### What is energy? What is physics?

In your high-school physics classes, you may have studied the dynamics of various physical systems, but you may have also heard that physics is the "general science", that is universally applicable. One may wonder if physics can be formulated as some general framework that can handle any type of system without more specific assumptions about such system, and what interesting conclusions can be drawn about physics in such a general setting.

Fundamentally, we seek to make predictions about some observables. You may imagine that all the observable information about a system is given by some abstract "phase space" of "state vectors" such that each observable is some function of the state vector. For example if the phase space is parameterized by position and momentum, then the position and momentum observables would be projection functions and various other observables can be written as compositions thereof of functions with these projections e.g. $E=\frac{p^2}{2m}+U(x,p)$.

The question of why position and momentum are sufficient to specify our phase space in so many real-life situations is a rather advanced one -- here's a paper that explains that this general "first-derivative sufficiency" in physics arises as a special case of something to do with Fisher information [1] -- but I don't understand Fisher information.

The Lie theory of dynamics

One may say that we are particularly interested in the time-evolution of such observables, $dA/dt$. If we can find a differential equation for this, then we can find the value of $A$ given some initial condition. But more generally we may not have an initial condition as such and maybe the problem we're trying to solve isn't even a differential equation as such but we just want to speak more abstractly about the dynamics of the system.

This should all be screaming "Lie groups" to you.

One may imagine that time evolution is the flow under a certain vector field.

More precisely, suppose the state of a system is given by some $\psi=(x, p)$, which represents the position and momentum of a particle at time $t$. Then the "position at time $t$" is an observable given by the function $X(\psi)=x$, the projection onto the $x$ co-ordinate -- but we may also think about the observable "position at time $t+\Delta t$", which is some function $\Phi^{\Delta t}X$ that projects onto a different co-ordinate system.

This co-ordinate transformation on the phase space is a type of canonical transformation -- we will soon define this generally, but it is important to realize that motion/time-evolution is a type of canonical transformation.

You may imagine that such canonical transformations form a Lie group, and as is standard the Lie algebra to this Lie group consists of vector fields (flows) on the phase space.

Noether and symplectic geometry

The question, then, is what sort of transformations count as "canonical transformations", or equivalently, what sort of vector fields are we interested in. What kind of "co-ordinate transformations" are acceptable?

We want to be as general as possible, so we do not wish to restrict what sort of paths are predicted on the phase space -- rather, we want to restrict the relationships between the predicted paths, i.e. how a distribution evolves under time evolution. We won't go into details of why this is true (because I do not fully understand it yet -- apparently it is called Liouville's theorem and has to do with "conservation of information", which has to do with probabilities inferred from some symmetries, see a priori probability), but we expect the vector fields to be solenoidal, i.e. have zero divergence. Analogous to how irrotational vector fields are precisely those that are gradients of functions, solenoidal vector fields are precisely those that are the symplectic gradients of functions (see reddit for a proof):

$$\nabla\cdot \vec{v}=0\leftrightarrow \exists g, \vec{v}=\frac{\partial g}{\partial y}\hat{x}-\frac{\partial g}{\partial x}\hat{y}$$

This "symplectic gradient" always points along the contours of $g$ -- thus, these flows conserve the quantity $g$. More generally, the change in some observable $f$ over the flow that is the symplectic gradient of $g$ is given by $\vec{v}\cdot\nabla g$, and is called the Poisson bracket of the two observables:

$$\{f,g\}=\frac{\partial g}{\partial y}\frac{\partial f}{\partial x}-\frac{\partial g}{\partial x}\frac{\partial f}{\partial y}$$

It should be intuitively clear that this Poisson bracket corresponds to the Lie bracket of their symplectic gradients -- if the vector fields commute, they must be constant under the flow of the other, etc.

The Hamiltonian and conservation of energy

The evolution of a classical Newtonian state is given by

$$\dot{x}=p/m$$

$$\dot{p}=F$$

We wish to find the observable $H$ such that this evolution is represented by the symplectic gradient of $H$, i.e. so that:

$$p/m=\{x, H\} = \frac{\partial H}{\partial p}\frac{\partial x}{\partial x}-\frac{\partial H}{\partial x}\frac{\partial x}{\partial p}=\frac{\partial H}{\partial p}$$

$$F = \{p, H\}=-\frac{\partial H}{\partial x}$$

Integrating, the quantity we want is

$$H=\frac{1}{2m}p^2-\int F dx$$

This quantity is called the "energy" of the system -- often we call $p^2/2m$ the "kinetic energy" and $U=-\int F dx$ the "potential energy". In particular, it is immediately clear that energy is conserved. In general, the physics of a system can be completely specified by some "Hamiltonian" $H$, this rather generalized definition of energy, whose symplectic gradient represents time translations.