Hamilton's principle/Applications/Geodesics

A geodesic is a stationary curve of arc length on a surface or in a curved space. Geodesics are the straight lines of curved geometry, and they are the setting in which this volume's central distinction is easiest to see: every great-circle arc on a sphere is a geodesic, and only the short ones are shortest.

The functional

Geodesics from a pole refocusing at the antipode.

On a surface with metric

$d{s}^2=\sum _{i,k}{g}_{ik}(q)d{q}_{i}d{q}_k\mathrm{(}\mathrm{1}\mathrm{)}$

the length of a curve $q(\lambda )$ is

$J[q]={\int }_{{\lambda }_0}^{{\lambda }_1}\sqrt{\sum _{i,k}{g}_{ik}{\dot{q}}_i{\dot{q}}_k}d\lambda \mathrm{(}\mathrm{2}\mathrm{)}$

with the dot denoting $d/d\lambda$.

The integrand is homogeneous of degree one in the velocities. That has a consequence worth naming before any calculation: (2) is unchanged under reparametrisation $\lambda \to p(\lambda )$. By Noether's second theorem on Noether's second theorem, invariance under a group of arbitrary functions forces an identity among the Lagrange expressions — here exactly her Weierstrass parametric example, $\sum {\psi }_i{\dot{q}}_i=0$. The Euler-Lagrange equations of (2) are therefore not independent, and the Hessian in the velocities is singular.

The standard remedy is to use the energy functional instead:

$E[q]=\frac{1}{2}\int \sum _{i,k}{g}_{ik}{\dot{q}}_i{\dot{q}}_{k}d\lambda \mathrm{(}\mathrm{3}\mathrm{)}$

whose integrand is quadratic and non-degenerate. Extremals of (3) are extremals of (2) traversed at constant speed, so nothing is lost, and the parametrisation degeneracy is fixed rather than carried. This is the practical reason (3) is used throughout differential geometry.

The geodesic equation

Geodesic behaviour on flat, spherical and saddle surfaces.

Apply the Euler-Lagrange equation of Several dependent variables to (3) with $L=\frac{1}{2}{g}_{ik}{\dot{q}}_i{\dot{q}}_k$. Then

$\frac{\partial L}{\partial {\dot{q}}_j}=g_{jk}{\dot{q}}_k,\frac{\partial L}{\partial q_j}=\frac{1}{2}{\partial }_j{g}_{ik}{\dot{q}}_i{\dot{q}}_k\mathrm{(}\mathrm{4}\mathrm{)}$

summation implied. The equation is

$\frac{d}{d\lambda }\left(g_{jk}{\dot{q}}_k\right)-\frac{1}{2}{\partial }_j{g}_{ik}{\dot{q}}_i{\dot{q}}_k=0\mathrm{(}\mathrm{5}\mathrm{)}$

Expanding the total derivative gives $g_{jk}{\ddot{q}}_k+{\partial }_i{g}_{jk}{\dot{q}}_i{\dot{q}}_k$. Symmetrising the middle term over $i$ and $k$, which is legitimate since it multiplies the symmetric ${\dot{q}}_i{\dot{q}}_k$, and contracting with the inverse metric $g^{mj}$,

${\ddot{q}}_m+{\Gamma }_{ik}^m{\dot{q}}_i{\dot{q}}_k=0\mathrm{(}\mathrm{6}\mathrm{)}$

with

${\Gamma }_{ik}^m=\frac{1}{2}{g}^{mj}({\partial }_i{g}_{jk}+{\partial }_k{g}_{ji}-{\partial }_j{g}_{ik})\mathrm{(}\mathrm{7}\mathrm{)}$

Equation (6) is the geodesic equation and (7) are the Christoffel symbols. They are not an extra structure imposed on the metric; they fall out of the variational problem, and this derivation is the shortest honest route to them.

The sphere

Take the unit sphere, $d{s}^2=d{\theta }^2+{\sin }^2\theta d{\phi }^2$, so $L=\frac{1}{2}({\dot{\theta }}^2+{\sin }^2\theta {\dot{\phi }}^2)$.

$\phi$ is cyclic, so by (7) of Several dependent variables its conjugate momentum is conserved:

$p_{\phi }={\sin }^2\theta \dot{\phi }=h=\text{const}\mathrm{(}\mathrm{8}\mathrm{)}$

$L$ carries no explicit $\lambda$, so the Beltrami identity gives

${\dot{\theta }}^2+{\sin }^2\theta {\dot{\phi }}^2=1\mathrm{(}\mathrm{9}\mathrm{)}$

taking the constant as 1, which fixes $\lambda$ as arc length. Eliminating $\dot{\phi }$ between (8) and (9),

${\dot{\theta }}^2=1-\frac{h^2}{{\sin }^2\theta }\mathrm{(}\mathrm{10}\mathrm{)}$

and dividing by (8) to get the orbit equation,

$\frac{d\phi }{d\theta }=\frac{h}{{\sin }^2\theta \sqrt{1-h^2/{\sin }^2\theta }}=\frac{h}{\sin \theta \sqrt{{\sin }^2\theta -h^2}}\mathrm{(}\mathrm{11}\mathrm{)}$

The substitution $u=\cot \theta$ gives $du=-d\theta /{\sin }^2\theta$ and turns (11) into

$d\phi =\frac{-hdu}{\sqrt{1-h^2-h^2{u}^2}}\mathrm{(}\mathrm{12}\mathrm{)}$

which integrates to an arccosine. Writing ${\alpha }^2=(1-h^2)/h^2$,

$\cot \theta =\alpha \cos (\phi -{\phi }_0)\mathrm{(}\mathrm{13}\mathrm{)}$

Multiplying (13) by $\sin \theta$ and expanding the cosine gives a relation linear in $\sin \theta \cos \phi$, $\sin \theta \sin \phi$ and $\cos \theta$ — that is, linear in the Cartesian coordinates. So the curve lies in a plane through the centre, and the geodesics of the sphere are exactly the great circles.

A shorter argument exists by symmetry: any point and direction lie in a unique plane through the centre, and reflection in that plane fixes both while mapping geodesics to geodesics. The calculation above is given because it produces (13) in a usable form and shows the machinery working.

Stationary is not shortest

Every great-circle arc satisfies (6). The long way round from London to Tokyo is a geodesic. It is not the shortest path.

The Jacobi analysis on the Jacobi condition and conjugate points gives the exact boundary. The accessory equation on the unit sphere is $u^{″}+u=0$, so the first conjugate point of any starting point is its antipode, at arc length $\pi$. Therefore:

• arcs shorter than $\pi$ are the unique shortest paths;
• arcs of exactly $\pi$ are stationary and non-unique — every meridian between the poles has the same length;
• arcs longer than $\pi$ are stationary and not shortest.

Nothing local distinguishes the three cases. They satisfy the same equation, at every point, with the same Legendre coefficient. Only the total arc length differs. This is the volume's counterexample in its most concrete form, and it is why the subject is called stationary action rather than least action.

Other surfaces

The plane and the cylinder have zero curvature. The accessory equation is $u^{″}=0$, which has no second zero, so there are no conjugate points and geodesics minimise without limit. On the cylinder they are helices, and the absence of conjugate points does not prevent several geodesics joining two points — the helix may wind any number of times. Uniqueness and minimality are different claims.

Negatively curved surfaces give $u^{″}-|K|u=0$, whose solutions grow, so again no conjugate points. Geodesics diverge and each is the unique minimiser.

Positively curved surfaces give the sphere's behaviour: geodesics reconverge and minimality fails past the refocusing point. Conjugate points are a symptom of positive curvature, and that is the geometric content of the Jacobi condition.

Where this goes

The Einstein-Hilbert action makes free fall a geodesic of spacetime, so (6) becomes the equation of motion of a body under gravity, and (7) is derived from the same variational principle. Conjugate points then acquire physical meaning as the refocusing of light by a gravitational lens.

Provenance: every equation on this page is derived here. No primary source is cited and none is claimed.

Next

Fermat's principle treats the optical analogue, where the same machinery gives Snell's law.