QMGreensite_merged

12 CHAPTER1. THECLASSICALSTATE

1.2 Euler-Lagrange and Hamilton’s Equations

Inbrief,theEuler-Lagrangeequationsarethesecond-orderformoftheequationsof
motion(1.1),whileHamilton’sequationsarethefirst-orderform(1.2).Ineitherform,
theequationsofmotioncanberegardedasaconsequenceofthePrincipleofLeast
Action. Wewillnowre-writethoseequationsinaverygeneralway,whichcan be
appliedtoanymechanicalsystem,includingthosewhicharemuchmorecomplicated
thanabaseball.
Webeginbywriting

S[{xi}]=

N∑− 1

n=0

!L[xn,x ̇n] (1.12)

where

L[xn,x ̇n]=

1

2

mx ̇^2 n−V(xn) (1.13)

andwhere

x ̇n≡

xn+1−xn

!

(1.14)

L[xn,x ̇n]isknownastheLagrangianfunction. Thentheprincipleofleastaction
requiresthat,foreachk, 1 ≤k≤N−1,

0 =

d

dxk

S[{xi}]=

N∑− 1

n=0

!

d

dxk

L[xn,x ̇n]

=!

∂

∂xk

L[xk,x ̇k]+

N∑− 1

n=0

!

∂L[xn,x ̇n]

∂x ̇n

dx ̇n

dxk

(1.15)

and,since

dx ̇n

dxk

=







1

! n=k−^1

−^1! n=k

0 otherwise

(1.16)

thisbecomes

∂

∂xk

L[xk,x ̇k]−

1

!

{

∂

∂x ̇k

L[xk,x ̇k]−

∂

∂x ̇k− 1

L[xk− 1 ,x ̇k− 1 ]

}

= 0 (1.17)

Recallingthatxn=x(tn),thislastequationcanbewritten
(
∂L[x,x ̇]
∂x

)

t=tn

−

1

!

{(

∂L[x,x ̇]

∂x ̇

)

t=tn

−

(

∂L[x,x ̇]

∂x ̇

)

t=tn−!

}

= 0 (1.18)

ThisistheEuler-Lagrangeequationforthebaseball. Itbecomessimplerwhenwe
takethe!→ 0 limit(the”continuum”limit).Inthatlimit,wehave

x ̇n=

xn+1−xn

!

→ x ̇(t)=

dx

dt

S=

N∑− 1

n=1

!L[xn,x ̇n] → S=

∫t 0 +∆t

t 0

dtL[x(t),x ̇(t)] (1.19)