Manifolds, Vectors, and 1-Forms

In the previous section, we saw that phase space is naturally equipped with an antisymmetric tensor $\omega _ {\mathsf{ab}}$ that is everywhere invertible and satisfies a Bianchi identity. We also saw that under canonical transformations, what we call position and momentum can get mixed up. For these reasons, it is natural to ask if classical mechanics can be formulated in a way that makes manifest this invariance under canonical transformations and a specific choice of coordinates on phase space. This is the language of symplectic geometry.

Manifolds and Coordinates

An $n$-dimensional manifold $M$ is a space such that, informally speaking, every small patch of it “looks” like $\mathbb{R}^{n}$. The surface of the Earth, for example, is a two-dimensional sphere, but in a small neighbourhood around you, it looks more or less flat. We formulate this mathematically as follows: around every point $x \in M$, mark out an open set $U \subset M$ that contains it, and map this open set to a subset of $ n $-dimensional flat space: \begin{equation} \varphi : U \longrightarrow \varphi(U) \subset \mathbb{R}^{n} \ . \end{equation} This map should be a invertible, and “sufficiently” differentiable, although we won’t be too careful specifying what that means. The pair $(U,\varphi)$ is called a coordinate chart, and a little thought should convince you that this is exactly how one sets up coordinates in the real world. If the standard coordinates on $\mathbb{R}^{n}$ are denoted $y^{\mathsf{a}}$, then the functions $y^{\mathsf{a}} : U \rightarrow \mathbb{R}$ are local coordinates on $M$.

Our definition of a manifold only required that it resemble flat space locally, and for this reason, a single coordinate chart will almost never suffice to cover the entire manifold. As in geography, one then covers the manifold with multiple (possibly overlapping) charts, called an atlas. Naturally, whenever two charts overlap, their coordinates should be related by smooth and invertible transition functions: \begin{equation} y^{\mathsf{a}} \longrightarrow y’^{\mathsf{a}}(y) \ . \end{equation} One may therefore approach the subject of manifolds in two complementary ways. The first is to consider the manifold as a unitary entity, which is then broken up into overlapping coordinate patches, sewn together with transition functions. The second is to go the opposite way, and consistently glue together little patches of flat space. Either of these descriptions of manifold make one thing abundantly clear: because of the many equivalent ways in which manifolds may be conceived, coordinates of points are simple labels assigned to them, and not intrinsic to the points themselves. This is our motivation for building up a description independent of these labels.

We have already seen the example of the two-dimensional sphere $ \mathbb{S}^ {2} $ as a manifold. In much the same way, the phase space of a system with $N$ degrees of freedom is a $2N$-dimensional manifold, whose local coordinates are the positions and momenta $y ^ {\mathsf{a} } = (q _ {a},p^{a})$.

Of course, a choice of coordinates is not set in stone. We may find, occasionally, that we have to change coordinate systems. This is captured mathematically by a smooth map $ \Phi : M \rightarrow M $ that is invertible — these are called diffeomorphisms. In the study of classical mechanics, as we have already seen in previous sections, the class of diffeomorphisms we are interested in typically also preserve the symplectic structure, or equivalently, the Poisson brackets don’t change. We’ll see over the course of this section that these are precisely the canonical transformations.

Vectors and 1-Forms

Given a manifold and functions on it, we now proceed to build up some structure. To every point $x \in M$, we associate an $n$-dimensional vector space $T _ {x}M$ called the tangent space. Its elements are called tangent vectors at $x$, and although it might seem odd, an especially useful definition of a tangent vector is as a derivative acting on smooth functions. Over the course of this section, we’ll see why this is a good definition. For now, note that derivatives can (formally) be added and multiplied by scalars, so they make up a vector space.

A tangent vector $X _ {x} \in T _ {x} M$ assigns to a smooth function $ f $ a number $X _ {x}[f]$, and obeys the Leibniz rule \begin{equation} X _ {x}[fg]=f(x)X _ {x}[g]+X _ {x}[f]g(x) \ . \end{equation} In local coordinates $y^{\mathsf{a}}$, a basis for the tangent space is supplied by the coordinate derivatives $ \partial _ {\mathsf{a} } $ we have encountered before, and a general tangent vector can be written as \begin{equation} X=X^{\mathsf{a}}\partial _ {\mathsf{a}} \ . \label{eq:coordinate-vector} \end{equation}

When this vector acts on a function, it returns the function $X[f]=X^{\mathsf{a}}\partial _ {\mathsf{a}}f$, and so $ X[f] $ can be thought of as a directional derivative. Under a change of coordinates, the basis vectors (i.e.~the coordinate derivatives) and the components transform oppositely: \begin{equation} \partial’ _ {\mathsf{a}}=\frac{\partial y^{\mathsf{b}}}{\partial y’^{\mathsf{a}}}\partial _ {\mathsf{b}} \quad \text{and} \quad X’^{\mathsf{a}}=\frac{\partial y’^{\mathsf{a}}}{\partial y^{\mathsf{b}}}X^{\mathsf{b}} \ , \end{equation} so that the vector $X$ itself remains unchanged.

Given two vector fields $X$ and $Y$, their Lie bracket is the vector field defined by \begin{equation} [X,Y][f]=X[Y[f]]-Y[X[f]] \ . \end{equation} In coordinates, its components are \begin{equation} [X,Y]^{\mathsf{a}} =X^{\mathsf{b}}\partial _ {\mathsf{b}}Y^{\mathsf{a}} -Y^{\mathsf{b}}\partial _ {\mathsf{b}}X^{\mathsf{a}} \ . \end{equation} The Lie bracket measures the failure of the flows generated by $X$ and $Y$ to commute.

The vector space dual to $T _ {x}M$ is denoted $T _ {x}^{\star}M$ and is called the cotangent space; its elements are called one-forms and are linear maps from tangent vectors to numbers: \begin{equation} \alpha _ {x}:T _ {x}M\longrightarrow\mathbb{R} \ . \end{equation} The basis of $T _ {x}^{\star}M$ dual to $\partial _ {\mathsf{a}}$ is denoted $\mathrm{d}y^{\mathsf{a}}$ and is defined by \begin{equation} \mathrm{d}y^{\mathsf{a}}\left(\partial _ {\mathsf{b}}\right) = \delta ^ {\mathsf{a}} {} _ {\mathsf{b}} \ . \end{equation}

More generally, a one-form will be some linear combination of these basis elements \begin{equation} \alpha=\alpha _ {\mathsf{a}}\,\mathrm{d}y^{\mathsf{a}} \ , \label{eq:coordinate-one-form} \end{equation} and its pairing with the vector in \eqref{eq:coordinate-vector} is the coordinate-independent number \begin{equation} \alpha(X)=\alpha _ {\mathsf{a}}X^{\mathsf{a}} \ . \end{equation} The most immediate example of a one-form is the differential of a function. Given $f:M\rightarrow\mathbb{R}$, its differential $\mathrm{d}f$ is defined by \begin{equation} \mathrm{d}f(X)=X[f] \ . \end{equation} In coordinates, this familiar object is \begin{equation} \mathrm{d}f=\partial _ {\mathsf{a}}f\,\mathrm{d}y^{\mathsf{a}} \ . \end{equation}

At this stage, our manifold doesn’t have enough structure for us to identify a vector with a $ 1 $-form. Depending on the kind of manifold one is working with, sometimes such an identification is possible. For example, in Riemannian geometry, we consider manifolds with a metric tensor, a symmetric $ 2 $-tensor that allows us to identify vectors and $ 1 $-forms. Practically speaking, this is the business of raising and lowering indices. On symplectic manifolds, the symplectic form will provide us with a different way of identifying vectors and $ 1 $-forms.

Before closing this section, let us briefly discuss mappings between manifolds. Let’s say that in local coordinate patches, $ M $ has coordinates $ y ^ {\mathsf{a} } $ and $ N $ has coordinates $ x ^ {i} $ respectively. We consider a function $ \Phi ^ {i} : M \rightarrow N $ takes as input the $ \text{dim } M $ coordinates on $ M $ and returns $ \text{dim } N $ coordinates (one for each $ i $) on $ N $. Now, a vector $ X $ on $ M $ can be written in local coordinates as \begin{equation} X = X ^ {\mathsf{a} } \frac{\partial }{\partial y ^ {\mathsf{a} } } = \left( X ^ {\mathsf{a} } \frac{\partial \Phi ^ {i} }{\partial y ^ {\mathsf{a} }} \right) \frac{\partial }{\partial x ^ {i} } \ , \end{equation} where in the second equality we have used the chain rule. Notice that the second form makes it look like a vector field that lives on $ N $. We say that we have pushed the vector forward using the map $ \Phi $, and this vector on $ N $ is now denoted $ \Phi _ {\star } X $.

Differential forms, on the other hand, move in the opposite direction. As before writing down a differential form $ \alpha $ on $ N $, we have: \begin{equation} \alpha = \alpha _ {i} \text{d} x ^ {i} = \alpha _ {i} \text{d} \Phi ^ {i} (y) = \left( \alpha _ {i} \frac{\partial \Phi ^ {i} }{\partial y ^ {\mathsf{a} } } \right)\text{d} y ^ {\mathsf{a} } \ , \end{equation} and just like before we have used the chain rule in the third equality. Notice that the last expression looks like a differential form that lives on $ M $. We say that the form $ \alpha $ has been pulled back using the map $ \Phi $, and the form on $ M $ is now denoted $ \Phi ^ {\star } \alpha $.

It is rather straightforward to see, based on the above expressions, that \begin{equation} (\Phi^{\star}\alpha)(X)=\alpha(\Phi _ {\star}X) \ . \label{eq:pullback-one-form} \end{equation} Notice that in our discussion of pushforwards and pullbacks, nowhere did we require that $ M $ and $ N $ be different manifolds. In fact, all this machinery goes through for ordinary diffeomorphisms, even canonical transformations. Going forward, we will often express any invariance properties under transformations in these terms, since they are explicitly coordinate-free.