QuantumPhysics.dvi

The variation is then computed using standard chain rule,

δS[q] = S[q+δq]−S[q]

=

∫t 2

t 1

dt

(

L(q+δq,q ̇+δq ̇;t)−L(q,q ̇;t)

)

=

∫t 2

t 1

dt

∑

i

(∂L

∂qi

δqi+

∂L

∂q ̇i

δq ̇i

)

(6.5)

Using now the fact that

δq ̇i=

d

dt

δqi (6.6)

integrating by parts and using the vanishing boundary conditions, we are left with

δS[q] =

∫t 2

t 1

dt

∑

i

(

−

d

dt

∂L

∂q ̇i

+

∂L

∂qi

)

δqi (6.7)

Finally, requiring that a path be stationary amounts to requiring that a small variation

around the path leaves the action unchangedδS[q] = 0, which clearly is equivalent to the

Euler-Lagrange equations.

In the simplest examples, the Lagrangian is the difference between kinetic and potential

energyV of the generalized positions,

LV=

1

2

∑N

i=1

miq ̇i^2 −V(q 1 ,···,qN) (6.8)

for which the Euler-Lagrange equations are

miq ̈i=−

∂V

∂qi

(6.9)

In a slightly more complicated example, we have the Lagrangian of a charged particle in

3 space dimensions in the presence of an electricE(r,t) and magnetic fieldB(r,t), which

derive from an electric potential Φ(r,t) and vector fieldA(r,t) as follows,

E=−∇~Φ−

∂A

∂t

B=∇×~ A (6.10)

The associated Lagrangian for a particle with electric chargee, massm, and positionris,

LA=

1

2

mr ̇^2 −eΦ(r,t) +eA(r,t)·r ̇ (6.11)

Check that the Euler-lagrange equation coincides with the Lorentzforce law

m ̈r=eE(r,t) +er ̇×B(r,t) (6.12)