Proof of Noether’s theorem
Örn Arnaldsson
Department of Applied Mathematics
University of Washington, Seattle
Amath 573, Autumn 2012, Final Project
1 / 117
A foreword on manifolds
The basic objects of this talk are manifolds.
For our purposes one can, sort of, get away with knowing
nothing about manifolds by just thinking of them as
Euclidean space.
So whenever an m dimensional manifold is mentioned, just
think of that object as an open set in the space Rm .
Smooth vector fields on manifolds just become smooth
vector fields on Rm in the classical sense.
The tangent space of a manifold M at x0 ∈ M becomes the
vector space of all vectors with origin at x0 .
2 / 117
A foreword on manifolds
The basic objects of this talk are manifolds.
For our purposes one can, sort of, get away with knowing
nothing about manifolds by just thinking of them as
Euclidean space.
So whenever an m dimensional manifold is mentioned, just
think of that object as an open set in the space Rm .
Smooth vector fields on manifolds just become smooth
vector fields on Rm in the classical sense.
The tangent space of a manifold M at x0 ∈ M becomes the
vector space of all vectors with origin at x0 .
3 / 117
A foreword on manifolds
The basic objects of this talk are manifolds.
For our purposes one can, sort of, get away with knowing
nothing about manifolds by just thinking of them as
Euclidean space.
So whenever an m dimensional manifold is mentioned, just
think of that object as an open set in the space Rm .
Smooth vector fields on manifolds just become smooth
vector fields on Rm in the classical sense.
The tangent space of a manifold M at x0 ∈ M becomes the
vector space of all vectors with origin at x0 .
4 / 117
A foreword on manifolds
The basic objects of this talk are manifolds.
For our purposes one can, sort of, get away with knowing
nothing about manifolds by just thinking of them as
Euclidean space.
So whenever an m dimensional manifold is mentioned, just
think of that object as an open set in the space Rm .
Smooth vector fields on manifolds just become smooth
vector fields on Rm in the classical sense.
The tangent space of a manifold M at x0 ∈ M becomes the
vector space of all vectors with origin at x0 .
5 / 117
A foreword on manifolds
The basic objects of this talk are manifolds.
For our purposes one can, sort of, get away with knowing
nothing about manifolds by just thinking of them as
Euclidean space.
So whenever an m dimensional manifold is mentioned, just
think of that object as an open set in the space Rm .
Smooth vector fields on manifolds just become smooth
vector fields on Rm in the classical sense.
The tangent space of a manifold M at x0 ∈ M becomes the
vector space of all vectors with origin at x0 .
6 / 117
A foreword on manifolds
The basic objects of this talk are manifolds.
For our purposes one can, sort of, get away with knowing
nothing about manifolds by just thinking of them as
Euclidean space.
So whenever an m dimensional manifold is mentioned, just
think of that object as an open set in the space Rm .
Smooth vector fields on manifolds just become smooth
vector fields on Rm in the classical sense.
The tangent space of a manifold M at x0 ∈ M becomes the
vector space of all vectors with origin at x0 .
7 / 117
Introduction
Noether’s theorem is a fundamental tool in modern theoretical
physics and the calculus of variations. The theorem roughly
states that each local symmetry group for the action
(Lagrangian) of a physical system gives rise to a conservation
law for the system.
Examples of this are
If the system is invariant under rotation (if the rotation
group is a symmetry group of the Lagrangian) then the
angular momentum of the system is conserved.
If the system is invariant under translations in time and
space, then Noether’s theorem promises conservation of
linear momentum, and energy, respectively.
8 / 117
Introduction
Noether’s theorem is a fundamental tool in modern theoretical
physics and the calculus of variations. The theorem roughly
states that each local symmetry group for the action
(Lagrangian) of a physical system gives rise to a conservation
law for the system.
Examples of this are
If the system is invariant under rotation (if the rotation
group is a symmetry group of the Lagrangian) then the
angular momentum of the system is conserved.
If the system is invariant under translations in time and
space, then Noether’s theorem promises conservation of
linear momentum, and energy, respectively.
9 / 117
Introduction
Noether’s theorem is a fundamental tool in modern theoretical
physics and the calculus of variations. The theorem roughly
states that each local symmetry group for the action
(Lagrangian) of a physical system gives rise to a conservation
law for the system.
Examples of this are
If the system is invariant under rotation (if the rotation
group is a symmetry group of the Lagrangian) then the
angular momentum of the system is conserved.
If the system is invariant under translations in time and
space, then Noether’s theorem promises conservation of
linear momentum, and energy, respectively.
10 / 117
Do not fret
Sophocles
To him who is in fear, everything rustles.
Much of the effort of this lecture will be devoted to setting
up the necessary machinery to tackle Noether’s theorem,
and the machinery can be overwhelming at first.
The ideas presented during this setup-phase, although
notation heavy, are mostly quite basic.
The pay-off from our toil will be apparent when the proof of
Noether’s theorem, from our geometric viewpoint, reduces
to nothing more that integration by parts!
11 / 117
Do not fret
Sophocles
To him who is in fear, everything rustles.
Much of the effort of this lecture will be devoted to setting
up the necessary machinery to tackle Noether’s theorem,
and the machinery can be overwhelming at first.
The ideas presented during this setup-phase, although
notation heavy, are mostly quite basic.
The pay-off from our toil will be apparent when the proof of
Noether’s theorem, from our geometric viewpoint, reduces
to nothing more that integration by parts!
12 / 117
Do not fret
Sophocles
To him who is in fear, everything rustles.
Much of the effort of this lecture will be devoted to setting
up the necessary machinery to tackle Noether’s theorem,
and the machinery can be overwhelming at first.
The ideas presented during this setup-phase, although
notation heavy, are mostly quite basic.
The pay-off from our toil will be apparent when the proof of
Noether’s theorem, from our geometric viewpoint, reduces
to nothing more that integration by parts!
13 / 117
Do not fret
Sophocles
To him who is in fear, everything rustles.
Much of the effort of this lecture will be devoted to setting
up the necessary machinery to tackle Noether’s theorem,
and the machinery can be overwhelming at first.
The ideas presented during this setup-phase, although
notation heavy, are mostly quite basic.
The pay-off from our toil will be apparent when the proof of
Noether’s theorem, from our geometric viewpoint, reduces
to nothing more that integration by parts!
14 / 117
Change of variables
Theorem
If F : Rm → Rn (m ≥ n) is a smooth function of maximal rank at
x0 . Then there exist local coordinates around x0 ∈ Rm and
F (x0 ) = y0 ∈ Rn in which F assumes the simple form
F (x1 , . . . , xm ) = (x1 , . . . , xn ).
Proof. First translate x0 and y0 to 0, in their respective domains.
Then, writing x = (x 0 , x 00 ), where x 0 ∈ Rn , x 00 ∈ Rm−n , make the
change of coordinates (x 0 , x 00 ) → (F (x 0 , x 00 ), x 00 ).
Remarks:
This is not a global result, of course.
It nevertheless demonstrates the power of "choosing the
right coordinates".
In some applications of Lie-symmetries, clever coordinate
transformations are used to reduce the order of a
differential equation.
15 / 117
Change of variables
Theorem
If F : Rm → Rn (m ≥ n) is a smooth function of maximal rank at
x0 . Then there exist local coordinates around x0 ∈ Rm and
F (x0 ) = y0 ∈ Rn in which F assumes the simple form
F (x1 , . . . , xm ) = (x1 , . . . , xn ).
Proof. First translate x0 and y0 to 0, in their respective domains.
Then, writing x = (x 0 , x 00 ), where x 0 ∈ Rn , x 00 ∈ Rm−n , make the
change of coordinates (x 0 , x 00 ) → (F (x 0 , x 00 ), x 00 ).
Remarks:
This is not a global result, of course.
It nevertheless demonstrates the power of "choosing the
right coordinates".
In some applications of Lie-symmetries, clever coordinate
transformations are used to reduce the order of a
differential equation.
16 / 117
Change of variables
Theorem
If F : Rm → Rn (m ≥ n) is a smooth function of maximal rank at
x0 . Then there exist local coordinates around x0 ∈ Rm and
F (x0 ) = y0 ∈ Rn in which F assumes the simple form
F (x1 , . . . , xm ) = (x1 , . . . , xn ).
Proof. First translate x0 and y0 to 0, in their respective domains.
Then, writing x = (x 0 , x 00 ), where x 0 ∈ Rn , x 00 ∈ Rm−n , make the
change of coordinates (x 0 , x 00 ) → (F (x 0 , x 00 ), x 00 ).
Remarks:
This is not a global result, of course.
It nevertheless demonstrates the power of "choosing the
right coordinates".
In some applications of Lie-symmetries, clever coordinate
transformations are used to reduce the order of a
differential equation.
17 / 117
Change of variables
Theorem
If F : Rm → Rn (m ≥ n) is a smooth function of maximal rank at
x0 . Then there exist local coordinates around x0 ∈ Rm and
F (x0 ) = y0 ∈ Rn in which F assumes the simple form
F (x1 , . . . , xm ) = (x1 , . . . , xn ).
Proof. First translate x0 and y0 to 0, in their respective domains.
Then, writing x = (x 0 , x 00 ), where x 0 ∈ Rn , x 00 ∈ Rm−n , make the
change of coordinates (x 0 , x 00 ) → (F (x 0 , x 00 ), x 00 ).
Remarks:
This is not a global result, of course.
It nevertheless demonstrates the power of "choosing the
right coordinates".
In some applications of Lie-symmetries, clever coordinate
transformations are used to reduce the order of a
differential equation.
18 / 117
Change of variables
Theorem
If F : Rm → Rn (m ≥ n) is a smooth function of maximal rank at
x0 . Then there exist local coordinates around x0 ∈ Rm and
F (x0 ) = y0 ∈ Rn in which F assumes the simple form
F (x1 , . . . , xm ) = (x1 , . . . , xn ).
Proof. First translate x0 and y0 to 0, in their respective domains.
Then, writing x = (x 0 , x 00 ), where x 0 ∈ Rn , x 00 ∈ Rm−n , make the
change of coordinates (x 0 , x 00 ) → (F (x 0 , x 00 ), x 00 ).
Remarks:
This is not a global result, of course.
It nevertheless demonstrates the power of "choosing the
right coordinates".
In some applications of Lie-symmetries, clever coordinate
transformations are used to reduce the order of a
differential equation.
19 / 117
The case m < n
Theorem
If m < n in the preceding theorem, there are coordinates in
which F becomes
F (x1 , . . . , xm ) = (x1 , . . . , xm , 0, . . . , 0).
Proof. Write F = (F1 , . . . , Fn ). As before, translate x0 and y0 to
the origin and assume that x → (F1 (x), . . . , Fm (x)) is of
maximal rank (by permuting coordinates if necessary). Then
there are coordinates in which
F (x) = (x1 , . . . , xm , Fm+1 (x), . . . , Fn (x)). The image of F in
these coordinates is the graph of the function
x → (Fm+1 (x), . . . , Fn (x)) a neighborhood of which can be
mapped to an open set around the origin such that the graph is
mapped into Rm × {0}.
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The case m < n
Theorem
If m < n in the preceding theorem, there are coordinates in
which F becomes
F (x1 , . . . , xm ) = (x1 , . . . , xm , 0, . . . , 0).
Proof. Write F = (F1 , . . . , Fn ). As before, translate x0 and y0 to
the origin and assume that x → (F1 (x), . . . , Fm (x)) is of
maximal rank (by permuting coordinates if necessary). Then
there are coordinates in which
F (x) = (x1 , . . . , xm , Fm+1 (x), . . . , Fn (x)). The image of F in
these coordinates is the graph of the function
x → (Fm+1 (x), . . . , Fn (x)) a neighborhood of which can be
mapped to an open set around the origin such that the graph is
mapped into Rm × {0}.
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Flows
Definition
For a smooth vector field ~v on a manifold M, its flow is the
unique function ϕ(ε, x), where x ∈ M and ε is in some open
interval containing 0, such that
d
ϕ(ε, x) = ~v (ϕ(ε, x)) and ϕ(0, x) = x.
dε
We will denote the flow as exp(ε~v )x = ϕ(ε, x)
Notice that by definition
d
exp(ε~v )x = ~v |exp(ε~v )x .
dε
By uniqueness of solutions to ordinary differential
equations exp(δ~v )(exp(ε~v )x) = exp(δ + ε)x
For a fixed x exp(ε~v )x is guaranteed to exists for small
enough ε by the basic existence theorems of ODEs.
22 / 117
Flows
Definition
For a smooth vector field ~v on a manifold M, its flow is the
unique function ϕ(ε, x), where x ∈ M and ε is in some open
interval containing 0, such that
d
ϕ(ε, x) = ~v (ϕ(ε, x)) and ϕ(0, x) = x.
dε
We will denote the flow as exp(ε~v )x = ϕ(ε, x)
Notice that by definition
d
exp(ε~v )x = ~v |exp(ε~v )x .
dε
By uniqueness of solutions to ordinary differential
equations exp(δ~v )(exp(ε~v )x) = exp(δ + ε)x
For a fixed x exp(ε~v )x is guaranteed to exists for small
enough ε by the basic existence theorems of ODEs.
23 / 117
Flows
Definition
For a smooth vector field ~v on a manifold M, its flow is the
unique function ϕ(ε, x), where x ∈ M and ε is in some open
interval containing 0, such that
d
ϕ(ε, x) = ~v (ϕ(ε, x)) and ϕ(0, x) = x.
dε
We will denote the flow as exp(ε~v )x = ϕ(ε, x)
Notice that by definition
d
exp(ε~v )x = ~v |exp(ε~v )x .
dε
By uniqueness of solutions to ordinary differential
equations exp(δ~v )(exp(ε~v )x) = exp(δ + ε)x
For a fixed x exp(ε~v )x is guaranteed to exists for small
enough ε by the basic existence theorems of ODEs.
24 / 117
Flows
Definition
For a smooth vector field ~v on a manifold M, its flow is the
unique function ϕ(ε, x), where x ∈ M and ε is in some open
interval containing 0, such that
d
ϕ(ε, x) = ~v (ϕ(ε, x)) and ϕ(0, x) = x.
dε
We will denote the flow as exp(ε~v )x = ϕ(ε, x)
Notice that by definition
d
exp(ε~v )x = ~v |exp(ε~v )x .
dε
By uniqueness of solutions to ordinary differential
equations exp(δ~v )(exp(ε~v )x) = exp(δ + ε)x
For a fixed x exp(ε~v )x is guaranteed to exists for small
enough ε by the basic existence theorems of ODEs.
25 / 117
Flows
Definition
For a smooth vector field ~v on a manifold M, its flow is the
unique function ϕ(ε, x), where x ∈ M and ε is in some open
interval containing 0, such that
d
ϕ(ε, x) = ~v (ϕ(ε, x)) and ϕ(0, x) = x.
dε
We will denote the flow as exp(ε~v )x = ϕ(ε, x)
Notice that by definition
d
exp(ε~v )x = ~v |exp(ε~v )x .
dε
By uniqueness of solutions to ordinary differential
equations exp(δ~v )(exp(ε~v )x) = exp(δ + ε)x
For a fixed x exp(ε~v )x is guaranteed to exists for small
enough ε by the basic existence theorems of ODEs.
26 / 117
The action of a vector field on a function
A smooth vector field on a manifold M has the following action
on smooth functions f : M → R, defined through its flow
d f (x) → ~v (f )(x) :=
f (exp(ε~v )x).
dε ε=0
Notice that ~v (f ) is again a smooth function.
Thinking of M as Euclidean space, this is just the classical
directional derivative ∇f · ~v .
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The action of a vector field on a function
A smooth vector field on a manifold M has the following action
on smooth functions f : M → R, defined through its flow
d f (x) → ~v (f )(x) :=
f (exp(ε~v )x).
dε ε=0
Notice that ~v (f ) is again a smooth function.
Thinking of M as Euclidean space, this is just the classical
directional derivative ∇f · ~v .
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The action of a vector field on a function
A smooth vector field on a manifold M has the following action
on smooth functions f : M → R, defined through its flow
d f (x) → ~v (f )(x) :=
f (exp(ε~v )x).
dε ε=0
Notice that ~v (f ) is again a smooth function.
Thinking of M as Euclidean space, this is just the classical
directional derivative ∇f · ~v .
29 / 117
Lie groups
Definition
A Lie group G is a group that also carries the structure of a
smooth manifold, such that the group operation
m : G × G → G, m(g, h) = g · h and inversion
i : G → G, i(g) = g −1 are smooth.
Examples of Lie groups:
The set of real numbers R, with addition as the group
operation and identity 0.
The unit circle S 1 = {(cos(θ), sin(θ)) : 0 ≤ θ ≤ 2π}, with
addition of arguments and identity (1, 0).
The group of all orthogonal n × n matrices
O(n) := {X ∈ GL(n) : XX T = I} with matrix multiplication
and identity I. This Lie group is 12 n(n − 1) dimensional.
30 / 117
Lie groups
Definition
A Lie group G is a group that also carries the structure of a
smooth manifold, such that the group operation
m : G × G → G, m(g, h) = g · h and inversion
i : G → G, i(g) = g −1 are smooth.
Examples of Lie groups:
The set of real numbers R, with addition as the group
operation and identity 0.
The unit circle S 1 = {(cos(θ), sin(θ)) : 0 ≤ θ ≤ 2π}, with
addition of arguments and identity (1, 0).
The group of all orthogonal n × n matrices
O(n) := {X ∈ GL(n) : XX T = I} with matrix multiplication
and identity I. This Lie group is 12 n(n − 1) dimensional.
31 / 117
Lie groups
Definition
A Lie group G is a group that also carries the structure of a
smooth manifold, such that the group operation
m : G × G → G, m(g, h) = g · h and inversion
i : G → G, i(g) = g −1 are smooth.
Examples of Lie groups:
The set of real numbers R, with addition as the group
operation and identity 0.
The unit circle S 1 = {(cos(θ), sin(θ)) : 0 ≤ θ ≤ 2π}, with
addition of arguments and identity (1, 0).
The group of all orthogonal n × n matrices
O(n) := {X ∈ GL(n) : XX T = I} with matrix multiplication
and identity I. This Lie group is 12 n(n − 1) dimensional.
32 / 117
Lie groups
Definition
A Lie group G is a group that also carries the structure of a
smooth manifold, such that the group operation
m : G × G → G, m(g, h) = g · h and inversion
i : G → G, i(g) = g −1 are smooth.
Examples of Lie groups:
The set of real numbers R, with addition as the group
operation and identity 0.
The unit circle S 1 = {(cos(θ), sin(θ)) : 0 ≤ θ ≤ 2π}, with
addition of arguments and identity (1, 0).
The group of all orthogonal n × n matrices
O(n) := {X ∈ GL(n) : XX T = I} with matrix multiplication
and identity I. This Lie group is 12 n(n − 1) dimensional.
33 / 117
Lie group actions
Definition
Let M be a smooth manifold. A local group of transformations
on M is given by a Lie group G, an open subset U, with
{e} × M ⊂ U ⊂ G × M,
which is the domain of definition of the group action, and a
smooth map Ψ : U → M such that Ψ(g, Ψ(h, x)) = Ψ(g · h, x),
when defined, Ψ(e, x) = x and such that if Ψ(g, x) ∈ U then
(g −1 , Ψ(g, x)) ∈ U.
Remark:
We will write g · x for the group action of g on x instead of
Ψ(g, x) when there is no danger of confusion.
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Lie group actions
Definition
Let M be a smooth manifold. A local group of transformations
on M is given by a Lie group G, an open subset U, with
{e} × M ⊂ U ⊂ G × M,
which is the domain of definition of the group action, and a
smooth map Ψ : U → M such that Ψ(g, Ψ(h, x)) = Ψ(g · h, x),
when defined, Ψ(e, x) = x and such that if Ψ(g, x) ∈ U then
(g −1 , Ψ(g, x)) ∈ U.
Remark:
We will write g · x for the group action of g on x instead of
Ψ(g, x) when there is no danger of confusion.
35 / 117
Example of a group action on a manifold
Let G = R and M be the two-dimensional torus T 2 . Let ω be a
fixed real number. Using angular coordinates (θ, ρ) for T 2 we
define the group action
Ψ(ε, (θ, ρ)) = (θ + ε, ρ + ωε) mod 2π.
As ε runs through all of G, the curve on T 2 drawn by Ψ starting
from some (θ, ρ) will be periodic if ω is rational, but dense in T 2
if ω is irrational.
The tangents of the curve ε → Ψ(ε, (θ, ρ)) form a smooth vector
field on T 2 . The curves arising in this way from group actions
are very important and we shall meet them again when we
discuss Lie-algebras.
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The flow as a local group action
The most important example of a local group action on a
manifold M is given by a smooth vector field ~v on M and its flow
exp(ε~v )x. Let G = R and define the group action as
Ψ(ε, x) = exp(ε~v )x.
The vector field ~v is called the infinitesimal generator of this
group action.
37 / 117
The flow as a local group action
The most important example of a local group action on a
manifold M is given by a smooth vector field ~v on M and its flow
exp(ε~v )x. Let G = R and define the group action as
Ψ(ε, x) = exp(ε~v )x.
The vector field ~v is called the infinitesimal generator of this
group action.
38 / 117
The Lie algebra of a Lie group
For a fixed g ∈ G we have a smooth map Rg : G → G,
Rg (h) = h · g as multiplication on the right with g. This function
has a differential dRg , a linear map between tangent spaces on
G. This is just the Jacobi matrix if we think of G as Euclidean
space.
Definition
The Lie algebra of a Lie group G is the collection of vector
fields ~v on G that are invariant under the differential dRg , that is
if dRg (~v |h ) = ~v |h·g for all g, h ∈ G. The Lie algebra is denoted
by g.
39 / 117
The Lie algebra of a Lie group
For a fixed g ∈ G we have a smooth map Rg : G → G,
Rg (h) = h · g as multiplication on the right with g. This function
has a differential dRg , a linear map between tangent spaces on
G. This is just the Jacobi matrix if we think of G as Euclidean
space.
Definition
The Lie algebra of a Lie group G is the collection of vector
fields ~v on G that are invariant under the differential dRg , that is
if dRg (~v |h ) = ~v |h·g for all g, h ∈ G. The Lie algebra is denoted
by g.
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More on the Lie algebra
Any right invariant vector field ~v is uniquely determined by its
value at the identity e, since ~v |g = dRg (~v |e ). Conversely, every
vector v , tangent to G at e gives rise to the right invariant vector
field ~v |g = dRg (v ), indeed with ~v defined in this way we have
dRg (~v |h ) = dRg (dRh (v )) = d(Rg ◦ Rh )(v ) = dRhg (v ) = ~v |hg ,
for all g, h ∈ G.
For this reason we identify the Lie algebra g of G with the
tangent space to G at the identity element e. We denote this
tangent space as TG|e . The elements of g = TG|e are called
the infinitesimal generators of G.
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More on the Lie algebra
Any right invariant vector field ~v is uniquely determined by its
value at the identity e, since ~v |g = dRg (~v |e ). Conversely, every
vector v , tangent to G at e gives rise to the right invariant vector
field ~v |g = dRg (v ), indeed with ~v defined in this way we have
dRg (~v |h ) = dRg (dRh (v )) = d(Rg ◦ Rh )(v ) = dRhg (v ) = ~v |hg ,
for all g, h ∈ G.
For this reason we identify the Lie algebra g of G with the
tangent space to G at the identity element e. We denote this
tangent space as TG|e . The elements of g = TG|e are called
the infinitesimal generators of G.
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The exponential map
Remember that TG|e = g. Define the exponential map
exp : g → G by putting ε = 1 in the flow on G generated by
~v ∈ g:
exp(~v ) := exp(~v )e.
Theorem
The differential of exp at the origin is the identity,
dexp|0 = I.
Proof. For an element ~v ∈ g, the exponential exp maps the
straight line curve ε → ε~v to the curve exp(ε~v )e, and the
tangent of this curve at ε = 0 is by definition ~v , so
dexp|0 (~v ) = ~v .
43 / 117
The exponential map
Remember that TG|e = g. Define the exponential map
exp : g → G by putting ε = 1 in the flow on G generated by
~v ∈ g:
exp(~v ) := exp(~v )e.
Theorem
The differential of exp at the origin is the identity,
dexp|0 = I.
Proof. For an element ~v ∈ g, the exponential exp maps the
straight line curve ε → ε~v to the curve exp(ε~v )e, and the
tangent of this curve at ε = 0 is by definition ~v , so
dexp|0 (~v ) = ~v .
44 / 117
The exponential map
Remember that TG|e = g. Define the exponential map
exp : g → G by putting ε = 1 in the flow on G generated by
~v ∈ g:
exp(~v ) := exp(~v )e.
Theorem
The differential of exp at the origin is the identity,
dexp|0 = I.
Proof. For an element ~v ∈ g, the exponential exp maps the
straight line curve ε → ε~v to the curve exp(ε~v )e, and the
tangent of this curve at ε = 0 is by definition ~v , so
dexp|0 (~v ) = ~v .
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The Lie algebra determines the Lie group
Since the differential of the exponential map is the identity at
the origin, by the inverse function theorem, the exponential map
is a diffeomorphism from an open neighborhood of 0 ∈ g to an
open neighborhood U, of e ∈ G.
Theorem
Let U be an open neighborhood of the identity e in a connected
Lie group G, and define U k := {g1 · · · gk : gi ∈ U}, then
G=
∞
[
Uk .
k =1
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The Lie algebra determines the Lie group
Since the differential of the exponential map is the identity at
the origin, by the inverse function theorem, the exponential map
is a diffeomorphism from an open neighborhood of 0 ∈ g to an
open neighborhood U, of e ∈ G.
Theorem
Let U be an open neighborhood of the identity e in a connected
Lie group G, and define U k := {g1 · · · gk : gi ∈ U}, then
G=
∞
[
Uk .
k =1
47 / 117
Proof
Proof. First replace U by the open set U ∩ i(U), where i is the
inversion on G. This is to make sure that g −1 ∈ U if g ∈ U.
Now, let Rg be multiplication on the right, as before. Notice that
Rg is one-to-one and onto and hence an open map and
therefore all the U k are open. If some h ∈ G is in none of the
U k we prove that the open set Rh (U) does not intersect any
U k . For if it did, for some g, g1 , . . . , gk ∈ U we could write
g · h = g1 · · · gn ⇔ h ∈ g −1 · g1 · · · gn ∈ U k +1 , a contradiction
S
k
since h was not in any of the U k . We have proved that ∞
k =1 U
is both open and closed in G and must therefore be all of G. 48 / 117
Significance for symmetries
The foregoing results show that each element g of a
(connected) Lie group G can be written as a product
g =exp(ε1~v1 ) · · · exp(εm ~vm )
= exp(ε1~v1 )e · · · exp(εm ~vm )e ,
for some εi ∈ R and ~vi ∈ g. Thus, to prove invariance of some
object under the entire Lie group action it is sufficient to prove
invariance under the flows exp(ε~v )e close to the identity. With a
little more work we can actually reduce this to proving
invariance for a basis {~v1 , . . . , ~vr } of g.
49 / 117
Significance for symmetries
The foregoing results show that each element g of a
(connected) Lie group G can be written as a product
g =exp(ε1~v1 ) · · · exp(εm ~vm )
= exp(ε1~v1 )e · · · exp(εm ~vm )e ,
for some εi ∈ R and ~vi ∈ g. Thus, to prove invariance of some
object under the entire Lie group action it is sufficient to prove
invariance under the flows exp(ε~v )e close to the identity. With a
little more work we can actually reduce this to proving
invariance for a basis {~v1 , . . . , ~vr } of g.
50 / 117
Infinitesimal action
Let G act on M via Ψ. Each ~v ∈ g gives rise to a vector field
ψ(~v ) on M, namely as the tangents of the flow Ψ(exp(ε~v )e, x)
through x ∈ M. In other words
ψ(~v )|x =
d Ψ(exp(ε~v )e, x) = dΨx (~v ),
dε ε=0
where Ψx (g) = Ψ(g, x).
The vector field ψ(~v ) on M will be identified with the invariant
vector field ~v on G, and will also be referred to as the
infinitesimal generator of G. Sophus Lie’s great insight was that
the invariance of objects under the group action of G could be
proven by showing the invariance under the flow of ψ(~v ).
51 / 117
Infinitesimal action
Let G act on M via Ψ. Each ~v ∈ g gives rise to a vector field
ψ(~v ) on M, namely as the tangents of the flow Ψ(exp(ε~v )e, x)
through x ∈ M. In other words
ψ(~v )|x =
d Ψ(exp(ε~v )e, x) = dΨx (~v ),
dε ε=0
where Ψx (g) = Ψ(g, x).
The vector field ψ(~v ) on M will be identified with the invariant
vector field ~v on G, and will also be referred to as the
infinitesimal generator of G. Sophus Lie’s great insight was that
the invariance of objects under the group action of G could be
proven by showing the invariance under the flow of ψ(~v ).
52 / 117
Infinitesimal action
Let G act on M via Ψ. Each ~v ∈ g gives rise to a vector field
ψ(~v ) on M, namely as the tangents of the flow Ψ(exp(ε~v )e, x)
through x ∈ M. In other words
ψ(~v )|x =
d Ψ(exp(ε~v )e, x) = dΨx (~v ),
dε ε=0
where Ψx (g) = Ψ(g, x).
The vector field ψ(~v ) on M will be identified with the invariant
vector field ~v on G, and will also be referred to as the
infinitesimal generator of G. Sophus Lie’s great insight was that
the invariance of objects under the group action of G could be
proven by showing the invariance under the flow of ψ(~v ).
53 / 117
Infinitesimal action
Let G act on M via Ψ. Each ~v ∈ g gives rise to a vector field
ψ(~v ) on M, namely as the tangents of the flow Ψ(exp(ε~v )e, x)
through x ∈ M. In other words
ψ(~v )|x =
d Ψ(exp(ε~v )e, x) = dΨx (~v ),
dε ε=0
where Ψx (g) = Ψ(g, x).
The vector field ψ(~v ) on M will be identified with the invariant
vector field ~v on G, and will also be referred to as the
infinitesimal generator of G. Sophus Lie’s great insight was that
the invariance of objects under the group action of G could be
proven by showing the invariance under the flow of ψ(~v ).
54 / 117
Invariance under group actions, Symmetry groups
Consider a smooth function F : M → Rm , defined on some
manifold M. Let G act on M. We say F is invariant under
the group action if for all g ∈ G and x ∈ M, we have
F (g · x) = F (x).
A subset S of the manifold M is said to be invariant if
g · x ∈ S for all g ∈ G and x ∈ S and G is called the
symmetry group of S.
A function F is invariant if and only if every level set
{x : F (x) = c} is invariant.
The zero-level set {x : F (x) = 0} is denoted by SF and is
called the subvariety of F .
55 / 117
Invariance under group actions, Symmetry groups
Consider a smooth function F : M → Rm , defined on some
manifold M. Let G act on M. We say F is invariant under
the group action if for all g ∈ G and x ∈ M, we have
F (g · x) = F (x).
A subset S of the manifold M is said to be invariant if
g · x ∈ S for all g ∈ G and x ∈ S and G is called the
symmetry group of S.
A function F is invariant if and only if every level set
{x : F (x) = c} is invariant.
The zero-level set {x : F (x) = 0} is denoted by SF and is
called the subvariety of F .
56 / 117
Invariance under group actions, Symmetry groups
Consider a smooth function F : M → Rm , defined on some
manifold M. Let G act on M. We say F is invariant under
the group action if for all g ∈ G and x ∈ M, we have
F (g · x) = F (x).
A subset S of the manifold M is said to be invariant if
g · x ∈ S for all g ∈ G and x ∈ S and G is called the
symmetry group of S.
A function F is invariant if and only if every level set
{x : F (x) = c} is invariant.
The zero-level set {x : F (x) = 0} is denoted by SF and is
called the subvariety of F .
57 / 117
Invariance under group actions, Symmetry groups
Consider a smooth function F : M → Rm , defined on some
manifold M. Let G act on M. We say F is invariant under
the group action if for all g ∈ G and x ∈ M, we have
F (g · x) = F (x).
A subset S of the manifold M is said to be invariant if
g · x ∈ S for all g ∈ G and x ∈ S and G is called the
symmetry group of S.
A function F is invariant if and only if every level set
{x : F (x) = c} is invariant.
The zero-level set {x : F (x) = 0} is denoted by SF and is
called the subvariety of F .
58 / 117
The great power of Lie theory
The great power of Lie theory lies in the crucial observation that
one can replace the nonlinear conditions for invariance of a
subset of function under a group action with the corresponding
invariance under the infinitesimal action of the infinitesimal
generators of the group action.
Theorem
Let G be a connected Lie group acting on a manifold M. A
smooth real valued function f : M → R is invariant under G if
and only if
ψ(~v )(f )(x) = 0, for all x ∈ M,
and every infinitesimal generator ~v = ψ(~v ) of G.
59 / 117
The great power of Lie theory
The great power of Lie theory lies in the crucial observation that
one can replace the nonlinear conditions for invariance of a
subset of function under a group action with the corresponding
invariance under the infinitesimal action of the infinitesimal
generators of the group action.
Theorem
Let G be a connected Lie group acting on a manifold M. A
smooth real valued function f : M → R is invariant under G if
and only if
ψ(~v )(f )(x) = 0, for all x ∈ M,
and every infinitesimal generator ~v = ψ(~v ) of G.
60 / 117
Proof
We will henceforth write ~v instead of ψ(~v ).
⇒ If f is invariant under G, then ε → f (exp(ε~v )e · x) is constant
for ε in some open interval around 0, differentiating and setting
ε = 0 gives ~v (f )(x) = 0.
⇐ Conversely, if ~v (f ) = 0 on all of M, then ε → f (exp(ε~v )e · x)
is constant and equal to x. This proves the invariance of f
under the action of flows exp(ε~v )e and hence under the entire
Lie group.
61 / 117
A system of algebraic equations
Consider a smooth function F : M → Rn , where M is an m
dimensional smooth manifold, m ≥ n. Let G be a local group of
transformations acting on M. We usually do not need the
invariance of the whole function F under G, but only the
invariance of the subvariety SF .
Theorem
Write F = (F1 , . . . , Fn ) and let F be of maximal rank on SF . The
subvariety SF is invariant under the group action on M if and
only if
~v (Fi )(x) = 0, i = 1, . . . , n
for all x ∈ SF .
62 / 117
A system of algebraic equations
Consider a smooth function F : M → Rn , where M is an m
dimensional smooth manifold, m ≥ n. Let G be a local group of
transformations acting on M. We usually do not need the
invariance of the whole function F under G, but only the
invariance of the subvariety SF .
Theorem
Write F = (F1 , . . . , Fn ) and let F be of maximal rank on SF . The
subvariety SF is invariant under the group action on M if and
only if
~v (Fi )(x) = 0, i = 1, . . . , n
for all x ∈ SF .
63 / 117
A system of algebraic equations
Consider a smooth function F : M → Rn , where M is an m
dimensional smooth manifold, m ≥ n. Let G be a local group of
transformations acting on M. We usually do not need the
invariance of the whole function F under G, but only the
invariance of the subvariety SF .
Theorem
Write F = (F1 , . . . , Fn ) and let F be of maximal rank on SF . The
subvariety SF is invariant under the group action on M if and
only if
~v (Fi )(x) = 0, i = 1, . . . , n
for all x ∈ SF .
64 / 117
Proof, sketch
Let F (x0 ) = 0 and choose coordinates around x0 in which
F (x1 , . . . , xm ) = (x1 , . . . , xn ).
Let ~v (x) = (ξ1 (x), . . . , ξm (x)) be any infinitesimal generator in
the new variables. The condition ~v (Fi ) = 0 means that
ξ1 (x) = · · · = ξn (x) = 0
whenever x ∈ {x1 = · · · = xn = 0} (this is SF is the new
variables). This means that the flow of ~v stays within
{x1 = · · · = xn = 0} near x0 which mean SF is invariant under
the flow of the infinitesimal generators.
65 / 117
A first look at characteristics
The following theorem will be important when we tackle
Noether’s theorem.
Theorem
Let F : M → Rn be of maximal rank on SF . A smooth function
f : M → R disappears on SF if and only if there are smooth
functions Q1 , . . . , Qn such that
f (x) = Q1 (x)F1 (x) + · · · + Qn (x)Fn (x),
for all x ∈ M. The functions Q1 , . . . , Qn are called the
characteristics of f on SF .
Proof (sketch). This is easy to prove locally by choosing local
coordinates in which F (x1 , . . . , xm ) = (x1 , . . . , xn ) and using the
multidimensional Taylor expansion. A global result is obtained
by partition of unity.
66 / 117
A first look at characteristics
The following theorem will be important when we tackle
Noether’s theorem.
Theorem
Let F : M → Rn be of maximal rank on SF . A smooth function
f : M → R disappears on SF if and only if there are smooth
functions Q1 , . . . , Qn such that
f (x) = Q1 (x)F1 (x) + · · · + Qn (x)Fn (x),
for all x ∈ M. The functions Q1 , . . . , Qn are called the
characteristics of f on SF .
Proof (sketch). This is easy to prove locally by choosing local
coordinates in which F (x1 , . . . , xm ) = (x1 , . . . , xn ) and using the
multidimensional Taylor expansion. A global result is obtained
by partition of unity.
67 / 117
Differential equations
We now know how to handle symmetry groups of subsets in
manifolds, specifically the subvarieties of smooth functions of
maximal rank. We can apply the same techniques to differential
equations by formulating them as algebraic. The tools needed
for this are jet spaces and prolongations, to be introduced
momentarily.
To accomplish this, we take a naïve approach to differential
equations, interpreting them as algebraic ones in all
independent variables. For example, the equation
ut + 6uux + uxx = 0 is seen as the subvariety of the smooth
map ∆ : R8 → R, where
∆(x, t, u, ut , ux , uxx , utt , uxt ) = ut + 6uux + uxx .
68 / 117
Differential equations
We now know how to handle symmetry groups of subsets in
manifolds, specifically the subvarieties of smooth functions of
maximal rank. We can apply the same techniques to differential
equations by formulating them as algebraic. The tools needed
for this are jet spaces and prolongations, to be introduced
momentarily.
To accomplish this, we take a naïve approach to differential
equations, interpreting them as algebraic ones in all
independent variables. For example, the equation
ut + 6uux + uxx = 0 is seen as the subvariety of the smooth
map ∆ : R8 → R, where
∆(x, t, u, ut , ux , uxx , utt , uxt ) = ut + 6uux + uxx .
69 / 117
Differential equations
We now know how to handle symmetry groups of subsets in
manifolds, specifically the subvarieties of smooth functions of
maximal rank. We can apply the same techniques to differential
equations by formulating them as algebraic. The tools needed
for this are jet spaces and prolongations, to be introduced
momentarily.
To accomplish this, we take a naïve approach to differential
equations, interpreting them as algebraic ones in all
independent variables. For example, the equation
ut + 6uux + uxx = 0 is seen as the subvariety of the smooth
map ∆ : R8 → R, where
∆(x, t, u, ut , ux , uxx , utt , uxt ) = ut + 6uux + uxx .
70 / 117
Differential equations
We now know how to handle symmetry groups of subsets in
manifolds, specifically the subvarieties of smooth functions of
maximal rank. We can apply the same techniques to differential
equations by formulating them as algebraic. The tools needed
for this are jet spaces and prolongations, to be introduced
momentarily.
To accomplish this, we take a naïve approach to differential
equations, interpreting them as algebraic ones in all
independent variables. For example, the equation
ut + 6uux + uxx = 0 is seen as the subvariety of the smooth
map ∆ : R8 → R, where
∆(x, t, u, ut , ux , uxx , utt , uxt ) = ut + 6uux + uxx .
71 / 117
Jet spaces
Consider smooth functions from X ⊆ Rp to U ⊆ Rq . Let M be
the manifold M = X × U = {(x1 , . . . , xp , u1 , . . . , uq )}, the
Cartesian product of X and U. The first jet space of M is
M (1) := M × U (1) where U (1) = Rpq represents all the first
derivatives of (u1 , . . . , uq ), that is
U (1) = {(
∂
uj )1≤i≤p,1≤j≤q } = {(uj,i )1≤i≤p,1≤j≤q }.
∂xi
∂
∂xi uj = uj,i as an independent
M (k +1) := M (k ) × U (k +1) where
Note that we are treating
variable. Recursively,
U (k +1) = Rqpk +1 and
pk +1
p+k
:=
,
k +1
so U (k +1) represents all the possible partial derivatives of u of
order k + 1. An element of M (k ) will be denoted (x, u (k ) ).
72 / 117
Jet spaces
Consider smooth functions from X ⊆ Rp to U ⊆ Rq . Let M be
the manifold M = X × U = {(x1 , . . . , xp , u1 , . . . , uq )}, the
Cartesian product of X and U. The first jet space of M is
M (1) := M × U (1) where U (1) = Rpq represents all the first
derivatives of (u1 , . . . , uq ), that is
U (1) = {(
∂
uj )1≤i≤p,1≤j≤q } = {(uj,i )1≤i≤p,1≤j≤q }.
∂xi
∂
∂xi uj = uj,i as an independent
M (k +1) := M (k ) × U (k +1) where
Note that we are treating
variable. Recursively,
U (k +1) = Rqpk +1 and
pk +1
p+k
:=
,
k +1
so U (k +1) represents all the possible partial derivatives of u of
order k + 1. An element of M (k ) will be denoted (x, u (k ) ).
73 / 117
Jet spaces
Consider smooth functions from X ⊆ Rp to U ⊆ Rq . Let M be
the manifold M = X × U = {(x1 , . . . , xp , u1 , . . . , uq )}, the
Cartesian product of X and U. The first jet space of M is
M (1) := M × U (1) where U (1) = Rpq represents all the first
derivatives of (u1 , . . . , uq ), that is
U (1) = {(
∂
uj )1≤i≤p,1≤j≤q } = {(uj,i )1≤i≤p,1≤j≤q }.
∂xi
∂
∂xi uj = uj,i as an independent
M (k +1) := M (k ) × U (k +1) where
Note that we are treating
variable. Recursively,
U (k +1) = Rqpk +1 and
pk +1
p+k
:=
,
k +1
so U (k +1) represents all the possible partial derivatives of u of
order k + 1. An element of M (k ) will be denoted (x, u (k ) ).
74 / 117
Jet spaces
Consider smooth functions from X ⊆ Rp to U ⊆ Rq . Let M be
the manifold M = X × U = {(x1 , . . . , xp , u1 , . . . , uq )}, the
Cartesian product of X and U. The first jet space of M is
M (1) := M × U (1) where U (1) = Rpq represents all the first
derivatives of (u1 , . . . , uq ), that is
U (1) = {(
∂
uj )1≤i≤p,1≤j≤q } = {(uj,i )1≤i≤p,1≤j≤q }.
∂xi
∂
∂xi uj = uj,i as an independent
M (k +1) := M (k ) × U (k +1) where
Note that we are treating
variable. Recursively,
U (k +1) = Rqpk +1 and
pk +1
p+k
:=
,
k +1
so U (k +1) represents all the possible partial derivatives of u of
order k + 1. An element of M (k ) will be denoted (x, u (k ) ).
75 / 117
Prolongations of functions
A smooth function f : X → U similarly has k th prolongation
pr(k ) f (x) such that (x, pr(k ) f (x)) ∈ M (k ) in the obvious way.
A k th order system of differential equations in p independent
variables and q dependent variables is given as a system
∆i (x, u (k ) ) = 0,
i = 1, . . . , r .
We shall assume that the ∆i are smooth in their arguments.
We can view this system as a mapping from M (k ) to Rr . A
solution to the differential equation is a function f , such that
∆i (x, pr(k ) f (x)) = 0,
i = 1, . . . , r
for all x is some open open subset of X .
76 / 117
Prolongations of functions
A smooth function f : X → U similarly has k th prolongation
pr(k ) f (x) such that (x, pr(k ) f (x)) ∈ M (k ) in the obvious way.
A k th order system of differential equations in p independent
variables and q dependent variables is given as a system
∆i (x, u (k ) ) = 0,
i = 1, . . . , r .
We shall assume that the ∆i are smooth in their arguments.
We can view this system as a mapping from M (k ) to Rr . A
solution to the differential equation is a function f , such that
∆i (x, pr(k ) f (x)) = 0,
i = 1, . . . , r
for all x is some open open subset of X .
77 / 117
Prolongations of functions
A smooth function f : X → U similarly has k th prolongation
pr(k ) f (x) such that (x, pr(k ) f (x)) ∈ M (k ) in the obvious way.
A k th order system of differential equations in p independent
variables and q dependent variables is given as a system
∆i (x, u (k ) ) = 0,
i = 1, . . . , r .
We shall assume that the ∆i are smooth in their arguments.
We can view this system as a mapping from M (k ) to Rr . A
solution to the differential equation is a function f , such that
∆i (x, pr(k ) f (x)) = 0,
i = 1, . . . , r
for all x is some open open subset of X .
78 / 117
Prolongations of group actions
Let G be a Lie group acting on the manifold M = X × U,
g · (x, u (k ) ) = (g · x, g · u (k ) ) = (x̃, ũ (k ) ). We can prolong the
action of G to the jet spaces M (k ) in the following way:
(k )
Let (x0 , u0 ) ∈ M (k ) and pick any smooth function, f ,
(k )
matching u0
(k )
at x0 (i.e. pr(k ) f (x0 ) = u0 ), for instance the
(k )
k th order Taylor polynomial at x0 matching u0 .
The action of g ∈ G, for g close enough to the identity e,
on the graph of f in M will give the graph of some other
function f̃ (at least on some open set around g · x0 = x˜0 ).
(k )
We define the prolonged action pr(k ) g on (x0 , u0 ) as
(x˜0 , pr(k ) f̃ (x˜0 )).
79 / 117
Prolongations of group actions
Let G be a Lie group acting on the manifold M = X × U,
g · (x, u (k ) ) = (g · x, g · u (k ) ) = (x̃, ũ (k ) ). We can prolong the
action of G to the jet spaces M (k ) in the following way:
(k )
Let (x0 , u0 ) ∈ M (k ) and pick any smooth function, f ,
(k )
matching u0
(k )
at x0 (i.e. pr(k ) f (x0 ) = u0 ), for instance the
(k )
k th order Taylor polynomial at x0 matching u0 .
The action of g ∈ G, for g close enough to the identity e,
on the graph of f in M will give the graph of some other
function f̃ (at least on some open set around g · x0 = x˜0 ).
(k )
We define the prolonged action pr(k ) g on (x0 , u0 ) as
(x˜0 , pr(k ) f̃ (x˜0 )).
80 / 117
Prolongations of group actions
Let G be a Lie group acting on the manifold M = X × U,
g · (x, u (k ) ) = (g · x, g · u (k ) ) = (x̃, ũ (k ) ). We can prolong the
action of G to the jet spaces M (k ) in the following way:
(k )
Let (x0 , u0 ) ∈ M (k ) and pick any smooth function, f ,
(k )
matching u0
(k )
at x0 (i.e. pr(k ) f (x0 ) = u0 ), for instance the
(k )
k th order Taylor polynomial at x0 matching u0 .
The action of g ∈ G, for g close enough to the identity e,
on the graph of f in M will give the graph of some other
function f̃ (at least on some open set around g · x0 = x˜0 ).
(k )
We define the prolonged action pr(k ) g on (x0 , u0 ) as
(x˜0 , pr(k ) f̃ (x˜0 )).
81 / 117
Prolongations of group actions
Let G be a Lie group acting on the manifold M = X × U,
g · (x, u (k ) ) = (g · x, g · u (k ) ) = (x̃, ũ (k ) ). We can prolong the
action of G to the jet spaces M (k ) in the following way:
(k )
Let (x0 , u0 ) ∈ M (k ) and pick any smooth function, f ,
(k )
matching u0
(k )
at x0 (i.e. pr(k ) f (x0 ) = u0 ), for instance the
(k )
k th order Taylor polynomial at x0 matching u0 .
The action of g ∈ G, for g close enough to the identity e,
on the graph of f in M will give the graph of some other
function f̃ (at least on some open set around g · x0 = x˜0 ).
(k )
We define the prolonged action pr(k ) g on (x0 , u0 ) as
(x˜0 , pr(k ) f̃ (x˜0 )).
82 / 117
Symmetry groups of differential equations
Definition
Let ∆(x, u (k ) ) be a system of k th order differential equations
with subvariety S∆ ∈ M (k ) . A group G acting on M is said to be
a symmetry group of the system ∆ if the k th prolongation of G
is a symmetry group of S∆ .
This definition is equivalent to saying that if f is a solution to the
system, then the function gotten from g · f , f̃ is also a solution.
Here, a "solution" means a solution on any open subset of X .
83 / 117
Symmetry groups of differential equations
Definition
Let ∆(x, u (k ) ) be a system of k th order differential equations
with subvariety S∆ ∈ M (k ) . A group G acting on M is said to be
a symmetry group of the system ∆ if the k th prolongation of G
is a symmetry group of S∆ .
This definition is equivalent to saying that if f is a solution to the
system, then the function gotten from g · f , f̃ is also a solution.
Here, a "solution" means a solution on any open subset of X .
84 / 117
Symmetry groups of differential equations
Definition
Let ∆(x, u (k ) ) be a system of k th order differential equations
with subvariety S∆ ∈ M (k ) . A group G acting on M is said to be
a symmetry group of the system ∆ if the k th prolongation of G
is a symmetry group of S∆ .
This definition is equivalent to saying that if f is a solution to the
system, then the function gotten from g · f , f̃ is also a solution.
Here, a "solution" means a solution on any open subset of X .
85 / 117
Prolongation of vector fields
Knowing how to prolong local group actions to higher order jet
spaces allows us to prolong vector fields on M = X × U.
Let ~v be a vector field on M and consider the local group action
of its flow, namely (remember G = R in this case)
Ψ(ε, (x, u)) = exp(ε~v )(x, u).
Now consider the prolonged local group action pr(k ) exp(ε~v ) on
M (k ) and define pr(k )~v as being
d (k )~
pr v |(x,u (k ) ) :=
pr(k ) exp(ε~v )(x, u (k ) ).
dε ε=0
86 / 117
Prolongation of vector fields
Knowing how to prolong local group actions to higher order jet
spaces allows us to prolong vector fields on M = X × U.
Let ~v be a vector field on M and consider the local group action
of its flow, namely (remember G = R in this case)
Ψ(ε, (x, u)) = exp(ε~v )(x, u).
Now consider the prolonged local group action pr(k ) exp(ε~v ) on
M (k ) and define pr(k )~v as being
d (k )~
pr v |(x,u (k ) ) :=
pr(k ) exp(ε~v )(x, u (k ) ).
dε ε=0
87 / 117
Prolongation of vector fields
Knowing how to prolong local group actions to higher order jet
spaces allows us to prolong vector fields on M = X × U.
Let ~v be a vector field on M and consider the local group action
of its flow, namely (remember G = R in this case)
Ψ(ε, (x, u)) = exp(ε~v )(x, u).
Now consider the prolonged local group action pr(k ) exp(ε~v ) on
M (k ) and define pr(k )~v as being
d (k )~
pr v |(x,u (k ) ) :=
pr(k ) exp(ε~v )(x, u (k ) ).
dε ε=0
88 / 117
The prolongation formula
Let ~v be a vector field on
M = X × U = {(x1 , . . . , xp , u1 , . . . , uq )}. We write ~v as
~v =
p
X
i=1
Where
∂
∂xi
q
X
∂
∂
ξi (x, u)
φj (x, u)
+
.
∂xi
∂uj
j=1
denotes the standard basis element with 1 in the
component of xi and 0 in all others and
∂
∂uj
is the p + jth
standard basis vector. Then the k th prolongation of ~v is
pr(k )~v = ~v +
q X
X
j=1 |J|≤k
φJj (x, u (k ) )
∂
.
∂uj,J
89 / 117
The prolongation formula
Let ~v be a vector field on
M = X × U = {(x1 , . . . , xp , u1 , . . . , uq )}. We write ~v as
~v =
p
X
i=1
Where
∂
∂xi
q
X
∂
∂
ξi (x, u)
φj (x, u)
+
.
∂xi
∂uj
j=1
denotes the standard basis element with 1 in the
component of xi and 0 in all others and
∂
∂uj
is the p + jth
standard basis vector. Then the k th prolongation of ~v is
pr(k )~v = ~v +
q X
X
j=1 |J|≤k
φJj (x, u (k ) )
∂
.
∂uj,J
90 / 117
The prolongation formula
Let ~v be a vector field on
M = X × U = {(x1 , . . . , xp , u1 , . . . , uq )}. We write ~v as
~v =
p
X
i=1
Where
∂
∂xi
q
X
∂
∂
ξi (x, u)
φj (x, u)
+
.
∂xi
∂uj
j=1
denotes the standard basis element with 1 in the
component of xi and 0 in all others and
∂
∂uj
is the p + jth
standard basis vector. Then the k th prolongation of ~v is
pr(k )~v = ~v +
q X
X
j=1 |J|≤k
φJj (x, u (k ) )
∂
.
∂uj,J
91 / 117
The prolongation formula
Where
φJj (x, u (k ) )
= DJ φj −
p
X
ξi uj,i +
p
X
ξi uj,J+i .
i=1
i=1
This calls for a few comments on the notation.
First, J is a multi-index (α1 , . . . , αp ) such that
∂J =
and |J| =
∂xα11
∂ |J|
α ,
· · · ∂xpp
Pp
i=1 |αi |.
92 / 117
The prolongation formula
Where
φJj (x, u (k ) )
= DJ φj −
p
X
ξi uj,i +
p
X
ξi uj,J+i .
i=1
i=1
This calls for a few comments on the notation.
First, J is a multi-index (α1 , . . . , αp ) such that
∂J =
and |J| =
∂xα11
∂ |J|
α ,
· · · ∂xpp
Pp
i=1 |αi |.
93 / 117
The prolongation formula
The operator DJ is called the total derivative, and works
like ∂J treating all u as actual functions of x. For example,
in the case p = q = 1
Dx (uux ) = ux2 + uuxx .
The element uj,J+i represents ∂xi uj,J = ∂xi ∂J uj .
94 / 117
The prolongation formula
The operator DJ is called the total derivative, and works
like ∂J treating all u as actual functions of x. For example,
in the case p = q = 1
Dx (uux ) = ux2 + uuxx .
The element uj,J+i represents ∂xi uj,J = ∂xi ∂J uj .
95 / 117
A useful reformulation
A useful reformulation of the prolongation formula is obtained
by defining the q-tuple
Qj (x, u
(1)
) := φj −
p
X
ξi uj,i .
i=1
Then the prolongation formula becomes, after rearrangement of
terms
pr(k )~v =
q X
X
DJ Qj
j=1 |J|≤k
= pr(k )~vQ +
p
q X
X
X
∂
∂
∂ +
ξi
+
uj,J+i
∂uj,J
∂xi
∂uj,J
1
i=1
p
X
j= |J|≤k
ξi Di .
i=1
96 / 117
A useful reformulation
A useful reformulation of the prolongation formula is obtained
by defining the q-tuple
Qj (x, u
(1)
) := φj −
p
X
ξi uj,i .
i=1
Then the prolongation formula becomes, after rearrangement of
terms
pr(k )~v =
q X
X
DJ Qj
j=1 |J|≤k
= pr(k )~vQ +
p
q X
X
X
∂
∂
∂ +
ξi
+
uj,J+i
∂uj,J
∂xi
∂uj,J
1
i=1
p
X
j= |J|≤k
ξi Di .
i=1
97 / 117
A useful reformulation
Where we have defined
~vQ =
q
X
Qj (x, u (1) )
j=1
and hence
(k )~
pr
vQ =
q X
X
j=1 |J|≤k
∂
,
∂uj
DJ Qj
∂
.
∂uj,J
Having the prolongation formula at our disposal, we are finally
in a position where we can attack Noether’s theorem. But first,
some calculus of variations.
98 / 117
A useful reformulation
Where we have defined
~vQ =
q
X
Qj (x, u (1) )
j=1
and hence
(k )~
pr
vQ =
q X
X
j=1 |J|≤k
∂
,
∂uj
DJ Qj
∂
.
∂uj,J
Having the prolongation formula at our disposal, we are finally
in a position where we can attack Noether’s theorem. But first,
some calculus of variations.
99 / 117
Lagrangians
Definition
Let Ω ⊆ X be a connected, open subset with smooth boundary
∂Ω. A variational problem consists of finding the extrema of a
functional
Z
Z
L(x, u (k ) )dx =
L(x, pr(k ) f (x))dx
L(u) =
Ω
Ω
where u = f (x) is a function of x.
A necessary condition for a function, f , to be an extremum of
the above variational problem, is that is satisfies the
Euler-Lagrange equations
Ej (L(x, pr(k ) f (x))) = 0,
P
where Ej = |J|≤k (−DJ ) ∂u∂j,J .
j = 1, . . . , q,
100 / 117
Lagrangians
Definition
Let Ω ⊆ X be a connected, open subset with smooth boundary
∂Ω. A variational problem consists of finding the extrema of a
functional
Z
Z
L(x, u (k ) )dx =
L(x, pr(k ) f (x))dx
L(u) =
Ω
Ω
where u = f (x) is a function of x.
A necessary condition for a function, f , to be an extremum of
the above variational problem, is that is satisfies the
Euler-Lagrange equations
Ej (L(x, pr(k ) f (x))) = 0,
P
where Ej = |J|≤k (−DJ ) ∂u∂j,J .
j = 1, . . . , q,
101 / 117
Transformation of a Lagrangian
Consider a change of variables on M = Ω × U:
x̃ = Λ(x, u),
ũ = Φ(x, u).
Let f : Ω → U be a smooth function and suppose that the
equation x̃ = Λ(x, f (x)) can be solved for all x in an open set
Ω0 . That is, there is some function γ, such that γ(x̃) = x, for all
x ∈ Ω0 . Let Ω̃0 = γ −1 (Ω0 ). This change of variables then
defines the transformed function on Ω̃0
f̃ (x̃) = Φ(γ(x̃), f (γ(x̃))).
Notice that γ is just the inverse of the function
x → Λ(x, f (x))
102 / 117
Transformation of a Lagrangian
Consider a change of variables on M = Ω × U:
x̃ = Λ(x, u),
ũ = Φ(x, u).
Let f : Ω → U be a smooth function and suppose that the
equation x̃ = Λ(x, f (x)) can be solved for all x in an open set
Ω0 . That is, there is some function γ, such that γ(x̃) = x, for all
x ∈ Ω0 . Let Ω̃0 = γ −1 (Ω0 ). This change of variables then
defines the transformed function on Ω̃0
f̃ (x̃) = Φ(γ(x̃), f (γ(x̃))).
Notice that γ is just the inverse of the function
x → Λ(x, f (x))
103 / 117
Transformation of a Lagrangian
Consider a change of variables on M = Ω × U:
x̃ = Λ(x, u),
ũ = Φ(x, u).
Let f : Ω → U be a smooth function and suppose that the
equation x̃ = Λ(x, f (x)) can be solved for all x in an open set
Ω0 . That is, there is some function γ, such that γ(x̃) = x, for all
x ∈ Ω0 . Let Ω̃0 = γ −1 (Ω0 ). This change of variables then
defines the transformed function on Ω̃0
f̃ (x̃) = Φ(γ(x̃), f (γ(x̃))).
Notice that γ is just the inverse of the function
x → Λ(x, f (x))
104 / 117
Transformation of a Lagrangian
Making the change of variables
(x, u) → (Λ(x, u), Φ(x, u)) = (x̃, ũ), the Lagrangian assumes a
new form L̃
Z
Z
(k )
L̃(x̃, pr f̃ (x))d x̃ =
L(x, pr(k ) f (x))dx.
Ω̃0
Ω0
Now making the change of variables
x → γ −1 (x) = Λ(x, f (x)) = x̃ gives
Z
Z
(k )
L̃(x̃, pr f̃ (x))detJΛ (x, f (x))dx =
Ω0
L(x, pr(k ) f (x))dx,
Ω0
where JΛ is the Jacobian of x → Λ(x, f (x)).
105 / 117
Transformation of a Lagrangian
Making the change of variables
(x, u) → (Λ(x, u), Φ(x, u)) = (x̃, ũ), the Lagrangian assumes a
new form L̃
Z
Z
(k )
L̃(x̃, pr f̃ (x))d x̃ =
L(x, pr(k ) f (x))dx.
Ω̃0
Ω0
Now making the change of variables
x → γ −1 (x) = Λ(x, f (x)) = x̃ gives
Z
Z
(k )
L̃(x̃, pr f̃ (x))detJΛ (x, f (x))dx =
Ω0
L(x, pr(k ) f (x))dx,
Ω0
where JΛ is the Jacobian of x → Λ(x, f (x)).
106 / 117
Transformation of a Lagrangian
Since the equality
Z
Z
(k )
L̃(x̃, pr f (x))detJΛ (x, f (x))dx =
Ω0
L(x, pr(k ) f (x))dx,
Ω0
holds on every open subset of Ω0 , it holds pointwise there
L̃(x̃, pr(k ) f̃ (x̃))detJΛ (x, f (x)) = L(x, pr(k ) f (x)).
Definition
A local group of transformations G acting on M = Ω × U is a
variational symmetry group if for any Ω0 ⊆ Ω, f defined on Ω0
such that g · f = f̃ is well defined on γ −1 (Ω0 ) = Ω̃0 we have
Z
Z
(k )
L(x̃, pr f̃ (x̃)d x̃ =
L(x, pr(k ) f (x)dx.
Ω̃0
Ω0
107 / 117
Transformation of a Lagrangian
Since the equality
Z
Z
(k )
L̃(x̃, pr f (x))detJΛ (x, f (x))dx =
Ω0
L(x, pr(k ) f (x))dx,
Ω0
holds on every open subset of Ω0 , it holds pointwise there
L̃(x̃, pr(k ) f̃ (x̃))detJΛ (x, f (x)) = L(x, pr(k ) f (x)).
Definition
A local group of transformations G acting on M = Ω × U is a
variational symmetry group if for any Ω0 ⊆ Ω, f defined on Ω0
such that g · f = f̃ is well defined on γ −1 (Ω0 ) = Ω̃0 we have
Z
Z
(k )
L(x̃, pr f̃ (x̃)d x̃ =
L(x, pr(k ) f (x)dx.
Ω̃0
Ω0
108 / 117
Infinitesimal invariance
Having defined symmetries of Lagrangians, we can prove an
analogue to the infinitesimal invariance of algebraic systems of
equations.
Theorem
A connected group of transformations G acting on M = Ω × U
is a variational symmetry group of the functional L if and only if
pr(k )~v (L) + LDivξ = 0
for all (x, u (k ) ) ∈ M (k ) and every infinitesimal generator
~v =
p
X
i=1
q
X
∂
∂
ξi (x, u)
+
φj (x, u)
.
∂xi
∂uj
j=1
Note that Divξ denotes the total divergence
Divξ = D1 ξ1 + . . . Dp ξp .
109 / 117
Infinitesimal invariance
Having defined symmetries of Lagrangians, we can prove an
analogue to the infinitesimal invariance of algebraic systems of
equations.
Theorem
A connected group of transformations G acting on M = Ω × U
is a variational symmetry group of the functional L if and only if
pr(k )~v (L) + LDivξ = 0
for all (x, u (k ) ) ∈ M (k ) and every infinitesimal generator
~v =
p
X
i=1
q
X
∂
∂
ξi (x, u)
+
φj (x, u)
.
∂xi
∂uj
j=1
Note that Divξ denotes the total divergence
Divξ = D1 ξ1 + . . . Dp ξp .
110 / 117
Conservation laws
Consider a system of differential equations ∆(x, u (k ) ) = 0. A
conservation law for this system is a divergence expression
DivP = 0,
that vanishes on all solutions f to the system.
Here, P(x, u (k ) ) is a p-tuple (P1 (x, u (k ) ), . . . , Pp (x, u (k ) )) and as
before Div is total divergence.
For evolution equations ut = N(x, u, ux , . . .), a conservation law
has the familiar form
Dt T + DivX = 0,
where T is the conserved density, and X is the associated flux.
111 / 117
Conservation laws
Consider a system of differential equations ∆(x, u (k ) ) = 0. A
conservation law for this system is a divergence expression
DivP = 0,
that vanishes on all solutions f to the system.
Here, P(x, u (k ) ) is a p-tuple (P1 (x, u (k ) ), . . . , Pp (x, u (k ) )) and as
before Div is total divergence.
For evolution equations ut = N(x, u, ux , . . .), a conservation law
has the familiar form
Dt T + DivX = 0,
where T is the conserved density, and X is the associated flux.
112 / 117
Conservation laws
Consider a system of differential equations ∆(x, u (k ) ) = 0. A
conservation law for this system is a divergence expression
DivP = 0,
that vanishes on all solutions f to the system.
Here, P(x, u (k ) ) is a p-tuple (P1 (x, u (k ) ), . . . , Pp (x, u (k ) )) and as
before Div is total divergence.
For evolution equations ut = N(x, u, ux , . . .), a conservation law
has the familiar form
Dt T + DivX = 0,
where T is the conserved density, and X is the associated flux.
113 / 117
Noether’s Theorem
Theorem
Suppose G is a local one-parameter
group of symmetries of the
Z
variational problem L[u] =
~v =
p
X
i=1
L(x, u (n) )dx. Let
q
X
∂
∂
ξ (x, u) i +
φ(x, u) α
∂u
∂x
i
α=i
be the infinitesimal generator of G, and
p
X
Qα (x, u) = φα −
ξ i uiα the corresponding characteristic of ~v .
i=1
Then Q = (Q1 , . . . , Qq ) is also the characteristic of a
conservation law for the Euler-Lagrange equations E(L) = 0.
114 / 117
Proof of Noether’s theorem
We have, by our reformulation of the prolongation formula
0 = pr(k )~v (L) + LDivξ
(k )~
= pr
vQ (L) +
p
X
ξi D i L + L
i=1
p
X
Di ξi
i=1
= pr(k )~vQ (L) + Div(Lξ).
115 / 117
Proof of Noether’s theorem
We can integrate the expression for pr(k )~vQ (L) by parts to
obtain
0 = pr(k )~vQ (L) +
X
DJ Qj
j,J
=
X
j,J
=
q
X
Qj · (−DJ )
∂
∂uj,J
∂
+ DivA
∂uj,J
Qj Ej (L) + DivA,
j=1
where A is some p-tuple of functions, whose explicit form is not
needed.
116 / 117
Q.E.D.
We therefore have
0 =pr(k )~vQ (L) + Div(Lξ)
pr(k )~vQ (L) =
q
X
Qj Ej (L) + DivA,
j=1
and hence
Div(−Lξ − A) = Q · E(L).
117 / 117