In physics the primary object that we are often want to understand is the Hamiltonian of a system, denoted by $H$. Let $M$ be the phase space of our system and so $H:M\to ℝ:(q,p)\mapsto H(q,p)$. $H$ gives rise to dynamics through Hamilton’s Equations

$$\frac{dq}{dt}=\frac{\partial H}{\partial p},\frac{dp}{dt}=-\frac{\partial H}{\partial p}.$$ (1)

While these equations can be thought of as just a system of PDES to be attacked, it is useful to think of them geometrically. By this I mean we think of the time derivatives appearing in (1) as components of a velocity. Let $m=(q,p)\in M$ and embed a parametric curve $\gamma :[-1,1]\to M:t\mapsto \gamma (t)$ in $M$. Trivialise a neighbourhood of $\gamma (0)$ so we have $\gamma (t)=(q(t),p(t))$, then $\dot{\gamma }$ is exactly the LHS components of Hamilton’s equations. Such curves whose velocities satisfy (1) are called integral curves of the equations. The fundamental theorem of vector fields guarantee the local existence of such curves.

Now that we are thinking about the LHS of Hamilton’s equations vectorially we might as well do the same with the RHS. By this I mean we should write (1) in a new way

$$(\dot{q},\dot{p})=(\frac{\partial H}{\partial p},-\frac{\partial H}{\partial p})=X_H(q,p)$$ (2)

where we have introduced a new object $X_H$, which is called the Hamiltonian vector field of $H$. The action of $X_H$ on $M$ generates a vector field whose integral curves are solutions to Hamilton’s equations. Those who like to think of equations of motion in terms of the Poisson bracket $\{H,(q,p)\}$ can think of $X_H$ One of the ways you can write the $X_H$ is by ”currying” the Poisson bracket to say $X_H=\{H,\mathrm{\_}\}$.

There are further geometric observations we can make about $X_H$. It is commonly said that canonical transformations (of which time evolution under Hamilton’s equations are an important example) ”preserve the area of phase space” - with $X_H$ in hand we can look at exactly what that means.

Let us take the area element of a 2$n$-dimensional phase space $M$, which we will call $\omega$, such an object is can always be written locally as

$$\omega =dp\wedge dq$$ (3)

To find out how the flows of $X_H$ affect $\omega$ we should take the Lie derivative, which, by the help of the Cartan homotopy formula is not too difficult, explicitly

(4)

In the Lie sense then we can say that Hamilton’s equations preserves the area element of phase space.

From the first line of (4) we can make the observation that, since $\omega$ is closed $X_H⌟\omega$ itself (being nonzero) must also be closed. By the Poincaré lemma we know that all closed differential forms are locally exact, i.e. locally we can take $X_H⌟\omega =df$. Recalling that

$$X_H⌟\omega =\frac{\partial H}{\partial p}dp+\frac{\partial H}{\partial p}dq$$ (5)

we see that the function $f$ introduced is the Hamiltonian (up to some additive constant killed by the exterior derivative). From this point we arrive at the formula usually used to define Hamiltonian vector fields

$$X_H⌟\omega +dH=0,$$ (6)

where we note that different authors choose different sign conventions for $dH$.

Hamiltonian vector are related to another class of important vector fields, the symplectic vector fields. While a Hamiltonian vector field we saw happens to preserve $\omega$, symplectic vector fields are by definition the vector fields $L_X\omega =d(X⌟\omega )=0$, note that we are not working in any particular local trivialisation here, we have made a global -purely geometric- statement. The existence of an exact form $dH$ defining a Hamiltonian vector field is what differentiates these two seemingly similar objects.