Inculcation: geodesic flows

Main article: inculcation-odes

What’s a geodesic flow? §

interpreting a lot of systems as geodesic flows and vice versa §

manifold	metric	system corresponding to geodesics
configuration space $Q$ be any manifold	$g_{ij}$ coming from a (non deg) quadratic Lagrangian $g_{ij}{\dot{q}}^i{\dot{q}}^j$ (with no potential term)	Lagrangian dynamics for $L=g_{ij}{\dot{q}}^i{\dot{q}}^j$
a surface in ${\mathbb{R}}^3$	induced by the metric in ${\mathbb{R}}^3$	force free (Newtonian) motion of the particle constrained on the surface, geodesic equation is literally ”$\underset{0}{\underbrace{\frac{F}{m}}}=a$” here
$SO(3)$ (configuration space of rigid body with one fixed point, say center of mass)	given by the torque-free Lagrangian of a rigid body	torque-free motion of the rigid body AKA Euler top
space $Q$ which can be ${\mathbb{R}}^n$ or any manifold	$n(q{)}^2{\delta }_{ij}$ where $n:Q\to (0,\infty )$ is refractive index	light rays in optics
configuration space $Q$ with metric $g$	$h_{ij}=\frac{2}{m}(E-V(q))g_{ij}$	particle in potential $V$ in space $(Q,g)$ with energy $E$
GR spacetime $M$ which can be any manifold	a metric $g$ with Lorenzian signature	(relativistic) free particle in the spacetime or light trajectories if it’s a null geodesic
spacetime $(M,g)$ cross $U(1)$ that is $M\times U(1)$	$h=\left[\begin{array}{cc}{h}_{ij}& h_{\theta i}\\ h_{i\theta }& h_{\theta \theta }\end{array}\right]=\left[\begin{array}{cc}{g}_{ij}+A_i{A}_j& -A_i\\ -A_i& 1\end{array}\right]$	Kaluz-Klein theory of particle of charge $q$ on a GR spacetime with electromagnetic field

^[Classical Mechanics]

More examples where the configuration space is specifically a group can be found in

Table of configuration spaces that are groups a metric and geodesic flows on them

Lecture1 slides of Geometric Fluid Dynamics, Fall 2021

dynamics that are isomorphic to geodesic flows §

• Kepler problem