Appendix 2: The Extremum Problem and the Euler-Lagrange Equations
77
It would be hard to exaggerate the utility of the Euler-Lagrange type of analysis
and the action concept in classical and quantum mechanics, classical and quantum
field theory, and essentially all of physics.
Appendix 3: Christoffel Connections as Fictitious Forces
The Christoffel connections are actually familiar objects in classical mechanics, but
they are seldom identified as such explicitly or seen from the geometrical point of
view. They give rise to the well-known fictitious forces encountered in non-cartesian
coordinate systems, rotating systems being a favorite example. To illustrate how
this works we will study the motion of a particle in a potential in 3-dimensional
space with a general coordinate system using the Lagrangian formulation of classical
mechanics. The manipulations are similar to those used in the preceding appendix
and for discussing geodesics in the text.
Let the particle have a trajectory in three dimensions, with the position is given
as a function of absolute (invariant) time by x
j
(t) in some coordinate system. Along
this trajectory the line element represents the Euclidean distance
ds
2
= g i j dx
i dx
j
.
(5.60)
Thus we may write the square of the velocity as
v
2
= g i j ˙
x
i
˙
x
j
, ˙
x
i
≡
dx
i
dt
.
(5.61)
For a particle moving in a potential field the Lagrangian is generally taken to be the
kinetic energy minus the potential energy,
L =
m
2
v
2
− V
x
k
=
m
2
g i j ˙
x
i
˙
x
j
− V
x
k
.
(5.62)
Note the similarity of this to the function T which we used in discussing geodesics.
Lagrangian mechanics is based on the postulate that the action, the integral of L, is
extremized for the correct trajectory. That is
δS = 0, S =
f
i
Ldt =
f
i
m
2
g i j ˙
x
i
˙
x
j
− V
x
k
dt.
(5.63)
Extremizing the action we are led to the Euler-Lagrange equations as in our derivation
of the geodesic equation, but now we also have a potential energy term. The EulerLagrange equations are obtained as usual, and are,
77
It would be hard to exaggerate the utility of the Euler-Lagrange type of analysis
and the action concept in classical and quantum mechanics, classical and quantum
field theory, and essentially all of physics.
Appendix 3: Christoffel Connections as Fictitious Forces
The Christoffel connections are actually familiar objects in classical mechanics, but
they are seldom identified as such explicitly or seen from the geometrical point of
view. They give rise to the well-known fictitious forces encountered in non-cartesian
coordinate systems, rotating systems being a favorite example. To illustrate how
this works we will study the motion of a particle in a potential in 3-dimensional
space with a general coordinate system using the Lagrangian formulation of classical
mechanics. The manipulations are similar to those used in the preceding appendix
and for discussing geodesics in the text.
Let the particle have a trajectory in three dimensions, with the position is given
as a function of absolute (invariant) time by x
j
(t) in some coordinate system. Along
this trajectory the line element represents the Euclidean distance
ds
2
= g i j dx
i dx
j
.
(5.60)
Thus we may write the square of the velocity as
v
2
= g i j ˙
x
i
˙
x
j
, ˙
x
i
≡
dx
i
dt
.
(5.61)
For a particle moving in a potential field the Lagrangian is generally taken to be the
kinetic energy minus the potential energy,
L =
m
2
v
2
− V
x
k
=
m
2
g i j ˙
x
i
˙
x
j
− V
x
k
.
(5.62)
Note the similarity of this to the function T which we used in discussing geodesics.
Lagrangian mechanics is based on the postulate that the action, the integral of L, is
extremized for the correct trajectory. That is
δS = 0, S =
f
i
Ldt =
f
i
m
2
g i j ˙
x
i
˙
x
j
− V
x
k
dt.
(5.63)
Extremizing the action we are led to the Euler-Lagrange equations as in our derivation
of the geodesic equation, but now we also have a potential energy term. The EulerLagrange equations are obtained as usual, and are,