Read on Twitter
One of my favorite insights from differential geometry is that often it's more useful to think about the evolution of a system as an entire trajectory instead of the motion of individual points. This perspective not only builds conceptual intuition but also guides algorithms! https://twitter.com/betanalpha/status/1336406678485540868
For example most introductions to Hamiltonian Monte Carlo talk about using Hamiltonian dynamics to move an initial point to a final point that is then accepted or rejected, but you can also consider the entire trajectory of intermediate points.
Generating a numerical trajectory and then sampling a point _along_ that trajectory in a way that compensates for numerical error is a much more general implementation of Hamiltonian Monte Carlo and also a much more performant one! This is for example how @mcmc_stan works.
Note that there's no notion of how fast you move along the trajectory, and hence no dependence on "speed' in the resulting algorithm. By working with the trajectory directly we not only generalize but also avoid irrelevant degrees of freedom.
Thinking about trajectories also motivates how to work with _time-dependent_ dynamics. By lifting to a space where time is a coordinate the evolution once again defines an invariant curve which can then be manipulated and numerically approximated using the usual tools.
Technically this is also how we work with higher-order ordinary differential equations -- adding variables lifts the system into a higher-dimensional one where the differential equation reduces to first-order dynamics that...define curves!
For example the solutions to the second-order differential equation d^2 x/dt^2 = f(x, t) are given by solutions to dx/dt = g(x, t) and dv/dt = d/dt(dx/dt) = h(x, t) which define curves in (x, v) space. This is why the solutions are always plotted in "phase space".
More generally instead of interpret constraints as modifications to motion one can interpret them as defining surfaces in a given space. These surfaces are much easier to analyze and motivate much more robust constraint-preserving algorithms.
