Hamiltonian Mechanics

3 Hamiltonian mechanics

In this chapter we start our investigation of the hamiltonian formalism. We have already seen glimpses of it throughout some of the exercises and examples in the previous chapter, but only scratched the surface. We will see how this, seemingly harmless, change of perspective will reveal much more about the geometric structure underlying classical dynamics.

In the lagrangian formulation for a system with \(n\) degrees of freedom and lagrangian function \(L=L(q, \dot q)\), we have seen that the equations of motion are encoded by the Euler-Lagrange equations, which are \(n\) second order differential equations in \(q\), requiring \(2n\) initial conditions, say \(q(0)\) and \(\dot q(0)\).

In the previous chapters, we saw that the \(n\) generalized momenta

\begin{equation} p_i = \pdv {L}{\dot q^i}, \quad i=1,\ldots ,n, \end{equation}

play a relevant role in the first integrals at the core of Noether theorem, and in particular in the conservation of energy. If we then look at the Euler-Lagrange equations and revisit them in terms of the momenta, they acquire the nice form

\begin{equation} \dot p_i = \pdv {L}{q^i}, \quad i=1,\ldots ,n. \end{equation}

Which seems to bring a nice symmetry between the coordinates and the momenta, and seems to allow us to reduce the problem to a first order system of equations.

Our current quest is then to find a function of \(q\) and \(p\) (and not of \(\dot q\)) which contains the same information as the lagrangian \(L\), allowing us to determine the unique evolution of \(q\) and \(p\), but bringing the two variables to more equal footings.

Theorem 1.6 and Theorem 1.7 provided us with some useful hints. Before entering into the gist of the matter, let’s reason in two dimensions and mimic our derivation of Theorem 1.6

3.1 The Legendre Transform in the euclidean plane

In this section we’ll be a bit sloppy, please bear with me. Let’s start considering the total derivative of an arbitrary function \(f(x,y)\),

\begin{equation} \dd f = \frac {\partial f}{\partial x} \dd x + \frac {\partial f}{\partial y} \dd y, \end{equation}

and defining a new function of three variables \(x,y,u\):

\begin{equation} g(x,y,u) = ux - f(x,y). \end{equation}

The total derivative of \(g\) is then given by

\begin{equation} \label {eq:hamdgex} \dd g = \dd (ux) - \dd f = u\dd x + x \dd u - \frac {\partial f}{\partial x} \dd x - \frac {\partial f}{\partial y} \dd y. \end{equation}

Even though \(u\) is a free variable, let’s assume for the sake of the argument that we choose it to be a specific function of \(x\) and \(y\), say

\begin{equation} u(x,y) = \frac {\partial f}{\partial x}. \end{equation}

Then (3.5) simplifies into

\begin{equation} \dd g = x \dd u - \frac {\partial f}{\partial y} \dd y, \end{equation}

i.e., \(g=g(y,u)\) could be considered a function of just \(y\) and \(u\). Inverting \(x = x(y,u)\), we could write down \(g\) explicitly as

\begin{equation} g(y,u) = u\, x(y,u) - f(x(y, u), y). \end{equation}

We have just described an operation which takes a function \(f(x,y)\) to a different function \(g(y,u)\) where \(u = \frac {\partial f}{\partial x}\) without losing any information: we can recover \(f(x,y)\) from \(g(y,u)\) by observing that

\begin{equation} \frac {\partial g}{\partial u} = x \quad \mbox {and}\quad \frac {\partial g}{\partial y} = - \frac {\partial f}{\partial y}, \end{equation}

which ensures that we can invert the transformation to get \(f(x,y) = \frac {\partial g}{\partial u}u - g\).

We have just described the Legendre transform. If you look carefully you may recognize it in Theorem 1.7.

The geometrical meaning of the Legendre transform is best captured with a picture¹, although I don’t believe this will justify it more or less than the description before. For a fixed \(y\), draw the curves \(f(x) = f(x,y)\) and \(ux\). For each slope \(u\), the value of \(g(u) = g(u,y) = \max _x (ux - f(x))\) measures the maximal distance between the two curves \(f(x)\) and \(ux\): finding the maximum of such distance means computing

\begin{equation} \frac {\partial }{\partial x}(ux - f(x)) = 0 \quad \rightarrow \quad u = \frac {\partial f}{\partial x}. \end{equation}

Note that in the computation above we are assuming that a maximum exists: this hints to the fact that the Legendre transformation can only be applied to convex functions.

One advantage of this geometric approach is that the procedure is meaningful also on non-convex functions, even if this comes at the cost of losing involutivity, i.e. the function is the inverse of itself only on the space of convex (or concave) functions. You can read more about this on convex optimization books², where is often called convex conjugation or Legendre-Fenchel transform.

In the next section we will explicit this discussion rigorously and in greater generality.

3.2 Intermezzo: cotangent bundle and differential forms

Again we will briefly present the basic concepts, you can refer to [J.M13] or [Ser20, Chapters 5-7] for further details.

Given an \(n\)-dimensional vector space \(V\) spanned by a basis \(\{e_1, \ldots , e_n\}\), we can define its dual space as the space \(V^* = \{f : V \to \mathbb {R} \; \mbox {linear}\}\) of linear functions from \(V\) to \(\mathbb {R}\). As a basis for the dual space we can define the dual basis \(\{e^1, \ldots , e^n\}\) as the linear maps such that \(e^i(e_j) = \delta ^i_j\). Since any linear function is determined by its action on a basis element, we can determine any \(\sigma \in V^*\) by computing its coefficients \(\sigma _i = \sigma (e_i)\). Then \(\sigma = \sigma _i e^i\).

We have seen that tangent vectors at a point \(q\) on a smooth manifold \(P\) are elements of a \(n\)-dimensional vector space, the tangent space \(T_x P\), and in coordinates they take the form \(v^i \frac {\partial }{\partial x^i}\). We can then use the theory of the previous paragraph to define the dual space to \(T_x P\), the cotangent space \(T^*_x P\). We call elements of the cotangent space at \(x\), covectors or one-forms at \(x\).

It is convenient to denote the dual basis to \(\{\frac {\partial }{\partial x^i} \mid 1\leq i \leq n\}\) by

\begin{equation} \{ \dd x^i \mid 1 \leq i \leq n \}, \quad \dd x^i\left (\frac {\partial }{\partial x^j}\right ) = \delta _j^i. \end{equation}

Even though we settled on the pushforward notation \(\phi _*\) for differentials, we have already seen the \(\dd \) appearing as alternative notation in the definition of the differential of a function. How is it related to one-forms? Observe that for \(f : P \to \mathbb {R}\), we have

\begin{align} \dd f_x : & T_x P \to T_{f(x)} \mathbb {R} \simeq \mathbb {R} \\ & v_x = \dot x^i \frac {\partial }{\partial x^i} \mapsto \dot x^i \frac {\partial f}{\partial x^i}. \end{align} Since derivations are linear maps, identifying the one-dimensional vector space \(T_{f(x)}\mathbb {R} \simeq \mathbb {R}\) with the reals (it is really just about sending the local basis of the tangent space to \(1\) as done in the equation just above), we can view \(\dd f_x\) as a linear map from \(T_x P \to \mathbb {R}\), that is as a differential one-form at \(x\). What are its coefficients? Let’s apply the recipe described at the beginning of the section and compute

\begin{equation} (\dd f_x)_i = \dd f_x\left (\frac {\partial }{\partial x^i}\right ) = \frac {\partial f(x)}{\partial x^i}. \end{equation}

That is, \(\dd f_x = \frac {\partial f(x)}{\partial x^i} \dd x^i\). As you can see, this is directly generalizing the same concepts from multivariable analysis.

Analogously to tangent bundles, we can glue cotangent spaces and define the cotangent bundle as \(T^*P := \cup _{x\in P} \{x\}\times T^*_x P\). Also the cotangent bundle is a \(2n\)-dimensional smooth manifold. The concept of differential forms extends to the cotangent bundle in a similar way as vector fields generalized the notion of tangent vectors to the level of the tangent bundle: a differential one-form on \(P\) is a map \(\sigma : x\in P \mapsto \sigma (x) \in T^*_x P\).

We are almost there, we need just a few more concepts. First of all we will need to talk about differential two-forms, these are a particular family of antisymmetric bilinear maps \(TP \times TP \to \mathbb {R}\). Given two one-forms \(\sigma \) and \(\nu \) we can define a two-form using an operation called wedge product: \(\sigma \wedge \nu \) is the antisymmetric bilinear map such that for all \(X,Y \in TP\),

\begin{equation} \sigma \wedge \nu (X,Y) := \sigma (X)\nu (Y) - \sigma (Y)\nu (X). \end{equation}

Any two-form can be written in terms of the basis \(\{\dd x^i\wedge \dd x^j\}\). Due to antisymmetry \(dx^i \wedge \dd x^j = - \dd x^j \wedge \dd x^i\), so one can simplify formulas by summing over ordered indices \(i<j\) and considering only half of the terms. Also due to antisymmetry, \(\dd x^i \wedge \dd x^i = 0\). As a linear map, we can associate to any two-form \(\sigma \) an antisymmetric matrix \(S\) whose coefficient are computed by evaluating \(\sigma \) over basis elements: \(S_{ij} = \sigma \left (\frac {\partial }{\partial x^i}, \frac {\partial }{\partial x^j}\right )\).

The contraction (or interior product) \(\iota _X\) of a one–form \(\theta = \theta _l(x)\dd x^l\) with a vector field \(X = X^l(x)\frac {\partial }{\partial x^l}\) is the natural pairing between the tangent and cotangent space:

\begin{equation} \iota _X \theta : P \to \mathbb {R}, \quad \iota _X \theta := \theta (X) = \theta _l(x) X^l(x). \end{equation}

Similarly, given a two-form \(\omega = \omega _{ij}(x) \dd x^i\wedge \dd x^j\), we can extend the contraction as the operation \(\iota _X\) that reduces it to a one–form by applying \(X\) as the first argument of \(\omega \), that is,

\begin{equation} \label {eq:contraction} \iota _X \omega := \omega (X, \cdot ) := \omega _{ij}(x)(X^i(x) \dd x^j - X^j(x) \dd x^i). \end{equation}

We will see in later chapters how this simplifies our lives.

The last thing to know, is that we can construct two-forms out of one-forms using so-called exterior products and exterior derivatives. For what concerns us, we will only need to know that \(\dd (\sigma _i \dd x^i) = \frac {\partial \sigma _i}{\partial x^j} \dd x^j \wedge \dd x^i\) and that \(\dd (\dd f) = 0\) for any \(f\).

We will also need to generalize the pull-back operation. Up until now we only encountered \(\Phi ^* : \mathcal {C}^\infty (T^*M) \to \mathcal {C}^\infty (T^*M)\), where \(\Phi \) is a diffeomorphism \(\Phi : T^*M \to T^*M\), defined in terms of right-composition \(\Phi ^* f = f \circ \Phi \). Note that if \(x = \Phi (y)\) then the pullback maps \(f(x)\) into \(\Phi ^* f (y) = f \circ \Phi (y)\), so you can effectively use the pullback to change coordinates. If \(\theta \) is a differential form on a manifold \(P\), then the diffeomorphism \(\Phi : P \to P\) induces a mapping on the space of differential forms on \(P\) defined \(\forall x\in P, X_j\in T_xP\) by

\begin{equation} (\Phi ^{*}\theta )_{x}(X_{1},\ldots ,X_{k})=\theta _{\Phi (x)}(\Phi _*(x)X_{1},\ldots ,\Phi _*(x)X_{k}). \end{equation}

We call the map \(\Phi ^*\) the pullback of \(\theta \). We will only be interested in \(k=1,2\).

The pullback of differential forms satisfies two useful identities that will be handy later on.

• 1. it is compatible with the exterior product, i.e., if \(\theta _1\) and \(\theta _2\) are differential forms on \(P\), then \(\Phi ^{*}(\theta _1 \wedge \theta _2) = \Phi ^{*}\theta _1 \wedge \Phi ^{*}\theta _2\);

• 2. it is compatible with the exterior derivative, i.e., if \(\theta \) is a differential form on \(P\), then \(\Phi ^{*}(\dd \theta ) = \dd (\Phi ^{*}\theta )\)

Finally, an important operation in differential geometry is the Lie derivative, the concept is quite general but we will only need it in the context of differential forms. Given a vector field \(X\) on a manifold \(P\) and the one–parameter group of diffeomorphisms \(\Phi _t\) that it generates, we define the Lie derivative of a differential form \(\theta \) with respect to \(X\) as

\begin{equation} \label {eq:lie-derivative} \mathcal {L}_X \theta = \dv {t}\Phi _t^*\theta \Big |_{t=0}. \end{equation}

The Lie derivative evaluates the change of a tensor field (including scalar function, vector field and differential forms), along the flow defined by another vector field.

The interior product is extremely useful when you want to compute something, as it relates the exterior derivative and Lie derivative of differential forms by the Cartan (magic) formula:

\begin{equation} \label {eq:cartan} \mathcal {L}_X \Theta = \dd (\iota _X \Theta ) + \iota _X (\dd {\Theta }). \end{equation}

With all this new equipment at hand, we are ready to delve into the hamiltonian side of mechanics.

3.3 The Legendre Transform

Let \(M\) be a smooth manifold. The evolution of the system with lagrangian \(L=L(q,\dot q)\) is given by the Euler-Lagrange equations: a system of second order equations on the tangent bundle \(TM\). Such dynamics can be reinterpreted in a particularly symmetric way as a dynamical system on the cotangent bundle \(T^*M\). Poisson and Hamilton have formalized the transformation that we described in the previous section and that realizes such symmetry.

It follows from Exercise 1.7 that the definition above does not depend on the choice of local coordinates.

If it is not yet clear to you how the momenta are associated to the cotangent bundle, I promise you will not have to wait long. But if you are really impatient you could already have a look at Remark 3.4.

It follows that for a non–degenerate lagrangian we can locally invert the equations

\begin{equation} p_i = \frac {\partial L(q,\dot q)}{\partial \dot q^i}, \; i=1,\ldots ,n, \end{equation}

and write

\begin{equation} \dot q^i = \dot q^i(q,p). \end{equation}

In this way, every non–degenerate function on \(TM\) defines a function on \(T^* M\).

Leaving all these remarks aside, the crucial point of this whole construction lies in the new shape taken by the Euler-Lagrange equations once we revisit them on the cotangent bundle.

We call hamiltonian system a system of differential equations of the form (3.31) and, as one can expect, we say that \(H\) is its corresponding hamiltonian function. Equations of the form (3.31) are usually called canonical equations.

3.4 Poisson brackets and first integrals of motion

One of the many advantages of the hamiltonian approach comes from an elegant algebraic description. This will play a central role in the study of conserved quantities, and provides a remarkable duality with the mathematical description of quantum mechanics.

From now on, we will denote by \(q = (q^1, \ldots , q^n)\) the local coordinates on \(M\), and

\begin{equation} (q^1, \ldots , q^n, p_1, \ldots , p_n)\in T^*M \end{equation}

the coordinates induced by \(q\) on \(T^*M\). We will, thus, write functions on the cotangent space as \(f=f(q,p)\).

Since 3.77 is a kind of weird definition, let’s look at some of its properties.

The only property requiring a proof is the Jacobi identity. Here I quote [Ton05]:

To prove this you need a large piece of paper and a hot cup of coffee. Expand out all 24 terms and watch them cancel one by one.

What Theorem 3.3 is telling us, is that Poisson’s bracket define the same algebraic structure as matrix commutators and differential operators. There is a deep relationship between what we are seeing here and Heisenberg’s and Schrödinger’s picture of quantum mechanics. This relationship is emphasized even more if we compute the Poisson bracket of the coordinate functions:

\begin{equation} \label {eq:coordcommutators} \big \{q^i,p_j\big \} = \delta ^i_j, \qquad \big \{q^i,q^j\big \} = \big \{p_i,p_j\big \} = 0, \qquad i,j = 1,\ldots ,n. \end{equation}

One crucial property of the Poisson bracket is enunciated by the following theorem.

We say that two functions \(H\) and \(F\) on \(T^* M\) commute, or are in involution, if

\begin{equation} \big \{H,F\big \} = 0. \end{equation}

Then Theorem 3.4 implies the following.

This corollary implies some further interesting corollaries.

Which is the hamiltonian version of the conservation of energy, see Theorem 1.7. In fact, we can reformulate this corollary as follows.

3.4.1 The symplectic matrix

Denote the \(2n\) coordinates on \(T^*M\) as

\begin{equation} x=(x^1, \ldots , x^{2n}) := (q^1, \ldots , q^n, p_1, \ldots , p_n). \end{equation}

With this notation, the hamiltonian system (3.31) takes the form

\begin{equation} \dot x^k = \big \{x^k, H\big \}, \qquad k=1,\ldots ,2n. \end{equation}

We can write the Poisson bracket of the coordinate functions (3.80) as

\begin{equation} \big \{x^k, x^l\big \} = J^{kl}, \qquad k,l = 1,\ldots ,2n, \end{equation}

where \(J^{kl}\) are the matrix elements of the following antisymmetric matrix, called the symplectic matrix,

\begin{equation} \label {eq:symmat} J = \left (J^{kl}\right )_{1\leq k,l\leq 2n} = \begin{pmatrix}0 & \Id _n \\ -\Id _n & 0\end {pmatrix}. \end{equation}

Then formula (3.77) for the Poisson bracket of two arbitrary functions takes the following form

\begin{equation} \label {eq:coordPB} \big \{f,g\big \} = J^{kl} \frac {\partial f}{\partial x^k}\frac {\partial g}{\partial x^l}, \end{equation}

and the hamiltonian system (3.31) can be rewritten as

\begin{equation} \dot x^k = J^{kl} \frac {\partial H}{\partial x^l}, \qquad k=1,\ldots ,2n, \end{equation}

or, more compactly,

\begin{equation} \label {eq:hamsysJ} \dot x = J \frac {\partial H}{\partial x}. \end{equation}

3.4.2 A brief detour on time-dependent hamiltonians

TODO: Imprecise and partially incorrect, rewrite

Let \(L=L(q,\dot q,t): TM\times \mathbb {R}\to \mathbb {R}\), \(q=(q^1,\ldots ,q^n)\in M\), be the time-dependent lagrangian of some mechanical system as in Exercise 3.3.

Also in this case we can arrive to canonical equations of the form (3.31), however these will be on the extended space \(T^*M\times \mathbb {R}^2\). The local coordinates \((q^1,\ldots ,q^n, p_1, \ldots ,p_n, q^{n+1}, p_{n+1})\) on \(T^*M\times \mathbb {R}^2\) are such that \(q^i, p_i\) remain the same for \(i\leq n\),

\begin{equation} q^{n+1} = t, \quad \mbox {and}\quad p_{n+1} = E, \end{equation}

where \(E\) is a new independent variable, and the new hamiltonian is

\begin{align} \hat H & = H(q^1, \ldots , q^n, p_1, \ldots , p_n, t) - p_{n+1}\dot q^{n+1} \\ & = H(q^1, \ldots , q^n, p_1, \ldots , p_n, t) - E. \end{align} Here, \(H\) is given by the usual formula

\begin{equation} H = \sum _{i=1}^n p_i \dot q^i - L(q, \dot q, t), \quad \dot q^i = \dot q^i(q,p,t). \end{equation}

The corresponding hamiltonian system is

\begin{align} & \dot q^i = \frac {\partial \hat H}{\partial p_i}, \quad \dot p_i = -\frac {\partial \hat H}{\partial q^i}, \quad i=1,\ldots ,n \\ & \dot t = \frac {\partial \hat H}{\partial E} = 1, \quad {\dot E} = -\frac {\partial \hat H}{\partial t}. \end{align} The first \(2n\) equations above are equivalent to the Euler-Lagrange equations for \(L\), the equation for \(q^{n+1} = t\) tells us that \(t\) is the same as in the original lagrangian and the final equation is simply the variation of the energy

\begin{equation} \dot E = \frac {\partial H}{\partial t}. \end{equation}

If we restrict ourselves to the zero levelset \(\hat H = H - E = 0\), then the hamiltonian system is equivalent to the Euler-Lagrange equations for \(L\) and the equation

\begin{equation} E = \sum _{i=1}^n p_i \dot q^i - L(q, \dot q, t), \end{equation}

which completes the recovering of the original dynamics.

The duality between time and energy and the reason they enter in this picture with a negative sign becomes clearer once one looks at variational principles from the hamiltonian point of view, coming up in the next section.

3.5 Variational principles of hamiltonian mechanics

Given a hamiltonian system \(h\), can we describe its solutions of \(q=q(t)\), \(p=p(t)\) in terms of a variational principle as we do in Lagrangian mechanics?

A natural way to approach this question is to go back to Hamilton’s variational principle, see Section 1.2, and use (3.55) to invert Legendre transformation. In fact, this is not in vane and one ends up showing the following theorem.

Note that, even though \(p\) and \(q\) play a rather symmetric role, we only fix the values of \(q\) at the endpoints \(t=t_1\) and \(t=t_2\), while \(p\) can be arbitrary.

Note that often the action (3.99) above is written as

\begin{equation} S = \int _{t_1}^{t_2} (p_i \dd q^i - H \dd t), \end{equation}

where is implied that on the curves \(q=q(t)\), \(p=p(t)\) one makes the substitution \(p_i \dd q^i = p_i(t) \dot q^i (t) \dd t\).

Interestingly, this is not the only way to formulate a hamiltonian variational approach. An alternative approach, the so-called Maupertuis variational principle considers the curves that satisfy conservation of energy, resulting in the following theorem.

An important application of Maupertuis principle is the determination of the projections \(q=q(t)\) on the configuration space \(M\) of the trajectories in the phase space \(T^*M\). To such end, one restricts the functional on the subset of curves that satisfy the equations

\begin{equation} \dot q^i = \frac {\partial H(q,p)}{\partial p_i}, \quad i=1,\ldots ,n. \end{equation}

With the right choice of time parametrization, this ends up being a constraint which is compatible with the variational problem. Solving the equations in the form \(p_i = p_i(q,\dot q)\) and using them in the energy constraint \(H(q, p(q,\dot q))=E\), we can find \(\dd t\) in terms of the coordinates \(q\) and their differentials \(\dd q\). Once we replace the values in (3.104), we remain with the variational problem

\begin{equation} \delta \int p_i\left (q, \frac {\partial q}{\partial t}\right ) \frac {\partial q^i}{\partial t} \dd t = 0, \end{equation}

which characterizes the trajectories in configuration space.

Let’s try to clarify this with some important examples.

3.6 Canonical transformations

Let \(M\) be a smooth manifold. In Section 3.4 we saw that, thanks to the Poisson bracket, we can endow \(\mathcal {C}^\infty (T^*M)\) with a Lie algebra structure, see Theorem 3.3.

In the coordinates \(x=(x^1, \ldots , x^{2n})\) on \(T^*M\) defined in Section 3.4.1, \(\Phi \) is described by \(2n\) functions of \(2n\) variables,

\begin{equation} x(x^1,\ldots ,x^{2n}) \mapsto \Phi (x) = \left (\Phi ^1(x), \ldots , \Phi ^{2n}(x)\right ), \end{equation}

and (3.129) takes the form (exercise)

\begin{equation} \label {eq:cantrafocoord} \frac {\partial \Phi ^k(x)}{\partial x^i} J^{ij} \frac {\partial \Phi ^l(x)}{\partial x^j} = J^{kl},\quad k,l=1,\ldots ,2n. \end{equation}

The following theorem should clarify why canonical transformations are important for us.

3.6.1 Hamiltonian vector fields

To investigate in more details the connections between canonical transformations and hamiltonian systems, it is convenient to introduce the infinitesimal version of (3.129).

In the rest of this section we will show that hamiltonian flows can be associated to hamiltonian vector fields which define one–parameter groups of canonical transformations. These will allow us to define hamiltonian symmetries and state the hamiltonian version of Noether theorem.

Given a function \(H\) on \(T^* M\), Leibnitz rule for the Poisson bracket (3.79) tells us that the mapping

\begin{equation} \label {eq:Hambrace} f \mapsto \big \{f,H\big \}, \quad f\in \mathcal {C}^\infty (T^*M), \end{equation}

is a first-order differential operator. Using the identification between derivations, i.e., first-order differential operators, and vector fields we can associate to the operator (3.150) a vector field \(X_H\) such that

\begin{equation} X_H f = \big \{f, H\big \}. \end{equation}

Such vector field \(X_H\) is called hamiltonian vector field generated by the hamiltonian \(H\). In local coordinates \(q,p\), the hamiltonian vector field \(X_H\) has the form

\begin{equation} X_H = \frac {\partial H}{\partial p_i} \frac {\partial }{\partial q^i} - \frac {\partial H}{\partial q^i}\frac {\partial }{\partial p_i}. \end{equation}

In the coordinates \(x = (x^1, \ldots , x^{2n})\) on \(T^*M\), the field has the compact representation

\begin{equation} X_H = J^{ij}\frac {\partial H}{\partial x^j}\frac {\partial }{\partial x^i}. \end{equation}

We have constructed a mapping of the space of smooth functions on \(T^*M\) into the space⁶ \(\Gamma (TT^* M)\) of smooth vector fields on \(T^*M\)

\begin{equation} \label {eq:antimapping} \mathcal {C}^\infty (T^*M) \to \Gamma (TT^* M), \quad H \mapsto X_H. \end{equation}

The Lie bracket of vector fields on a smooth manifold \(N\) [J.M13],

\begin{equation} [X,Y] f := X(Y f) - Y(X f), \qquad f\in \mathcal {C}^\infty (N), \quad X,Y\in \Gamma (TN), \end{equation}

defines a Lie algebra structure on the space \(\Gamma (TN)\) of vector fields on \(N\). In our case, \(N = T^* M\) and it seems perfectly reasonable to ask if and how this bracket can be related to Poisson bracket. Jacobi identity comes to the rescue, leading to the following beautiful relation.

A crucial property of hamiltonian vector fields is that they are infinitesimal symmetries of the Poisson bracket.

Even though we only proved the local existence of the hamiltonian generator there are many examples of globally defined generators of the group – as was the case for the generators of symmetries in the Lagrangian setting.

3.6.2 The hamiltonian Noether theorem

With all these results at hand we are finally able to give a meaningful definition of hamiltonian symmetries. Note that, until now, in this chapter we have been discussing symmetries associated to the Poisson structure on the space \(C^\infty (T^*M)\) of smooth real-valued functions on the cotangent bundle. We have not yet considered the symmetries of a specific hamiltonian function. To recover Noether we need to introduce precisely such concept.

Theorem 3.14 then implies that symmetries map solutions of the hamiltonian system to other solutions.

We can finally state the hamiltonian counterpart of Noether theorem.

3.7 The symplectic structure on the cotangent bundle

There is a beautiful geometric description that underlies everything that we have seen so far. This not only allows a coordinate free description of Hamiltonian mechanics, but provides the framework to develop many of our future discussions on integrability and perturbation theory. For a deeper geometric investigation than what we can discuss in this course, you can refer to [Sil08].

In fact, we have already encountered the symplectic form in disguise twice. Recall equations (3.90) and (3.172). Especially in the latter case, we have been in front of a mapping that relates

\begin{equation} \dd H = \left (\frac {\partial H}{\partial q^i}, \frac {\partial H}{\partial p_i}\right ) \leftrightarrow X_H = \left (\frac {\partial H}{\partial p_i}, - \frac {\partial H}{\partial q^i}\right ), \end{equation}

that is, a mapping that associates a vector field \(X_H\) to a differential \(\dd H\) via the matrix \(J\).

Since \(J\) is antisymmetric, we can describe antisymmetric bilinear maps in terms of two-forms and we can generate one-forms out of two-forms with the contraction, it may be curious to see if we can actually use these concepts to connect the dots. We look for an antisymmetric form, say \(\alpha \), on a cotangent bundle \(T^*M\) such that \(\iota _{X_H} \alpha \) can be related to \(\dd H\). Let’s compute it!

In terms of the basis elements \(dq^i\wedge dq^j\), \(dp_i\wedge dp_j\), \(dp_i\wedge dq^j\), we have

\begin{align} \iota _{X_H} \alpha &= \alpha _{ij} (\dd q^i(X_H) \dd q^j - \dd q^j(X_H) \dd q^i) \\ &\quad + \alpha ^{ij} (dp_i(X_H) \dd p_j - dp_j(X_H) \dd p_i) \\ &\quad + \alpha ^i_j( dp_i(X_H) \dd q^j - dq^j(X_H) \dd p_i) \\ &= \alpha _{ij}\left (\frac {\partial H}{\partial p_i} \dd q^j - \frac {\partial H}{\partial p_j}\dd q^i\right ) \\ &\quad + \alpha ^{ij}\left (-\frac {\partial H}{\partial q^i} \dd p_j + \frac {\partial H}{\partial q^j} \dd p_i\right ) \\ &\quad + \alpha ^i_j\left (-\frac {\partial H}{\partial q^i} \dd q^j - \frac {\partial H}{\partial p_j} \dd p_i\right ), \end{align} where the sum was over all indices \(i\leq j\). This should be equal to

\begin{equation} dH = \frac {\partial H}{\partial q^i} \dd q^i + \frac {\partial H}{\partial p_i} \dd p_i. \end{equation}

That is, \(\alpha ^{ij}=\alpha _{ij} = 0\) and \(\alpha ^i_j = -\delta ^i_j\). Rewriting everything explicitly, we have computed

\begin{equation} \iota _{X_H}(\dd p_i\wedge \dd q^i) = - \dd H. \end{equation}

As it turns out, \(\dd p_i \wedge \dd q^i\) is a symplectic form, and the equation above is an alternative definition for the hamiltonian vector field (see Exercise 3.12).

Without further ado, let’s dig into it. Instead of working on the cotangent bundle \(T^*M\) over a smooth manifold \(M\), we will consider at first a generic smooth manifold \(P\).

Let \(\omega \) be a two–form on \(P\), that is, for each \(x\in P\), the map \(\omega _x: T_x P \times T_x P \to \mathbb {R}\) is a skew-symmetric bilinear map on the tangent space to \(P\) at \(x\). As we mentioned, in terms of coordinates and linear algebra, bilinear means that it corresponds to some \(\dim P\times \dim P\) matrix \(\Omega (x) = \left (\omega _{ij}(x)\right )_{1\leq i,j\leq \dim P}\) and skew-symmetric means that the matrix \(\Omega (x)\) is antisymmetric. I.e. \(\omega \), seen as a matrix, acts on vectors \(X,Y\) as \(X^T \Omega Y\).

If we go back to our example of an \(n\)-dimensional manifold \(M\), we can construct a symplectic structure on its cotangent bundle \(T^*M = P\) using the coordinates \(q,p\) associated to the local coordinates \(q^1, \ldots , q^n\) on \(M\). Then, the symplectic form is defined by the following formula

\begin{equation} \label {eq:canonical2form} \omega = \dd p_i \wedge \dd q^i, \end{equation}

where as usual we are summing the repeated indices.

Note that \(\Omega \) is the inverse to the matrix \(J = (J^{ij})\) introduced in (3.87).

The proof is not hard, but it is more convenient if we first prove the following lemma.

The tautological one–form, also called canonical one–form, Liouville one–form, the Poincaré one–form and possibly in many other ways, provides a natural way to endow \(T^*M\) with a symplectic structure.

With these new concepts at hand, we have defined two natural geometric structures on \(T^* M\): the Poisson bracket and the symplectic structure. In coordinates, these two structures arise from two antisymmetric tensors: the \((2,0)\)-tensor \(J^{ij}\) from the Poisson bracket (a bilinear form on the cotangent space) and the \((0,2)\)-tensor \(\omega _{ij}\) of the symplectic form (a bilinear form on the tangent space). You may have observed by now that these two tensors are strictly related: their associated matrices are inverse of each other:

\begin{equation} \label {eq:omegaJinverse} \omega _{il}J^{lj} = \delta _i^j, \quad i,j = 1,\ldots , 2n. \end{equation}

In canonical coordinates \(q,p\), these matrices take respectively the constant form of (3.87) and (3.202).

In the rest of this section we will construct a mapping between the notions that we introduced in the previous chapters with the Poisson bracket and the new language of symplectic manifolds. As we will see ,this will allow us to discover a range of new interesting properties.

3.7.1 Symplectomorphisms and generating functions

We start by looking back at the canonical transformations.

If you have looked carefully, you may have noticed that the proof above carefully avoids any use of the structure of cotangent bundle. It is then worth asking ourselves what happens if we consider the tautological one–form \(\eta \) instead of \(\omega \).

The function \(S\) appearing in (3.228) is called generating function of the symplectic transformation. The name stems from the fact that we can effectively use \(S\) to construct canonical transformations and effectively compute the corresponding transformed positions or momentums. Put slightly differently, equation (3.228) is telling us that canonical transformations can be identified with a generating function. We will leave it as a vague remark for the time being but we will come back to it, and study its consequences, in more details once we start discussing integrable systems. For now, lets keep drawing parallels between the Poisson bracket and the symplectic formalism.

We have shown in Theorem 3.16 that the hamiltonian flow is an infinitesimal symmetry of the Poisson bracket, which in particular means that it defines a one–parameter group of canonical transformations. The following result comes, thus, as an immediate consequence of Theorem 3.22.

We can now see the implications of the converse part of Theorem 3.16.

The reason of name of the transformation may become clearer if we observe that (3.230) means that, if \(\Phi _t\) is the flow of \(X\),

\begin{equation} \lim _{t\to 0} \frac {\Phi _t^*\omega - \omega }{t} = 0. \end{equation}

That is, \(\Phi _t^*\omega = \omega \) along the flow of \(X\).

Checking that hamiltonian vector fields satisfy this property is a matter of applying Cartan’s magic formula once:

\begin{equation} \mathcal {L}_{X_H} \omega = \dd (\iota _{X_H} \omega ) + \iota _{X_H}(\dd \omega ) = \dd (-\dd H) = 0. \end{equation}

Where we used the fact that \(\omega \) is closed and Exercise 3.12.

With this new baggage at hand, we can compare again Theorem 3.16 and Theorem 3.22 to extend the previous result with the following theorem.

3.7.2 Darboux Theorem

There seems to be more to symplectic manifolds that just cotangent bundles. However, as manifolds locally look like euclidean spaces, the following theorem tells us that symplectic manifolds locally look like cotangent bundles.

Here we will present a very hands-on proof, on the line of [J.M13, Problem 22-19]. For a more elegant approach, due to Moser and Weinstein, refer to [J.M13, Chapter 22] or [Kna18, Chapter 10.3].

3.7.3 Liouville theorem

The symplectic formalism allows to prove in a rather immediate fashion a crucial property of hamiltonian systems: the conservation of phase space volume along the flow. This is called Liouville theorem, and will be at the center of attention in this section. To start with, let’s prove the following lemma.

This lemma immediately implies the invariance of phase space volume with respect to canonical transformations (or symplectomorphisms). Let \(D\subset T^*M\), we define the volume of \(D\) as the integral

\begin{equation} \label {eq:symplvolume} \Vol (D) :=\int _D \Omega , \quad \Omega = \frac 1{n!} \omega ^n = \dd p_1 \wedge \dd q^1 \wedge \cdots \wedge \dd p_n \wedge \dd q^n, \end{equation}

provided that such integral exists. We call measurable the subsets \(D\) for which \(\Vol (D)\) is well defined.

The renown Liouville theorem is a special case of the theorem we just proved. For an alternative treatment of Liouville theorem, please refer to [Arn89, Chapter 3.16].

More general invariants can be found considering other products \(\omega ^k\) of the symplectic two–form for \(k\leq n\) and their integrals with respect to closed submanifolds or cycles. These are called integral invariants of Poincaré-Cartan. We will not treat them further in these notes, but we refer interested readers to [Arn89, Chapter 9.44].

A further interesting point of view on the meaning and consequences of Liouville theorem can be found in [Ton05, Chapters 4.2.1-4.2.2].

An important consequence of Theorem 3.29 is that we can apply a large corpus of methods from ergodic theory and, more generally, measure preserving dynamical systems to hamiltonian systems.

These are methods that apply to systems with finite measure, however this is not an uncommon property of hamiltonian systems: for a mechanical system with energy \(E = T + U\), due to energy conservation and the positivity of \(T\), the accessible phase space is contained in the spatial region \(U(q) < E\). If, for example, \(U\) is bounded from below and unbounded from above then this is a region of phase space with bounded volume. You can read more about this in [Arn89, Chapter 3.16.C-D].