1.2. EULER-LAGRANGEANDHAMILTON’SEQUATIONS 13
wheretheLagrangianfunctionforthebaseballis
L[x(t),x ̇(t)]=1
2
mx ̇^2 (t)−V[x(t)] (1.20)andtheEuler-Lagrangeequation,inthecontinuumlimit,becomes
∂L
∂x(t)−
d
dt∂L
∂x ̇(t)= 0 (1.21)
FortheLagrangianofthebaseball,eq. (1.20),therelevantpartialderivativesare
∂L
∂x(t)= −
dV[x(t)]
dx(t)
∂L
∂x ̇(t)= mx ̇(t) (1.22)which,whensubstitutedintoeq. (1.21)give
m∂^2 x
∂t^2+
dV
dx= 0 (1.23)
ThisissimplyNewton’slawF=ma,inthesecond-orderformofeq.(1.1).
We now wantto rewrite the Euler-Lagrange equationin first-orderform. Of
course, we already know the answer, whichis eq. (1.2), but letus ”forget” this
answerforamoment,inordertointroduceaverygeneralmethod. Thereasonthe
Euler-Lagrangeequationissecond-orderinthetimederivativesisthat∂L/∂x ̇isfirst-
orderinthetimederivative. Soletusdefinethemomentumcorrespondingtothe
coordinatextobe
p≡∂L
∂x ̇(1.24)
Thisgivespas afunctionof xandx ̇,but, alternatively, wecan solve forx ̇ as a
functionofxandp,i.e.
x ̇=x ̇(x,p) (1.25)
Next,weintroducetheHamiltonianfunction
H[p,x]=px ̇(x,p)−L[x,x ̇(x,p)] (1.26)Sincex ̇isafunctionofxandp,Hisalsoafunctionofxandp.
ThereasonforintroducingtheHamiltonianisthatits firstderivativeswithre-
specttoxandphavearemarkableproperty;namely,onatrajectorysatisfyingthe
Euler-Lagrangeequations,thexandpderivativesofHareproportionaltothetime-
derivativesofpandx.Toseethis,firstdifferentiatetheHamiltonianwithrespectto
p,
∂H
∂p= x ̇+p∂x ̇(x,p)
∂p−
∂L
∂x ̇∂x ̇(p,x)
∂p
= x ̇ (1.27)