Geodesics and their initial-value problem

Try

A great circle on a sphere curves in R³, yet its acceleration points entirely normal to the sphere. Its tangent projection is zero. This is the intrinsic notion of straight motion: a geodesic satisfies D_tγ′=0.

The parameter matters. An affinely parametrized geodesic has constant speed. Traversing the same image with an arbitrary nonlinear clock introduces tangential acceleration and need not satisfy that equation, even though the unparametrized path remains a geodesic image.

Predict

Predict whether every geodesic is globally shortest. A great-circle arc longer than half a circumference is beaten by the complementary arc. The equation is local and differential; global minimizing behavior needs additional arguments.

Predict whether one initial point and one tangent velocity determine a geodesic. The coordinate equation is a smooth second-order ODE, so local existence and uniqueness apply.

Experiment

Compare geodesic equations in two metrics.

Metric	Coordinate equations
dx²+dy²	x″=0, y″=0
dr²+r²dθ²	r″−rθ′²=0; θ″+2r′θ′/r=0

The second row describes the same flat plane away from the polar origin. A Euclidean straight line can have nonzero r″ because the coordinates themselves bend. Verify that r²θ′ is constant by differentiating it and using the angular equation.

Understand

Apply the covariant derivative formula to V=γ′, whose components are u′k. This gives u″k+Γ^k_(ij)u′i u′j=0. Smooth connection coefficients yield a local solution for specified position and velocity. Coordinate changes preserve the geometric equation even though the individual Christoffel coefficients change.

Metric compatibility proves constant speed: d||γ′||²/dt=2⟨D_tγ′,γ′⟩=0. Under a new clock t=φ(s), covariant acceleration becomes φ″γ′+(φ′)²D_tγ′. Thus only affine parameter changes automatically preserve the affine geodesic equation for a nonconstant geodesic.

For a surface of revolution with metric ds²+r(s)²dθ², the angular equation implies d(r²θ′)/dt=0. This is Clairaut’s conserved quantity. For unit speed it can also be written r sin β constant, where β is the angle with the meridian direction. The meridian component is s′ and the parallel component is rθ′, explaining the angle convention.

Apply

A constant-speed great circle is a sphere geodesic because its ambient acceleration is radial and hence normal. On a cylinder, unrolling converts geodesics locally to straight lines: vertical generators, horizontal circles and helices arise from different line slopes. This classification concerns the local equation, not global minimizing lengths after wrapping.

Repair “A geodesic has zero acceleration.” It has zero covariant acceleration. Its ambient acceleration can be nonzero and encodes the surface’s normal curvature.

For optional depth, a stationary-radius parallel on a surface of revolution can be geodesic; the meridian equation demands r′(s)=0 when θ′≠0. This gives the equator on a sphere. The poles need separate charts, so no conclusion should be drawn by dividing by r=0.

Continue

Write the metric, initial velocity and parameter convention before integrating geodesic equations. Continue to energy variation to derive the same equation from a stationary functional.