Madhusudhan Raman

We have defined $ p $-forms to be multilinear maps that take as input $ p $ vectors and return numbers. We might, of course, just a feed a $ p $-form $ \alpha $ a single vector $ X $, in which we get a $ (p-1) $-form. This is called a contraction or interior product and is denoted $\iota _ {X}\alpha$. By convention, we insert $X$ into the first slot: \begin{equation} (\iota _ {X}\alpha)(X _ {2},\cdots,X _ {p}) =\alpha(X,X _ {2},\cdots,X _ {p}) \ . \label{eq:contraction-def} \end{equation} For the component form of $\alpha$, we can determine the components of this contraction to be \begin{equation} \iota _ {X}\alpha =\frac{1}{(p-1)!}X^{\mathsf{b}} \alpha _ {\mathsf{b a} _ {2}\cdots\mathsf{a} _ {p}} \,\mathrm{d}y^{\mathsf{a} _ {2}}\wedge\cdots\wedge \mathrm{d}y^{\mathsf{a} _ {p}} \ . \end{equation} The contraction obeys the graded Leibniz rule: \begin{equation} \iota _ {X}(\alpha\wedge\beta) =(\iota _ {X}\alpha)\wedge\beta +(-1)^{p}\alpha\wedge(\iota _ {X}\beta) \ , \label{eq:contraction-leibniz} \end{equation} where $\alpha$ has degree $p$.

Another operation that is useful to define is the Lie derivative, which describes the change of a geometric object along the flow generated by a specific vector field. To develop this concept, it is useful to first define the flow associated to a vector field. Let $ X $ be a smooth vector field on a manifold $ M $. In the neighbourhood of every point $ x \in M $ there exists a curve $ \gamma _ {x} : \mathbb{R} \rightarrow M$ such that \begin{equation} \frac{\text{d} }{\text{d} t} \gamma _ {x} (t) = X _ {\gamma _ {x} (t)} \quad \text{such that} \quad \gamma _ {x} (0) = x \ . \end{equation} That is, the curve is defined in such a way that the tangent to the curve $ \gamma _ {x} (t) $ is the vector field $ X $ at that point on the curve. This is called an integral curve.

The flow $ \Phi _ {t}: M \rightarrow M $ corresponding to the vector field $ X $ is defined such that $ \Phi _ {t} (x) = \gamma _ {x} (t) $, so \begin{equation} \frac{\text{d} }{\text{d} t} \bigg\vert _ {t=0} \Phi _ {t} (x) = X _ {x} \ . \end{equation} Flows are therefore maps that move points along integral curves of vector fields. They may only exist locally, although there are circumstances where flows extend globally, in which case they are said to be complete.

Let $\Phi _ {t}$ be the flow of $X$. For a differential form $\alpha$, the Lie derivative is defined by \begin{equation} \mathcal{L} _ {X}\alpha =\left.\frac{\mathrm{d}}{\mathrm{d}t}\right\vert _ {t=0} \Phi _ {t}^{\star}\alpha \ . \label{eq:lie-derivative-flow} \end{equation} This definition makes the geometric meaning of the Lie derivative clear: $\mathcal{L} _ {X}\alpha=0$ if and only if the form is invariant under the infinitesimal flow of $X$.

The Lie derivative of one vector field along another, in fact, recovers the Lie bracket introduced earlier: \begin{equation} \mathcal{L} _ {X}Y=[X,Y] \ . \end{equation} More generally, in calculations with forms, the Lie derivative is most expediently calculated by using Cartan’s magic formula: \begin{equation} \mathcal{L} _ {X} =\mathrm{d}\iota _ {X}+\iota _ {X}\mathrm{d} \ . \label{eq:cartan-identity} \end{equation} For example, when acting on a function $f$, the second term is absent and $\mathcal{L} _ {X}f=\iota _ {X}\mathrm{d}f=X[f]$. Cartan’s magic formula also implies \begin{equation} \left[\mathcal{L} _ {X},\mathrm{d}\right]=0 \ , \end{equation} because of the nilpotence of the exterior derivative, and so exterior differentiation commutes with transport along a flow.

The Lie derivative obeys the ordinary Leibniz rule \begin{equation} \mathcal{L} _ {X}(\alpha\wedge\beta) =(\mathcal{L} _ {X}\alpha)\wedge\beta +\alpha\wedge(\mathcal{L} _ {X}\beta) \ , \end{equation} and is related to contraction through \begin{equation} \left[\mathcal{L} _ {X},\iota _ {Y}\right] =\iota _ {[X,Y]} \ . \end{equation}

We now apply these results to Hamiltonian dynamics. The non-degeneracy of $\omega$ implies that, to every function $f$, we can associate a unique vector field $X _ {f}$ as: \begin{equation} \iota _ {X _ {f}}\omega=-\mathrm{d}f \ . \label{eq:hamiltonian-vector-field} \end{equation} The sign on the right-hand side is a matter of convention. This is called the Hamiltonian vector field generated by $f$, which we’ll have more to say about in later sections. In coordinates, \eqref{eq:hamiltonian-vector-field} becomes \begin{equation} X _ {f}^{\mathsf{a}} =\omega^{\mathsf{ab}}\partial _ {\mathsf{b}}f \ . \end{equation} Let us write this out in coordinates more explicitly. In Darboux coordinates, we can decompose: \begin{equation} X _ {f}=A _ {a}\partial _ {q _ {a}} +B^{a}\partial _ {p^{a}} \ . \end{equation} Since $\omega=\mathrm{d}p^{a}\wedge\mathrm{d}q _ {a}$, we have \begin{equation} \iota _ {X _ {f}}\omega =B^{a}\mathrm{d}q _ {a} -A _ {a}\mathrm{d}p^{a} =-\mathrm{d}f \ , \end{equation} and therefore on comparing the coefficients in the above equation, we may conclude that \begin{equation} A _ {a}=\frac{\partial f}{\partial p^{a}} \quad \text{and} \quad B^{a}=-\frac{\partial f}{\partial q _ {a}} \ . \end{equation}

Notice that from the above discussion, we may conclude that the Poisson bracket can be written in a number of equivalent ways, which we record for reference: \begin{equation} \left\lbrace f,g\right\rbrace =X _ {g}[f] =-\omega(X _ {f},X _ {g}) \ . \label{eq:poisson-index-free} \end{equation}

For $f=H$, the integral curves of $X _ {H}$ obey \begin{equation} \dot{q} _ {a}=\frac{\partial H}{\partial p^{a}} \quad \text{and} \quad \dot{p}^{a}=-\frac{\partial H}{\partial q _ {a}} \ , \end{equation} so Hamilton’s equations say simply that time evolution is the flow generated by $X _ {H}$, the Hamiltonian vector field associated to the Hamiltonian itself.

Hamiltonian flows are easily seen to preserve the symplectic form. Indeed, using Cartan’s magic formula, the closure of $\omega$, and \eqref{eq:hamiltonian-vector-field}, \begin{equation} \mathcal{L} _ {X _ {f}}\omega =\mathrm{d}(\iota _ {X _ {f}}\omega) +\iota _ {X _ {f}}\mathrm{d}\omega =-\mathrm{d}^{2}f =0 \ . \label{eq:hamiltonian-preserves-symplectic} \end{equation} Thus, Hamiltonian vector fields generate infinitesimal canonical transformations.

It is useful at this point to distinguish between two kinds of vector fields here. We have already seen that Hamiltonian vector fields are defined by \eqref{eq:hamiltonian-vector-field}, which says that $ \iota _ {X} \omega $ is (up to an irrelevant sign) an exact form. On the other hand, we can define symplectic vector fields to be such that $ \iota _ {X} \omega $ is closed: \begin{equation} \text{d} \left( \iota _ {X} \omega \right) = \mathcal{L}_ {X} \omega = 0 \ , \end{equation} where in the second equality we have rewritten the closure condition on symplectic vector fields in terms of the Lie derivative. Since locally every closed form is exact, all Hamiltonian vector fields are symplectic. The converse, however, is not true. Once again, the distinction between them is one that indicates non-trivial topology of $ M $.

Essentially the same calculation establishes Liouville’s theorem on the preservation of phase space volumes (once again!) for Hamiltonian evolution: \begin{equation} \mathcal{L} _ {X _ {H}}\operatorname{vol} _ {\omega}=0 \ . \end{equation}

Integration and Stokes’ Theorem

Differential forms are natural objects to integrate over submanifolds of appropriate dimension. One way to think about this is to think of the differential of a function $ f $ of a single variable, say $ x $, which we denote $ \text{d} f $. Written out “in components” (although, in this case, the number of components is just one) we might write \begin{equation} \text{d} f = \frac{\partial f}{\partial x} \text{d} x \ , \end{equation} which allows us to see that the form on the right hand side is precisely the sort of object that we might integrate along the real line! More generally, a $p$-form may be integrated over an oriented $p$-dimensional (sub)manifold. For example, if $\gamma:[t _ {i},t _ {f}]\rightarrow M$ is a one-dimensional curve and $\alpha$ is a one-form, then \begin{equation} \int _ {\gamma}\alpha =\int _ {t _ {i}}^{t _ {f}}\gamma^{\star}\alpha \ , \end{equation} where on the right-hand side, we are integrating the pullback of $ \alpha $ constructed using the map $ \gamma $ between $ t _ {i} $ and $ t _ {f} $. If $\alpha=\alpha _ {\mathsf{a}}\mathrm{d}y^{\mathsf{a}}$, this is the familiar line integral \begin{equation} \int _ {\gamma}\alpha =\int _ {t _ {i}}^{t _ {f}}\mathrm{d}t\, \alpha _ {\mathsf{a}}(y(t)) \frac{\text{d} y^{\mathsf{a}}(t) }{\text{d} t} \ . \end{equation} The use of the pullback makes the definition manifestly independent of the coordinates used on $M$ or of the parametrisation used for the curve.

More generally, we have Stokes’ theorem, which we now state. Let $\Sigma$ be an oriented $p$-dimensional manifold with boundary $\partial\Sigma$. For every $(p-1)$-form $\alpha$ with suitable support, Stokes’ theorem states \begin{equation} \int _ {\Sigma}\mathrm{d}\alpha =\int _ {\partial\Sigma}\alpha \ . \label{eq:stokes-theorem} \end{equation} The above equation combines efficiently a number of theorems of vector calculus. When $p=1$, it is simply the fundamental theorem of calculus. In two and three dimensions, after some rearranging of furniture, it reproduces the Stokes’ and Green’s theorems concerning curls and divergences respectively. I would like to quote Spivak’s Calculus on Manifolds here, since I think it captures something beautiful and general about Stokes’ theorem:

Stokes’ theorem shares three important attributes with many fully evolved major theorems:

  1. It is trivial.
  2. It is trivial because the terms appearing in it have been properly defined.
  3. It has significant consequences.

Since this entire chapter was little more than a series of definitions which made the statement and proof of Stokes’ theorem possible, the reader should be willing to grant the first two of these attributes to Stokes’ theorem. The rest of the book is devoted to justifying the third…

Applying this to symplectic geometry, suppose that $\omega$ is exact on a region containing a surface $\Sigma$, so that $\omega=\mathrm{d}\theta$. Then by Stokes’ theorem: \begin{equation} \int _ {\Sigma}\omega =\int _ {\partial\Sigma}\theta \ . \label{eq:symplectic-stokes} \end{equation} In Darboux coordinates, $\theta=p^{a}\mathrm{d}q _ {a}$, and the right-hand side is the familiar phase-space integral $\oint p^{a}\mathrm{d}q _ {a}$, perhaps familiar from semiclassical quantisations of Bohr and Sommerfeld. The left-hand side shows that this is the symplectic area enclosed by the curve. We’ll use this later, when we discuss geometric quantisation and the role that topology plays.

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