Appendix 3: Christoffel Connections as Fictitious Forces
79
where f (ρ) is a smooth function of ρ. Obtain the geodesic equations and show
that rays ϕ = const. are geodesics. This should also be intuitively obvious.
5.5 Go through the derivation of the geodesic equation for a curve with negative
line element as mentioned briefly in the text, leading to (5.28).
5.6 All humans to date have lived on or near the surface of the spherical earth, so it is
a good idea to study connections and geodesics for a spherical surface. Indeed
the word geodesic derives from “dividing the earth” in Greek. Write down the
metric in terms of spherical coordinates with constant radius From the metric
write down the function T defined in (5.32) to be used as a Lagrangian. From
this T write down the Euler-Lagrange equations which describe a geodesic on
the surface. For a sphere these are also called great circles.
5.7 Continue working on the sphere. From the Euler-Lagrange equations in Exercise 5.6 show that the equator and longitude lines are geodesics but latitude
lines are not. This should also be obvious.
5.8 How many affine connections are there on the spherical surface? Compare the
Euler-Lagrange equations with the geodesic equations in standard form (5.28)
and identify the affine connections using the procedure we used in Example
5.3.
5.9 Use classical Lagrangian mechanics to study the motion of a particle in a
2-dimensional plane, with a central potential energy field, using polar coordinates; that is, write down the Euler-Lagrange equations. For the case of zero
force note that one obtains the equation of a simple straight line. Is the physical
meaning clear?
5.10 The Gauss-Bonnet Theorem relates the angle of rotation of a vector parallel
displaced along geodesics on a closed curve to the area enclosed by the curve.
Find a reference on this theorem and verify it for the sphere shown in Fig. 5.4
in which the closed curve is a triangle with all right angles. This theorem
provides one way to define curvature.
79
where f (ρ) is a smooth function of ρ. Obtain the geodesic equations and show
that rays ϕ = const. are geodesics. This should also be intuitively obvious.
5.5 Go through the derivation of the geodesic equation for a curve with negative
line element as mentioned briefly in the text, leading to (5.28).
5.6 All humans to date have lived on or near the surface of the spherical earth, so it is
a good idea to study connections and geodesics for a spherical surface. Indeed
the word geodesic derives from “dividing the earth” in Greek. Write down the
metric in terms of spherical coordinates with constant radius From the metric
write down the function T defined in (5.32) to be used as a Lagrangian. From
this T write down the Euler-Lagrange equations which describe a geodesic on
the surface. For a sphere these are also called great circles.
5.7 Continue working on the sphere. From the Euler-Lagrange equations in Exercise 5.6 show that the equator and longitude lines are geodesics but latitude
lines are not. This should also be obvious.
5.8 How many affine connections are there on the spherical surface? Compare the
Euler-Lagrange equations with the geodesic equations in standard form (5.28)
and identify the affine connections using the procedure we used in Example
5.3.
5.9 Use classical Lagrangian mechanics to study the motion of a particle in a
2-dimensional plane, with a central potential energy field, using polar coordinates; that is, write down the Euler-Lagrange equations. For the case of zero
force note that one obtains the equation of a simple straight line. Is the physical
meaning clear?
5.10 The Gauss-Bonnet Theorem relates the angle of rotation of a vector parallel
displaced along geodesics on a closed curve to the area enclosed by the curve.
Find a reference on this theorem and verify it for the sphere shown in Fig. 5.4
in which the closed curve is a triangle with all right angles. This theorem
provides one way to define curvature.