14 CHAPTER1. THECLASSICALSTATE
wherewehaveapplied(1.24).Next,differentiatingHwithrespecttox,
∂H
∂x= p∂x ̇(x,p)
∂x−
∂L
∂x−
∂L
∂x ̇∂x ̇(p,x)
∂x
= −∂L
∂x(1.28)
UsingtheEuler-Lagrangeequation(1.21)(andthisiswheretheequationsofmotion
enter),wefind
∂H
∂x= −
d
dt∂L
∂x ̇
= −dp
dt(1.29)
Thus,withthehelpoftheHamiltonianfunction,wehaverewrittenthesingle2nd
orderEuler-Lagrangeequation(1.21)asapairof1storderdifferentialequations
dx
dt=
∂H
∂p
dp
dt= −
∂H
∂x(1.30)
whichareknownasHamilton’sEquations.
Forabaseball,theLagrangianisgivenbyeq. (1.20),andthereforethemomentum
is
p=∂L
∂x ̇=mx ̇ (1.31)Thisisinvertedtogive
x ̇=x ̇(p,x)=p
m(1.32)
andtheHamiltonianis
H = px ̇(x,p)−L[x,x ̇(x,p)]= pp
m−
[
1
2m(p
m)^2 −V(x)]=
p^2
2 m+V(x) (1.33)Note that the Hamiltonian forthe baseball issimply the kinetic energy plus the
potentialenergy; i.e. theHamiltonian isanexpressionforthe totalenergyofthe
baseball.SubstitutingHintoHamilton’sequations,onefinds
dx
dt=
∂
∂p[
p^2
2 m+V(x)]
=p
m
dp
dt= −
∂
∂x[
p^2
2 m+V(x)]
=−dV
dx(1.34)
whichissimplythefirst-orderformofNewton’sLaw(1.2).