Is this the pullback of a Hamiltonian flow?

Question

In this reference just in the beginning the author gives the theorem (Theorem 1) of the conservation of a Hamiltonian flow $\phi_t$. According to it this means that $$ \frac{d}{dt}\phi_t^* H = 0$$

I am confused on what the $\phi_t^*$ is and how it acts on the Hamiltonian function $H$ (let's assume it is a potential in classical mechanics). Additionally I would like to ask what is a "level set" of $H$.

Answer 1

The Hamiltonian vector field defines a flow on a symplectic manifold, called a symplectomorphism. A diffeomorphism $f: (M,\omega) \rightarrow (M',\omega')$ between two symplectic manifolds defines a symplectomorphism where $f^*\omega'=\omega$. The $f^*$ is the pullback of $f$.

Assuming that the Hamiltonian function H is not constant on any open set we simply need to plot the level curves giving the solutions of the system. These solutions will live on these level sets.

Answer 2

If $\phi:M\rightarrow N$ is a differentiable map, and $f$ is a real function defined on $N$, then the pullback $\phi^*f$ is defined as $$ \phi^*f=f\circ\phi. $$

Since $\phi_t$ is the Hamiltonian flow, what $\phi^*_tH$ means is $$ H\circ\phi_t $$.

If we insert the dependence on position, then $$ H\circ\phi_t(p)=H(\phi_t(p)), $$but what $\phi_t(p)$ is, that we take that integral curve of the Hamiltonian vector field $X_H$ that starts at $p$, and move on that curve to a parameter value of $t$. So, if we take $\phi_t(p)$ as a function of $t$, then it IS the integral curve that starts at $p$.

Now, according to these all, $$\left.\frac{d}{dt}\phi_t^*H\right|_{t=0}=\mathcal{L}_{X_H}H=\left.\frac{d}{dt}(H\circ\phi_t)\right|_{t=0}=X_H(H), $$ where the last equality comes from the fact, that $H\circ\phi_t$ is basically the Hamiltonian composed with the integral curves of $X_H$, and this expression's $t$-derivative is the directional derivative along these curves, which is by definition, the vector field's action on the function.

Now, by the definition of the differential of functions, $$X_H(H)=dH(X_H), $$ but $X_H$ is defined as $\omega(\cdot,X_H)=dH$, thus $$ dH(X_H)=\omega(X_H,X_H)=0, $$ since $\omega$ (the symplectic form) is skew-symmetric.

What this all means, that since the flow $\phi_t$ describes the dynamics of the hamiltonian system, the system's dynamical evolution leaves the Hamiltonian invariant: $\mathcal{L}_{X_H}H=0$, thus, as we move along the dynamical system's trajectories, $H$ doesn't change. $H$ in physics is usually the energy, so this is a statement about conservation of energy.

A level set is a set of points along which a function is constant. So if $E$ is a constant, then the set of all $p$s, for which $H(p)=E$ is a level set of $H$. Since $H$ is constant along the trajectories of the dynamical system, the dynamical system moves on level sets of $H$.

Physical meaning is that for a system like this, initial value data determines the total energy of the system, that doesn't change along the motion. The Hamiltonian is the energy function, so this means the initial values choose a submanifold of the phase space, and the dynamics associated with those initial values will not leave that submanifold.