#+begin_quote Philosophy is written in that great book which ever lies before our eyes---I mean the Universe---but we cannot understand it if we do not learn the language and grasp the symbols in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles, and other geometrical figures without whose help it is impossible to comprehend a single word of it, without which one wanders in vain through a dark labyrinth.
Galileo Galilei [8] #+end_quote
Differential geometry is a mathematical language that can be used to express physical concepts. In this introduction we show a typical use of this language. Do not panic! At this point we do not expect you to understand the details of what we are showing. All will be explained as needed in the text. The purpose is to get the flavor of this material.
At the North Pole inscribe a line in the ice perpendicular to the Greenwich Meridian. Hold a stick parallel to that line and walk down the Greenwich Meridian keeping the stick parallel to itself as you walk. (The phrase "parallel to itself" is a way of saying that as you walk you keep its orientation unchanged. The stick will be aligned East-West, perpendicular to your direction of travel.) When you get to the Equator the stick will be parallel to the Equator. Turn East, and walk along the Equator, keeping the stick parallel to the Equator. Continue walking until you get to the 90◦E meridian. When you reach the 90°E meridian turn North and walk back to the North Pole keeping the stick parallel to itself. Note that the stick is perpendicular to your direction of travel. When you get to the Pole note that the stick is perpendicular to the line you inscribed in the ice. But you started with that stick parallel to that line and you kept the stick pointing in the same direction on the Earth throughout your walk --- how did it change orientation?
The answer is that you walked a closed loop on a curved surface. As seen in three dimensions the stick was actually turning as you walked along the Equator, because you always kept the stick parallel to the curving surface of the Earth. But as a denizen of a 2-dimensional surface, it seemed to you that you kept the stick parallel to itself as you walked, even when making a turn. Even if you had no idea that the surface of the Earth was embedded in a 3-dimensional space you could use this experiment to conclude that the Earth was not flat. This is a small example of intrinsic geometry. It shows that the idea of parallel transport is not simple. For a general surface it is necessary to explicitly define what we mean by parallel.
If you walked a smaller loop, the angle between the starting orientation and the ending orientation of the stick would be smaller. For small loops it would be proportional to the area of the loop you walked. This constant of proportionality is a measure of the curvature. The result does not depend on how fast you walked, so this is not a dynamical phenomenon.
Denizens of the surface may play ball games. The balls are constrained to the surface; otherwise they are free particles. The paths of the balls are governed by dynamical laws. This motion is a solution of the Euler-Lagrange equations[fn:1] for the free-particle Lagrangian with coordinates that incorporate the constraint of living in the surface. There are coefficients of terms in the EulerLagrange equations that arise naturally in the description of the behavior of the stick when walking loops on the surface, connecting the static shape of the surface with the dynamical behavior of the balls. It turns out that the dynamical evolution of the balls may be viewed as parallel transport of the ball's velocity vector in the direction of the velocity vector. This motion by parallel transport of the velocity is called /geodesic motion/.
So there are deep connections between the dynamics of particles and the geometry of the space that the particles move in. If we understand this connection we can learn about dynamics by studying geometry and we can learn about geometry by studying dynamics. We enter dynamics with a Lagrangian and the associated Lagrange equations. Although this formulation exposes many important features of the system, such as how symmetries relate to conserved quantities, the geometry is not apparent. But when we express the Lagrangian and the Lagrange equations in differential geometry language, geometric properties become apparent. In the case of systems with no potential energy the Euler-Lagrange equations are equivalent to the geodesic equations on the configuration manifold. In fact, the coefficients of terms in the Lagrange equations are Christoffel coefficients, which define parallel transport on the manifold. Let's look into this a bit.
We write the Lagrange equations in functional notation[fn:2] as follows:
$$ D\left(\partial_{2} L \circ \Gamma[q]\right) - \partial_{1} L \circ \Gamma[q]=0. $$
In SICM [19], Section 1.6.3, we showed that a Lagrangian describing the free motion of a particle subject to a coordinate-dependent constraint can be obtained by composing a free-particle Lagrangian with a function that describes how dynamical states transform given the coordinate transformation that describes the constraints.
A Lagrangian for a free particle of mass m and velocity v is just its kinetic energy, \(mv^2/2\). The procedure =Lfree= implements the free Lagrangian:[fn:3]
#+begin_src clojure
(defn Lfree [mass]
(fn [[_ _ v]]
(* 1/2 mass (square v))))
#+end_src