Mathematical Principles of Theoretical Physics

2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 91

Remark 2.39.The correspondencebetween symmetries and conservation laws in the Noether
Theorem holds true as well for discrete transformation groups, such as the reflections of time
and space.

Proof of Theorem2.38.Let(u,ut)be a solution of the variational equation ofLgiven by
(2.5.49):

(2.5.53)

d

dt

(

δL(u,u ̇)

δu ̇

)

=

δ

δu

L(u,u ̇).

By the invariance (2.5.50),(Aλu,Aλu ̇)are also solutions of (2.5.53). Then it follows from
(2.5.50) that

(2.5.54) 0 =

∂L(Aλu,Aλu ̇)

∂ λ

=

〈δL

∂Φ

,

dΦ

dλ

〉

+

〈δL

δΦ ̇

,

dΦ ̇

dλ

〉

,

whereΦ=Aλu. Since(Aλu,Aλu ̇)satisfies (2.5.53), we have

(2.5.55)

δ

δΦ

L(Aλu,Aλu ̇) =

d

dt

(

δL(Aλu,Aλu ̇)

δΦ ̇

)

.

Inserting (2.5.55) in (2.5.54) we deduce that for anyΦ=Aλuand anyλ∈R^1 ,

0 =

〈d

dt

(

δL

δΦ ̇

)

,

dΦ

dλ

〉

+

〈∂L

∂Φ ̇

,

d

dt

(

dΦ

dλ

)〉

=

d

dt

〈δL

δΦ ̇

,

dΦ

dλ

〉

,

which implies that
d
dt
I(u,u ̇) =

d

dt

〈δL

δΦ ̇

,

dΦ

dλ

〉∣

∣

∣

λ= 0

= 0 ,

whereAλu=uifλ=0. The proof is complete.

We now give two examples to show how to apply the Noether theorem to a specific
physical problem.

Example 2.40.Consider anN-body motion, such as a system ofNplanets. Letmkandxkbe
the mass and the coordinates of thek-th body. The Lagrange density is

(2.5.56) L=

1

2

N

∑

k= 1

mk|x ̇k|^2 −∑

i 6 =j

V(|xi−xj|).

Note thatxis theuin Theorem2.38. LetGbe the translation group:

G={Aλ|Aλ:R^3 →R^3 ,Aλx=x+λ~r},

where~ris a given vector. It clear that (2.5.56) is invariant under the transformation ofG. By

δL

δx ̇

=

(

∂L

∂x ̇ 1

,···,

∂L

∂x ̇N

)

= (m 1 x ̇ 1 ,···,mNx ̇N),

d

dλ

∣

∣

∣

λ= 0

Aλx=

d

dλ

∣

∣

∣

λ= 0

(x+λ~r) = (~r,···,~r

︸︷︷ ︸

N

).