Noether’s Theorem
Emmy Noether: 1882 - 1935
Outline:
• History of the calculus of variations: the Principle of Least Action
• The Calculus of Variations in a nutshell, some applications
• Action functionals and Euler-Lagrange equations
• Noether’s Theorem; Transformation groups, symmetries, conservation laws
• Some ideas on the proof
• Some examples of conservations laws derived from symmetries
Origins of the Variational Calculus: The Principle of
Least Action in Physics
Plato ( 400 B.C.): the architect of the world created the cosmos by reducing
the primal state of chaos into order...
Aristotle ( 350 B.C.): motion follows the 'simplest' patterns; the minimum
movement is the swiftest...
Hero of Alexandria ( 125 B.C.): showed that when a ray of light is
reected by a mirror, the path actually taken from the light source to the
observer's eye is shorter than any other possible path so reected.
William of Ockham ( 1325): It is futile to employ many principles when
it is possible to employ fewer. (i.e., apply the razor...)
Newton ( 1680): We are to admit no more causes of natural things than
such as are both true and sucient to explain their appearances.
Leibniz ( 1680): ours is the best of all possible worlds, exhibiting the
greatest simplicity in its premises and the greatest wealth in its phenomena.
Malbrange (philosopher, 1675): the economy of nature.
Fermat ( 1650): The principle of least time in optics (reection or refraction).
Maupertuis ( 1744): Nature changes in such as way as to make the action
the least possible. (collisions of particles, refraction)
John Bernoulli ( 1696): brachistochrone problem.
Euler and Lagrange ( 1750): formal development of calculus of variations.
Hamilton ( 1850): Hamiltonian formulation of Lagrange's variational
problem, and the canonical equations of mechanics.
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The Calculus of Variations:
Variational Derivative, Euler-Lagrange Equations
Functional S : X ! , X a space of functions ' on
Action density S (u; ) : m !
R
p
S [']
R
=
Z
R
R
m
R
m
m
S
(
'
(
x
)
;
r
'
(
x
))
d
x;
R
Derivative of S at ' in the direction ;
S ['](
) =
=
=
=
0
since
d
S [' + " ]j"=0
d"
Z
d
S (' + " ; r' + "r
d"
Z
(S + S r ) dmx
u
)j"=0 dmx
p
Z
(Su r S )
dm x
p
S r = r (S ) (r S ) ; integration by parts
p
and
m
r
(
S
)
d
x =
Z
p
p
Z
@
S
p
ds
p
= 0; divergence theorem and = 0 on @ If S ['] = 0, i.e., S [']( ) = 0 for all (test) functions , then ' must
satisfy the
equations;
@'
@S m @ @S
Su('; r') r S ('; r') = 0 = @'
;
@'i =
@x @'
@x
0
0
Euler-Lagrange
X
p
i=1
i
i
i
Classical mechanics: Particle position (t); S ( (t); _ (t)). Euler - Lagrange
equations are;
@S
d @S
@
dt @ _
q
q
!
q
q
2
q
: S ['] =
m
S
(
'
(
x
)
;
r
'
(
x
))
d
x
@S m @ @S
Euler-Lagrange (E-L) equations:
Action functional
Z
X
@'
Action density
1 mj _ j2
2
q
S
Commonly known as...
 = rV ( )
Newton's equation
' = 0
Laplace equation
V (q )
q
1 (@ 2')2 + 1 jr'j2 + F (')
2 t
2
- 2 Im _
q
ij
g Rij
' + F (') = 0 nonlinear wave equation
@t2 '
h 2
2 + V j j2 ih
@t
jr
j
2m
1 jA_ j2 jr Aj2
2
@xi @'i
E-L equations
jr'j2
h
i=1
p
0
h 2
A
2m + V
Schrodinger equation
A = 0;
rA=0
Maxwell equations
g Rij
1g R = 0
2 ij
Einstein equation
=
0
@
1
!
3
Noether's Theorem
Action functional; S : X ! ; S ['] = S ('(x); rm'(x)) dmx;
One parameter group of transformations; T : X ! X; 2
T is a
of S if S ['] = S ['] for all '; where ' T'.
(=) T is a symmetry of the Euler-Lagrange equations, but not conversely)
Z
R
R
symmetry group
i.e., If '~(~x; t~) = T(') and ~ = T(
) then
~ '~(~x) dmx~ = (S ('(x); r'(x)) dmx = S [']
S [~
'] = ~ S '~(~x); r
Z
Z
Examples of symmetries: translations in space and time, rotations in space,
Lorentz transformations (rotations in space and time), etc.
If T is a symmetry of S , then the solutions to the associated
Euler-Lagrange equations satisfy a conservation law, i.e. there is some function E (t), (E (t) = E ('; r') dx), such that dtd E (t) = 0.
(Here we are distinguishing one coordinate as time t and the rest as spatial
variables, so we have m = m 1 and dmx = dt dm 1x in the above
formulae.)
Noether:
Z
R
R
R
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Point particles
qi (t): mi qi (t)
Symmetry
= rV (q(t))
Conserved quantity
translation in time energy E (t) = 21
X
i
(i.e., dtd E (t) = 0)
mi jq_ i j2 (t) + V (q(t))
translation in space linear momentum (t) =
P
rotation in space
i
angular momentum (t) =
L
Virial Theorem: 12
Wave equation
X
'(x; t): @t2 '
X
i
mq_i2 (t)
=
mq_ i (t)
X
i
qi
(t) mi i(t)
q
_
K V (q (t))
' +2f (') = 0; f2(u) = V (u)
@
@
@t2 2 ; @
@2
0
X
i
t
Symmetry
xi
Conserved quantity
1
2
2
energy E (t) = n
(@t '(x; t)) + jr'(x; t)j + V ('(x; t)) dn x
2
ZR
= n e('(x; t)) dn x
Z
translation in time
R
linear momentum P(t) =
translation in space
Z
Rn
rotation in space (about r) angular momentum L(t) =
Lorentz boost direction r
r (t) =
Z T
1
Virial Theorem: Tlim
!1
T
T
Z
Rn
Z
Rn
@t '(x; t)r'(x; t) dn x
Z
Rn
r ^ r'(x; t)
@t '(x; t) dn x
fe('(x; t))r t(r r'(x; t))@t'(x; t)g dnx
1 (@ '(x; t)2 + 1 jr'(x; t)j2 + V ('(x; t)) dn x dt = 0
2 t
2
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The idea of Noether's proof
Action functional; S : X ! ; S ['] = S ('(x); rm'(x)) dmx;
One parameter group of transformations; T : X ! X; 2
T is a
of S if S ['] = S ['] for all '; where ' T'.
(=) T is a symmetry of E-L, but not conversely)
For any transformation group T, by the Chain Rule,
Z
R
R
symmetry group
d
S [' ]
d
=0
= S ['](A');
where
0
A=
d T ; ' = '=0
d =0
for '(x; t): Div(e; ) = rm (e; ) = @te + r = 0,
are functions of '(x; t). Note rm = (@t; r).
Conservation law
p
where e;
p
Recall;
S ['](A') = (Su (A') + S
p
p
This is because if we dene E (t) = x e('(x; t); r'(x; t)) dx, then integrating the equation
R
R
@t e = r p over all space x we obtain dtd E (t) = x @t e dx = x r p dx = 0 since
the last integral vanishes (Divergence Theorem and the hypothesis that ' vanishes on the
boundary of x ). We are also singling out one variable as time t so that = [ T; T ] x
(often x = Rn ).
R
0
rm(A')) dmx
If T is a symmetry of S , then for any ' ;
Z
p
Noether:
(Su(A') + S rm(A')) = Div(Q);
p
where Q is explicitly given from A
(Su(A') + S rm(A')) Div(Q) = 0.
Now, we can write this as;
(Su rm S ) (A') + Div (S (A') Q) = 0
So,
p
p
p
(integration by parts; Sp rm (A') = rm (Sp (A')) (rm Sp )(A'))
If in addition ' is also a critical point of S , then the rst term on
the left vanishes and we obtain the conservation law;
Div (S (A')
p
6
Q) = 0
Some References
These notes: http://www.sfu.ca/rpyke/presentations.html
Notes on Symmetries, Conservation Laws, and Virial Relations;
http://www.sfu.ca/rpyke/research/index.html
Calculus of Variations, I.M. Gelfand, S.V. Fomin.
Variational Principles in Dynamics and Quantum Theory, Wolfgang Yourgrau, Stanley Mandelstam.
Applications of Lie Groups to Dierential Equations, Peter Olver.
Calculus of Variations, with applications to physics and engineering, Robert Weinstock.
The Variational Principles of Mechanics, Cornelius Lanczos.
The Calculus of Variations, C. Caratheodory, 1935.
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