That’s just a parabola, the potential for a simple harmonic oscillator.

Parallel transport is the process by which we can start with a vector defined at some point and move it down a specified path, doing our best at each step to keep it parallel to its previous orientation.

the momentum is pushed around by changes in the potential energy, and the position is pushed around by changes in the kinetic energy.

In particle physics and quantum field theory we have internal symmetries that “rotate” different parts of a quantum field into one another.

Today, most physicists would come down on the side of Newton, treating space as a thing in itself, for a couple of reasons. First, the space in between objects is not empty; it is filled with fields of various sorts. Second, space (as part of spacetime) has a life of its own; Einstein showed that the geometry of space responds to energy and can

mechanics; in other words, phase space is the set of all possible states the system could be in.

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Hamiltonian mechanics. Its central idea is to elevate “momentum” to a concept with an existence of its own, independent of “velocity.”

There is one dimension of time, and correspondingly a single conserved quantity: energy. But there are three dimensions of space, and we can translate separately in any one of those three directions.