Mathematical Principles of Theoretical Physics

90 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

Now, we discuss the symmetry from the viewpoint of action functional and group action.
The key feature is then characterized by the Noether theorem. LetF:R^2 →R^1 be a function
(a finite dimensional functional) defined by

(2.5.46) F(u) =x^2 +y^2 foru= (x,y)∈R^2.

All the transformation matrices in (2.5.44) constitute a group, denoted bySO( 2 ), called the
orthogonal group:

(2.5.47) SO( 2 ) =

{

Aθ|Aθ=

(

cosθ sinθ

−sinθ cosθ

)

,θ∈R^1

}

.

The symmetry here is the invariance of the function (2.5.46) under theSO( 2 )group action:

F(Aθu) =F(u) ∀Aθ∈SO( 2 ).

The generalization to the Lagrangian dynamics is what we shall introduce in this subsection.

Definition 2.37.Let G be a group, X be a Banach space, and F:X→R^1 a continuous
functional. Let G be a group acting on X:

(2.5.48)

Au∈X ∀u∈XandA∈G,

A(Bu) = (AB)u ∀u∈X,A,B∈G.

The functional F is called invariant under the action (2.5.48) of G, if

F(Au) =F(u) ∀A∈G.

We consider a Lagrange action defined on a function spaceX:

(2.5.49) L(u,u ̇) =

∫T

0

L(u,u ̇)dt foru∈X,

where the Lagrange densityLis defined by

L(u,u ̇) =







L(x,x ̇) for an N-body system,

∫

Ω

g(u,u ̇,Du,···,Dmu)dx otherwise.

The following theorem is the well-known Noether theorem, which provides a correspon-
dence between symmetries and conservation laws in the Lagrangian system (2.5.49).

Theorem 2.38(Noether Theorem).Let G={Aλ|λ∈R^1 }be a parameterized group, and
L(u,u ̇)be the Lagrange action given by (2.5.49). If L is invariant under the group action of
G:

(2.5.50) L(u,u ̇) =L(Aλu,Aλu ̇) ∀Aλ∈G,

then the system has a conserved quantity induced by G, expressed as

(2.5.51) I(u,u ̇) =

〈δL

δu ̇

,

dAλ(u)

dλ

∣

∣

∣

λ= 0

〉

for Aλ∈G.

In other words,

(2.5.52)

d

dt

I(u,u ̇) = 0 for any solutions u ofδL= 0.