Exercise List #2

Classical Field Theory

October 23, 2009

1. Derive the Euler–Lagrange equations for the following Lagrangians:

• L = g

µν

˙

x

µ

˙

x

ν

,

• L =

pg

µν

˙

x

µ

˙

x

ν

,

• L =

1

2e

g

µν

˙

x

µ

˙

x

ν

−

1
2

em

2

. e ≡ e(λ) is a sort of ‘auxiliary coordinate’, so it also

depends on the affine parameter and yields an additional ELE.

Each of those may be solved in two cases: either the metric g is constant, or

∂g

µν

∂x

κ

≡

∂

κ

g

µν

≡ g

µν,κ

6= 0. For instructive purposes, in the last example also consider a case

of nontrivial g(e) dependence.