
set o f
d i f f e o m o r f isms o f t h e r n a n i f o l d . We should then s u b s t i t u t e an z r g l z r i r ?
r - d i m e n s i o n a l L i e group, w i t h
r g r e a t e r t h a n one, f o r t h e s e t R o f r e a l
nurnbers w h i c h parameter$ t h e s e t o f d i f f e o r m o r f i s m s .
I n t h i s way, u s i n g
t h e sane procedure as b e f o r e , we s h a l l a r r i v e ar a s e t o f
conservation
laws which we s h a l l conclude t o be t h e same as í o r , a t t h e w o r s t , a l i n ear c o m b i n a t i o n o f ) t h e r c o n s e r v a t i o n laws we s h o u l d o b t a i n by
taking
r one- dimensional L i e groups which r e p r o d u c e ( l o c a l l y ) t h e r - d i m e n s i o n a l one.
We a r e g r a t e f u l l t o CNPq f o r r e s e a r c h g r a n t s .
REFERENCES
1 . V.I.
A r n o l d , Mathernatical Methods of CZassica2 Mechanics
-Verlag,
New York,
(Springer-
1978).
2 . A . Trautman, P. P i r a n i , H. Bondi, Lectures on
(Gordon and ~ r e a c h , New York, 1964).
General R e z a t i v i t u
Revista Brasileira de Flsica, Vol. 16, no 4, 1986
Resumo
Damos uma prova do teorema de Noether em uma v a r i e d a d e l i s a M,
a qual não usa a h i p ó t e s e de que a f a m í l i a a um parâmetro de difeomorf.ismos de M, { h s , s € R ) = C, que d e i x a a Lagrangeana i n v a r i a n t e , s e j a um
grupo. Também d i s c u t i m o s o s i g n i f i c a d o f i s i c o da r e s t r i ç ã o de que C sej a um grupo de d i f e o m o r f ismos a um parâmetro.