15.2 Hamiltonian vector fields and canonical transformations

15.2 Hamiltonian vector fields and canonical transformations

Concept links · terms present in this machine draft; source roles are unverified: linear map · Lie algebra · Lie bracket

Our interest is not in general vector fields, but in vector fields corresponding to Hamilton’s equations for some Hamiltonian function $f,e.g.$ , the case

$$F_q=\frac{\partial f}{\partial p},F_p=-\frac{\partial f}{\partial q}$$

We call such vector fields Hamiltonian vector fields, defining:

Definition (Hamiltonian vector field)

$A$ vector field on $M={\mathbf{R}}^2$ given by

$$\frac{\partial f}{\partial p}\frac{\partial }{\partial q}-\frac{\partial f}{\partial q}\frac{\partial }{\partial p}=-\{f,\cdot \}$$

for some function $f$ on $M={\mathbf{R}}^2$ is called a Hamiltonian vector field and will be denoted by $X_f$ . In higher dimensions, Hamiltonian vector fields will be those of the form

$$X_f=\sum _{j=1}^d\left(\frac{\partial f}{\partial p_j}\frac{\partial }{\partial q_j}-\frac{\partial f}{\partial q_j}\frac{\partial }{\partial p_j}\right)=-\{f,\cdot \}\text{\hspace{1em}(15.2)}$$

for some function $f$ on $M={\mathbf{R}}^{2d}$

The simplest non-zero Hamiltonian vector fields are those for $f\mathrm{a}$ linear function. For $c_q,c_p$ constants, if

$$f=c_{q}q+c_{p}p$$

then

$$X_f=c_p\frac{\partial }{\partial q}-c_q\frac{\partial }{\partial p}$$

and the map

$$f\to X_f$$

is the isomorphism of $\mathcal{M}$ and M of equation 14.6.

For example, taking $f=p$ , we have $\begin{array}{r}{X}_p=\frac{\partial }{\partial q}\end{array}$ . The exponential map for this vector field satisfies

$$q(\exp (t{X}_p)(m))=q(m)+t,p(\exp (t{X}_p)(m))=p(m)\text{\hspace{1em}(15.3)}$$

Similarly, for $f=q$ one has $X_q=-\frac{\partial }{\partial p}$ and

$$q(\exp (t{X}_q)(m))=q(m),p(\exp (t{X}_q)(m))=p(m)-t\text{\hspace{1em}(15.4)}$$

Quadratic functions $f$ give vector fields $X_f$ with components linear in the coordinates. An important example is the case of the quadratic function

$$h=\frac{1}{2}(q^2+p^2)$$

which is the Hamiltonian function for a harmonic oscillator, a system that will be treated in much more detail beginning in chapter 22. The Hamiltonian vector field for this function is 2

$$X_h=p\frac{\partial }{\partial q}-q\frac{\partial }{\partial p}$$

The trajectories satisfy

$$\frac{dq}{dt}=p,\frac{dp}{dt}=-q$$

and are given by

$$q(t)=q(0)\cos t+p(0)\sin t,p(t)=p(0)\cos t-q(0)\sin t$$

The exponential map is given by clockwise rotation through an angle $t$

$$\begin{array}{rl}q(\exp (t{X}_h)(m))& =q(m)\cos t+p(m)\sin t\\ p(\exp (t{X}_h)(m))& =-q(m)\sin t+p(m)\cos t\end{array}$$

The vector field $X_h$ and the trajectories in the qp plane look like this

Figure 15.1: Hamiltonian vector field for a simple harmonic oscillator.

and describe a periodic motion in phase space.

The relation of vector fields to the Poisson bracket is given by (see equation 15.2)

$$\{f_1,f_2\}=X_{f_2}(f_1)=-X_{f_1}(f_2)$$

and in particular

$$\{q,f\}=\frac{\partial f}{\partial p},\{p,f\}=-\frac{\partial f}{\partial q}$$

The definition we have given here of $X_f$ (equation 15.2) carries with it a choice of how to deal with a confusing sign issue. Recall that vector fields on $M$ form a Lie algebra with Lie bracket the commutator of diferential operators. A natural question is that of how this Lie algebra is related to the Lie algebra of functions on $M$ (with Lie bracket the Poisson bracket).

The Jacobi identity implies

$$\{f,\{f_1,f_2\}\}=\{\{f,f_1\},f_2\}+\{\{f_2,f\},f_1\}=\{\{f,f_1\},f_2\}-\{\{f,f_2\},f_1\}$$

so

$$X_{\{f_1,f_2\}}=X_{f_2}{X}_{f_1}-X_{f_1}{X}_{f_2}=-[X_{f_1},X_{f_2}]\text{\hspace{1em}(15.5)}$$

This shows that the map $f{X}_f$ of equation 15.2 that we defined between these Lie algebras is not quite a Lie algebra homomorphism because of the - sign in equation 15.5 (it is called a Lie algebra “antihomomorphism”). The map that is a Lie algebra homomorphism is

$$f\to -X_f\text{\hspace{1em}(15.6)}$$

To keep track of the minus sign here, one needs to keep straight the diference between

• The functions on phase space $M$ are a Lie algebra, with a function $f$ acting on the function space by the adjoint action

$$ad(f)(\cdot )=\{f,\cdot \}$$

and

• The functions $f$ provide vector fields $X_f$ acting on functions on $M$ , where

$$X_f(\cdot )=\{\cdot ,f\}=-\{f,\cdot \}$$

As a simple example, the function $p$ satisfies

$$\{p,F(q,p)\}=-\frac{\partial F}{\partial q}$$

so

$$\{p,\cdot \}=-\frac{\partial (\cdot )}{\partial q}=-X_p$$

Note that acting on functions with $p$ in this way is the Lie algebra version of the representation of the translation group on functions induced from the translation action on the position (see equations 10.1 and 10.2).

It is important to note that the Lie algebra homomorphism 15.6 from functions to vector fields is not an isomorphism, for two reasons:

• It is not injective (one-to-one), since functions $f$ and $f+C$ for any constant $C$ correspond to the same $X_f$

• It is not surjective since not all vector fields are Hamiltonian vector fields $(i.e.$ , of the form $X_f$ for some $f)$ . One property that a vector field $X$ must satisfy in order to possibly be a Hamiltonian vector field is

$$X\{g_1,g_2\}=\{X{g}_1,g_2\}+\{g_1,X{g}_2\}\text{\hspace{1em}(15.7)}$$

for $g_1$ and $g_2$ on $M$. This is the Jacobi identity for $f,g_1,g_2$ , when $X=X_f$

Digression. For a general symplectic manifold $M,$ the symplectic two-form $\omega$ gives us an analog of Hamilton’s equations. This is the following equality of one-forms, relating a Hamiltonian function $h$ and a vector field $X_h$ determining time evolution of trajectories in $M$

$$i_{X_h}\omega =\omega (X_h,\cdot )=dh$$

(here $i_X$ is interior product with the vector field $X)$ . The Poisson bracket in this context can be defined as

$$\{f_1,f_2\}=\omega (X_{f_1},X_{f_2})$$

Recall that a symplectic two-form is defined to be closed, satisfying the equation $d\omega =0,$ which is then a condition on a three-form d$\omega$. Standard diferential form computations allow one to express d$\omega$ $\cdot (X_{f_1},X_{f_2},X_{f_3})$ in terms of Poisson brackets of functions $f_1,f_2,f_3$ , and one finds that $d\omega =0$ is the Jacobi identity for the Poisson bracket.

The theory of “prequantization” (see $[52],[41])$ enlarges the phase space $M$ to a $U(1)$ bundle with connection, where the curvature of the connection is the symplectic form $\omega$ . Then the problem of lack of injectivity of the Lie algebra homomorphism

$$f\to -X_f$$

is resolved by instead using the map

$$f\to -{\nabla }_{X_f}+if\text{\hspace{1em}(15.8)}$$

where ${\nabla }_X$ is the covariant derivative with respect to the connection. For details of this, see $[52]or[41]$

In our treatment of functions on phase space $M.$ , we have always been taking such functions to be time-independent. M can be thought of as the space of trajectories of a classical mechanical system, with coordinates $q,p$ having the interpretation of initial conditions $q(0),p(0)$ of the trajectories. The exponential maps exp $(t{X}_h)$ give an action on the space of trajectories for the Hamiltonian function $h,$ , taking the trajectory with initial conditions given by $n\in M$ to the time-translated one with initial conditions given by $\exp (t{X}_h)(m)$ . One should really interpret the formula for Hamilton’s equations

$$\frac{df}{dt}=\{f,h\}$$

as meaning

$$\frac{d}{dt}f(\exp (t{X}_h)(m)){|}_{t=0}=\{f(m),h(m)\}$$

for each $m\in M$

Given a Hamiltonian vector field $X_f$ , the maps

$$\exp (t{X}_f):M\to M$$

are known to physicists as “canonical transformations”, and to mathematicians as “symplectomorphisms”. We will not try and work out in any more detail how the exponential map behaves in general. In chapter 16 we will see what happens for $f$ an order-two homogeneous polynomial in the $q_j,p_j$ . In that case the vector field $X_f$ will take linear functions on $M$ to linear functions, thus acting on $\mathcal{M}.$ , in which case its behavior can be studied using the matrix for the linear transformation with respect to the basis elements $q_j,p_j$

Digression. The exponential map exp(tX) can be defined as above on a general manifold. For a symplectic manifold $M$, Hamiltonian vector fields $X_f$ will have the property that they preserve the symplectic form, in the sense that

$$\exp (t{X}_f{)}^{\ast }\omega =\omega$$

This is because

$${\mathcal{L}}_{X_f}\omega =(d{i}_{X_f}+i_{X_f}d)\omega =d{i}_{X_f}\omega =d\omega (X_f,\cdot )=ddf=0\text{\hspace{1em}(15.9)}$$

where ${\mathcal{L}}_{X_f}$ is the Lie derivative along $X_f$

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