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)