Chapter 11

Poisson Brackets, Canonical Transformations and Action–Angle Variables

Theory, worked examples and 19 previous-year questions with full solutions — IIT JAM, GATE, JEST and CSIR NET-JRF, 2020 onwards.

Theory

11.1 Time evolution in phase space and the Poisson bracket

Hamilton’s equations (Chapter 10) describe the motion of a system as the flow of a point through the \(2n\)-dimensional phase space of the variables \((q_1,\dots,q_n,p_1,\dots,p_n)\). Every measurable quantity of the system, such as its energy, its angular momentum or the position of one of its particles, is a function on this space, possibly depending on time as well. We call such a function \(f(q_i,p_i,t)\) a dynamical variable. In this chapter we ask how dynamical variables change in time, which changes of phase-space variables preserve the form of Hamilton’s equations, and how periodic motion can be described by especially convenient variables.

(a) The total time derivative of a dynamical variable

Along a trajectory the variables \(q_i\) and \(p_i\) change with time, so by the chain rule

\[\frac{df}{dt}=\sum_i\left(\frac{\partial f}{\partial q_i}\dot q_i+\frac{\partial f}{\partial p_i}\dot p_i\right)+\frac{\partial f}{\partial t}.\]

Substituting Hamilton’s equations \(\dot q_i=\partial H/\partial p_i\) and \(\dot p_i=-\partial H/\partial q_i\) gives

\[\frac{df}{dt}=\sum_i\left(\frac{\partial f}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial H}{\partial q_i}\right)+\frac{\partial f}{\partial t}.\]

The combination of derivatives that appears here occurs so often that it is given its own symbol. For any two dynamical variables \(A(q_i,p_i,t)\) and \(B(q_i,p_i,t)\) the Poisson bracket is defined as

\[\boxed{\{A,B\}=\sum_{i=1}^{n}\left(\frac{\partial A}{\partial q_i}\frac{\partial B}{\partial p_i}-\frac{\partial A}{\partial p_i}\frac{\partial B}{\partial q_i}\right)}.\]

The time evolution of any dynamical variable then takes the compact form

\[\boxed{\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}}.\]

Some books write the bracket as \([A,B]\), and a few define it with the opposite sign; in this book \(\{A,B\}\) always has the sign shown above, so that \(\{q,p\}=+1\).

(b) Constants of the motion

A dynamical variable \(f\) is a constant of the motion (or integral of the motion) if \(df/dt=0\) along every trajectory. By the result just obtained this requires

\[\{f,H\}+\frac{\partial f}{\partial t}=0.\]

If \(f\) does not depend explicitly on time, the condition reduces to \(\{f,H\}=0\): a time-independent quantity is conserved if and only if its Poisson bracket with the Hamiltonian vanishes. A quantity that depends explicitly on time can still be conserved, provided its explicit time dependence exactly cancels the change produced by the motion; Example 11.2 gives a simple case. Applied to \(f=H\) itself, the result gives

\[\frac{dH}{dt}=\{H,H\}+\frac{\partial H}{\partial t}=\frac{\partial H}{\partial t},\]

because \(\{H,H\}=0\) by antisymmetry. This reproduces the result of Chapter 10: \(H\) is conserved whenever it does not depend explicitly on time.

(c) Hamilton’s equations in Poisson-bracket form

Taking \(f=q_i\) and \(f=p_i\), neither of which depends explicitly on time, we obtain

\[\boxed{\dot q_i=\{q_i,H\},\qquad \dot p_i=\{p_i,H\}}.\]

Indeed, \(\{q_i,H\}=\sum_j\left(\dfrac{\partial q_i}{\partial q_j}\dfrac{\partial H}{\partial p_j}-\dfrac{\partial q_i}{\partial p_j}\dfrac{\partial H}{\partial q_j}\right)=\dfrac{\partial H}{\partial p_i}\), and similarly \(\{p_i,H\}=-\partial H/\partial q_i\). In this form the coordinates and momenta appear completely symmetrically, and the equations of motion of every dynamical variable, including \(q_i\) and \(p_i\) themselves, are given by one formula.

11.2 Properties of the Poisson bracket

(a) Algebraic properties

Directly from the definition, for any dynamical variables \(A\), \(B\), \(C\), any constants \(a\), \(b\) and any differentiable function \(f\), the Poisson bracket has the following properties. It is antisymmetric, \(\{A,B\}=-\{B,A\}\), so that \(\{A,A\}=0\); more generally \(\{A,f(A)\}=0\), since \(\partial f(A)/\partial q_i=f'(A)\,\partial A/\partial q_i\) and the two terms of the bracket cancel. It is linear in each argument, \(\{A,bB+cC\}=b\{A,B\}+c\{A,C\}\), and a constant has a vanishing bracket with everything. It obeys the product (Leibniz) rule

\[\{A,BC\}=\{A,B\}C+B\{A,C\},\qquad \{AB,C\}=\{A,C\}B+A\{B,C\},\]

which follows from the product rule for derivatives. Finally it satisfies the Jacobi identity

\[\boxed{\{A,\{B,C\}\}+\{B,\{C,A\}\}+\{C,\{A,B\}\}=0}.\]

The Jacobi identity can be verified by expanding all twelve terms of second derivatives: each product of a second derivative of one function with first derivatives of the other two appears twice with opposite signs. Together with antisymmetry and linearity it makes the dynamical variables a Lie algebra under the Poisson bracket. We also note the useful rule for partial derivatives with respect to a parameter, \(\dfrac{\partial}{\partial t}\{A,B\}=\left\{\dfrac{\partial A}{\partial t},B\right\}+\left\{A,\dfrac{\partial B}{\partial t}\right\}\).

(b) The fundamental Poisson brackets

The brackets of the canonical variables among themselves are

\[\boxed{\{q_i,q_j\}=0,\qquad \{p_i,p_j\}=0,\qquad \{q_i,p_j\}=\delta_{ij}},\]

where \(\delta_{ij}=1\) for \(i=j\) and \(0\) otherwise. These fundamental brackets are the starting point of most calculations, because the product rule reduces any bracket of polynomials to them. Two consequences are used constantly: for any function \(f(q)\) of the coordinates and \(g(p)\) of the momenta,

\[\{q_i,g(p)\}=\frac{\partial g}{\partial p_i},\qquad \{p_i,f(q)\}=-\frac{\partial f}{\partial q_i}.\]

For a particle with position \(\mathbf r=(x,y,z)\) and momentum \(\mathbf p\), the angular momentum \(\mathbf L=\mathbf r\times\mathbf p\) has components such as \(L_z=xp_y-yp_x\). Using the product rule,

\[\{x,L_z\}=\{x,xp_y\}-\{x,yp_x\}=-y\{x,p_x\}=-y,\qquad \{y,L_z\}=x\{y,p_y\}=x,\]

and in the same way \(\{p_x,L_z\}=-p_y\) and \(\{p_y,L_z\}=p_x\). Writing \(x_1,x_2,x_3\) for \(x,y,z\), the general results are

\[\{x_i,L_j\}=\epsilon_{ijk}x_k,\qquad \{p_i,L_j\}=\epsilon_{ijk}p_k,\qquad \boxed{\{L_i,L_j\}=\epsilon_{ijk}L_k},\]

with summation over \(k\) and \(\epsilon_{ijk}\) the Levi-Civita symbol. For example \(\{L_x,L_y\}=L_z\). Since the bracket of \(L^2\) with each component vanishes, \(\{L^2,L_i\}=0\).

(c) Poisson’s theorem

If \(f\) and \(g\) are two constants of the motion, then their Poisson bracket \(\{f,g\}\) is also a constant of the motion. This is Poisson’s theorem. For time-independent \(f\) and \(g\) the proof is a single application of the Jacobi identity:

\[\{\{f,g\},H\}=-\{\{g,H\},f\}-\{\{H,f\},g\}=-\{0,f\}-\{0,g\}=0.\]

If \(f\) and \(g\) depend explicitly on time, we combine the Jacobi identity with the rule for \(\partial/\partial t\) of a bracket:

\[\frac{d}{dt}\{f,g\}=\{\{f,g\},H\}+\frac{\partial}{\partial t}\{f,g\}=\left\{\frac{df}{dt},g\right\}+\left\{f,\frac{dg}{dt}\right\}=0.\]

The theorem sometimes yields new integrals of the motion. For instance, if \(L_x\) and \(L_y\) are conserved, then so is \(\{L_x,L_y\}=L_z\). Often, however, the bracket is merely a constant, a function of the integrals already known, or zero.

(d) Poisson brackets and quantum commutators

The algebra of Poisson brackets carries over to quantum mechanics. In Dirac’s correspondence the classical bracket is replaced by the commutator of the corresponding operators according to

\[\{A,B\}\;\longrightarrow\;\frac{1}{i\hbar}[\hat A,\hat B],\]

so that \(\{x,p\}=1\) becomes \([\hat x,\hat p]=i\hbar\) and the classical equation \(df/dt=\{f,H\}+\partial f/\partial t\) becomes the Heisenberg equation of motion. For polynomials of low degree, symmetrized in \(x\) and \(p\) where necessary, a classical bracket calculation and the corresponding commutator calculation therefore give the same result apart from the factor \(i\hbar\). Several examination questions exploit this parallel.

11.3 Liouville's theorem

Consider not a single system but a large number of identical copies of it, an ensemble, whose members start from different initial conditions. Their phase points fill a region of phase space, and we describe them by a density \(\rho(q,p,t)\), defined so that \(\rho\,d^nq\,d^np\) is the number of members in the volume element \(d^nq\,d^np\). As time goes on each member moves along its own trajectory. Members are neither created nor destroyed, so \(\rho\) obeys the equation of continuity in the \(2n\)-dimensional space,

\[\frac{\partial\rho}{\partial t}+\sum_i\left[\frac{\partial}{\partial q_i}(\rho\,\dot q_i)+\frac{\partial}{\partial p_i}(\rho\,\dot p_i)\right]=0.\]

The phase-space velocity \((\dot q_i,\dot p_i)=(\partial H/\partial p_i,-\partial H/\partial q_i)\) has zero divergence,

\[\sum_i\left(\frac{\partial\dot q_i}{\partial q_i}+\frac{\partial\dot p_i}{\partial p_i}\right)=\sum_i\left(\frac{\partial^2H}{\partial q_i\,\partial p_i}-\frac{\partial^2H}{\partial p_i\,\partial q_i}\right)=0,\]

whether or not \(H\) depends explicitly on time. Expanding the derivatives of the products in the continuity equation, the terms containing the divergence drop out, and we are left with

\[\frac{\partial\rho}{\partial t}+\sum_i\left(\frac{\partial\rho}{\partial q_i}\dot q_i+\frac{\partial\rho}{\partial p_i}\dot p_i\right)=0,\]

that is,

\[\boxed{\frac{d\rho}{dt}=\frac{\partial\rho}{\partial t}+\{\rho,H\}=0}.\]

This is Liouville’s theorem: the density of phase points is constant along the flow, so that the ensemble moves through phase space like an incompressible fluid. An equivalent statement is that the volume of any region of phase space is conserved as each of its points moves according to Hamilton’s equations, although the shape of the region may become greatly distorted. For a stationary ensemble, \(\partial\rho/\partial t=0\), Liouville’s theorem requires \(\{\rho,H\}=0\), which is satisfied, for example, by any density that depends on the phase variables only through \(H\); this is the foundation of the microcanonical and canonical ensembles of statistical mechanics.

The conservation of phase volume can also be seen directly. Over a short time \(\delta t\) a phase point moves from \((q,p)\) to \(Q=q+\dot q\,\delta t\), \(P=p+\dot p\,\delta t\). For one degree of freedom the Jacobian of this map is

\[\frac{\partial(Q,P)}{\partial(q,p)}=\begin{vmatrix}1+\dfrac{\partial^2H}{\partial q\,\partial p}\delta t&\dfrac{\partial^2H}{\partial p^2}\delta t\\[2mm]-\dfrac{\partial^2H}{\partial q^2}\delta t&1-\dfrac{\partial^2H}{\partial p\,\partial q}\delta t\end{vmatrix}=1+O(\delta t^2),\]

so the area element is unchanged to first order in \(\delta t\), and hence for all times. In the language of Section 11.5, the motion itself is a continuous succession of canonical transformations, and canonical transformations preserve phase-space volume. Liouville’s theorem holds for any Hamiltonian system; it fails only when the motion is not Hamiltonian, as for a system with frictional forces, whose phase points crowd together as the energy is dissipated.

11.4 The Jacobian

Changes of variable in phase space are conveniently described by Jacobians, which we now review. If \(u\) and \(v\) are functions of two independent variables \(x\) and \(y\), the determinant

\[\frac{\partial(u,v)}{\partial(x,y)}=\begin{vmatrix}\dfrac{\partial u}{\partial x}&\dfrac{\partial u}{\partial y}\\[2mm]\dfrac{\partial v}{\partial x}&\dfrac{\partial v}{\partial y}\end{vmatrix}\]

is called the Jacobian of \(u\), \(v\) with respect to \(x\), \(y\), sometimes written \(J\!\left(\frac{u,v}{x,y}\right)\). For three functions \(u,v,w\) of \(x,y,z\) the Jacobian \(\partial(u,v,w)/\partial(x,y,z)\) is the \(3\times3\) determinant of the partial derivatives in the same arrangement, and similarly for any number of variables. Its geometrical meaning is that of a scale factor for volumes: \(du\,dv=\left\lvert\dfrac{\partial(u,v)}{\partial(x,y)}\right\rvert dx\,dy\). In polar coordinates, for instance, \(x=r\cos\theta\), \(y=r\sin\theta\) and

\[\frac{\partial(x,y)}{\partial(r,\theta)}=\begin{vmatrix}\cos\theta&-r\sin\theta\\ \sin\theta&r\cos\theta\end{vmatrix}=r,\]

which is the familiar factor in \(dx\,dy=r\,dr\,d\theta\).

Jacobians have three properties that make them easy to manipulate. First, the Jacobian of an inverse transformation is the reciprocal of the original one,

\[\frac{\partial(u,v,w)}{\partial(x,y,z)}\cdot\frac{\partial(x,y,z)}{\partial(u,v,w)}=1.\]

Second, they obey a chain rule: if \(u,v\) are functions of \(r,s\), and \(r,s\) are functions of \(x,y\), then

\[\frac{\partial(u,v)}{\partial(x,y)}=\frac{\partial(u,v)}{\partial(r,s)}\cdot\frac{\partial(r,s)}{\partial(x,y)},\]

which is the multiplication theorem for determinants applied to the chain rule for partial derivatives. Third, if \(u_1,u_2,u_3\) are not given explicitly but are defined implicitly by three relations \(f_k(u_1,u_2,u_3,x_1,x_2,x_3)=0\), then

\[\frac{\partial(u_1,u_2,u_3)}{\partial(x_1,x_2,x_3)}=(-1)^3\,\frac{\partial(f_1,f_2,f_3)/\partial(x_1,x_2,x_3)}{\partial(f_1,f_2,f_3)/\partial(u_1,u_2,u_3)},\]

the factor being \((-1)^n\) for \(n\) variables. Finally, the functions \(u_1,u_2,u_3\) of \(x_1,x_2,x_3\) are functionally dependent, that is, connected by some relation \(F(u_1,u_2,u_3)=0\), if and only if their Jacobian vanishes identically.

For one degree of freedom the Poisson bracket of two functions is itself a Jacobian:

\[\{A,B\}=\frac{\partial A}{\partial q}\frac{\partial B}{\partial p}-\frac{\partial A}{\partial p}\frac{\partial B}{\partial q}=\frac{\partial(A,B)}{\partial(q,p)}.\]

This observation links the two ideas, and it is the key to the theory of canonical transformations.

11.5 Canonical transformations

(a) Definition

In the Lagrangian formulation any choice of generalized coordinates is admissible, and a change of coordinates \(Q_i=Q_i(q,t)\) (a point transformation) leaves the form of Lagrange’s equations unchanged. In the Hamiltonian formulation the coordinates and momenta are independent variables, and it is natural to consider more general transformations that mix them,

\[Q_i=Q_i(q,p,t),\qquad P_i=P_i(q,p,t),\]

together with their inverses \(q_i=q_i(Q,P,t)\), \(p_i=p_i(Q,P,t)\). Not every such transformation is useful. We are interested only in those for which the new variables are again canonical, that is, for which there exists a function \(K(Q,P,t)\) such that the equations of motion in the new variables have Hamilton’s form,

\[\dot Q_i=\frac{\partial K}{\partial P_i},\qquad \dot P_i=-\frac{\partial K}{\partial Q_i}.\]

Such a transformation is called a canonical transformation, and \(K\) plays the role of the Hamiltonian in the new variables. The point of making a canonical transformation is to choose new variables in which \(K\) is as simple as possible, ideally one in which all the \(Q_i\) are cyclic, for then all the \(P_i\) are constants and the \(Q_i\) increase linearly with time.

(b) The Poisson-bracket test

For a transformation that does not involve time, the condition for it to be canonical can be stated simply in terms of Poisson brackets. A transformation \((q,p)\to(Q,P)\) is canonical if and only if the new variables satisfy the fundamental brackets when these are evaluated with respect to the old variables:

\[\boxed{\{Q_i,Q_j\}_{q,p}=0,\qquad \{P_i,P_j\}_{q,p}=0,\qquad \{Q_i,P_j\}_{q,p}=\delta_{ij}}.\]

To see why this is sufficient, consider one degree of freedom and a time-independent transformation, and set \(K(Q,P)=H(q,p)\). Then

\[\dot Q=\frac{\partial Q}{\partial q}\dot q+\frac{\partial Q}{\partial p}\dot p=\frac{\partial Q}{\partial q}\frac{\partial H}{\partial p}-\frac{\partial Q}{\partial p}\frac{\partial H}{\partial q}=\{Q,H\}_{q,p}.\]

Now express \(H\) as a function of \((Q,P)\) and use the chain rule, \(\partial H/\partial q=(\partial K/\partial Q)(\partial Q/\partial q)+(\partial K/\partial P)(\partial P/\partial q)\) and similarly for \(\partial H/\partial p\). The terms in \(\partial K/\partial Q\) cancel and we obtain

\[\dot Q=\frac{\partial K}{\partial P}\,\{Q,P\}_{q,p},\qquad\text{and likewise}\qquad \dot P=-\frac{\partial K}{\partial Q}\,\{Q,P\}_{q,p}.\]

Hence Hamilton’s form is preserved, for every Hamiltonian, precisely when \(\{Q,P\}_{q,p}=1\). Since for one degree of freedom \(\{Q,P\}_{q,p}=\partial(Q,P)/\partial(q,p)\), the condition is equivalently

\[\boxed{\frac{\partial(Q,P)}{\partial(q,p)}=\frac{\partial Q}{\partial q}\frac{\partial P}{\partial p}-\frac{\partial Q}{\partial p}\frac{\partial P}{\partial q}=1}.\]

A canonical transformation therefore preserves phase-space area, \(dQ\,dP=dq\,dp\), and for \(n\) degrees of freedom it preserves the volume element \(\prod_i dq_i\,dp_i=\prod_i dQ_i\,dP_i\)

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Chapter 11 Poisson Brackets, Canonical Transformations and Action–Angle Variables