Classical Mechanics

Herbert Goldstein, Charles P. Poole Jr., John L. Safko

Chapter 9 Canonical Transformations - all with Video Answers

Educators

Chapter Questions

01:41

Problem 1

One of the attempts at combining the two sets of Hamilton's equations into one trics to take $q$ and $p$ as forming a complex quantity. Show directly from Homilton's equations of motion that for a system of one degree of freedom the transformation
\[
Q=q+1 p, \quad P=Q^{*}
\]
is not canonical if the Hamiltomian is left unaltered. Can you find another set of coordinates $Q^{\prime}, P^{\prime}$ that are related to $Q, P$ by a change of scale only, and that are canonical?

Prabhu Ramji

Numerade Educator

03:15

Problem 2

Show that the transformation for a system of one degree of freedom.
\[
\begin{array}{l}
Q=q \cos \alpha-p \sin \alpha \\
P=q \sin \alpha+p \cos \alpha
\end{array}
\]
satisfies the symplectic condition for any value of the parameter $\alpha$. Find a penerating function for the transformation. What is the physical significance of the transformation for $\alpha=0 ?$ For $\alpha-\pi / 2 ?$ Does your generating function work for both of these cases

Manisha Sarker

Numerade Educator

01:44

Problem 3

In Section 8.4 some of the problems of treating time as one of the canonical variables are discussed. If we are able to sidestep these difficulties, show that the equations of transformation in which $t$ is considered a canonical variable reduce to Eqs. (9.14) if in fixt the transformation does not affect the time scale.

Manik Pulyani

Numerade Educator

03:15

Problem 4

Show directly that the transformation
\[
Q=\log \left(\frac{1}{q} \sin p\right), \quad P=q \cot p
\]
is canonical

Manisha Sarker

Numerade Educator

02:12

Problem 5

Show directly that for a system of one degree of freedom the transformation
\[
Q=\arctan \frac{\alpha q}{p}, \quad P=\frac{\alpha q^{2}}{2}\left(1+\frac{p^{2}}{\alpha^{2} q^{2}}\right)
\]
is canonical, where $\alpha$ is an arbitrary constant of suitable dimensions.

Nick Johnson

Numerade Educator

01:48

Problem 6

The transformation cquations hetween two sets of coordinates are
\[
\begin{array}{l}
Q=\log \left(1+q^{1 / 2} \cos p\right) \\
P=2\left(1+q^{1 / 2} \cos p\right) q^{1 / 2} \sin p
\end{array}
\]
(a) Show directly from these transformation equations that $Q, P$ are canonical variables if $q$ and $p$ are.
(b) Show that the function that gencrates this transformation is
\[
F_{3}=-\left(e^{Q}-1\right)^{2} \tan p
\]

Manik Pulyani

Numerade Educator

02:44

Problem 7

(a) If each of the four types of generating functions crist for a given canonical transformation, use the Legendre transformation to derive relations between them.
(b) Find a generating function of the $F_{4}$ type for the identify transformation and of the $F_{3}$ type for the exchange transformation.
(c) For an orthogonal point transformation of $q$ in a system of $n$ degrees of freedom, show that the new mornenta are likewise given by the orthogonal transformation of an $n$ -dimensional vector whose components are the old momenta plus a gradient in configuration space.

James Kiss

Numerade Educator

11:08

Problem 8

Prove directly that the transformation Derivations
\[
\begin{array}{ll}
Q_{1}=q_{1}, & P_{1}=p_{1}-2 p_{2} \\
Q_{2}=p_{2}, & P_{2}=-2 q_{1}-q_{2}
\end{array}
\]
is canonical and find a generating function.

Melanie Richey

Numerade Educator

18:30

Problem 9

(a) For a single particle show directly (that is, by direct evaluation of the Poisson brackets), that if $u$ is a scalar function only of $r^{2}, p^{2},$ and $\mathbf{r} \cdot \mathbf{p} .$ then
\[
[u, \mathbf{L}]=0
\]
(b) Similarly show directly that if $\mathbf{F}$ is a vector function.
\[
\mathbf{F}=u \mathbf{r}+\mathbf{v} \mathbf{p}+w(\mathbf{r} \times \mathbf{p})
\]
where $\psi, v,$ and $w$ are scalar functions of the same type as in part (a), then
\[
\left[F_{i}, L_{j}\right]=\epsilon_{i j k} F_{k}
\]

João Gabriel Alencar Caribé

Numerade Educator

02:03

Problem 10

Find under what conditions
\[
Q=\frac{\alpha p}{x}, \quad P=\beta x^{2}
\]
where $\alpha$ and $\beta$ are constants, represents a canonical transformation for a system of one degree of freedom, and obtain a suitable generating function. Apply the transformation to the solution of the linear harmonic oscillates.

James Kiss

Numerade Educator

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Problem 11

Determine whether the transformation
\[
\begin{array}{ll}
Q_{1}=q_{1} q_{2}, & P_{1}=\frac{P_{1}-p_{2}}{q_{2}-q_{1}}+1 \\
Q_{2}=q_{1}+q_{2} & P_{2}=\frac{q_{2} p_{2}-q_{1} p_{1}}{q_{2}-q_{1}}-\left(q_{2}+q_{1}\right)
\end{array}
\]
is canonical.

Victor Salazar

Numerade Educator

02:02

Problem 12

Show that the direct conditions for a canonical condition are given immediately by the symplectic condition expressed in the form
\[
\left|M-M^{-1}\right|
\]

Adriano Chikande

Numerade Educator

04:11

Problem 13

The set of restricted canconical transformations has a group-property. Verify this statement once using the imariance of Hamilton's principle under canonical transformal bon $(\mathrm{cf} . \mathrm{Eq} .(9.11)),$ and again using the symplectic condition.

Gideon Idumah

Numerade Educator

01:48

Problem 14

Prove that the transformation
\[
\begin{array}{ll}
Q_{1}=q_{1}^{2}, & Q_{2}=q_{2} \sec p_{2} \\
P_{1}-\frac{P_{1} \cos p_{2}-2 q_{2}}{2 q_{1} \cos p_{2}}, & p_{2}=\sin p_{2}-2 q_{1}
\end{array}
\]
is canonicul, by any method you choose. Find a suitable generating function that will Ioad to this transformation.

Manik Pulyani

Numerade Educator

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Problem 15

(a) Using the fundamental Poisson brackets find the values of $\alpha$ and $\beta$ for which the equations
\[
Q=q^{\alpha} \cos f p, \quad P=q^{a} \sin \beta p
\]
represent a canonical transformation.
(b) For what values of $\alpha$ and $\beta$ do these equations represent an extended canonical transformation? Find a generating function of the $F_{3}$ form for the transformation.
(c) On the basis of part (b), can the transformation equations be modified so that they describe a canonical transformation for all values of $\beta ?$

Nick Johnson

Numerade Educator

01:29

Problem 16

For a symmetric rigid body. obtain formulas for evaluating the Poisson brackets
\[
[\phi, f(\theta, \phi, \psi)], \quad[\psi, f(\theta, \phi, \psi)]
\]
where $\theta, \phi,$ and $\psi$ are the Eulcr angles, and $f$ is any arbitrary function of the Euler angles

Ajay Singhal

Numerade Educator

03:54

Problem 17

Show that the Jacobi identity is satisfied if the Poisson bracket sign stands for the contmutator of two square matrices:
\[
[\mathbf{A}, \mathbf{B}]=\mathbf{A B}-\mathbf{B A}
\]
Show also that for the same representation of the Poison hracket that
$| \mathbf{A}, \mathbf{B C}]-[\mathbf{A}, \mathbf{B}|\mathbf{C}+\mathbf{B}| \mathbf{A}, \mathbf{C}]$

Lucas Finney

Numerade Educator

10:50

Problem 18

Prove $\mathrm{Eg}$. (9.83) using the symplectic matrix notation for the Lagringe and Poisson hrackets.

Chris Trentman

Numerade Educator

01:53

Problem 19

Verify the annlog of the Jacobi identity for Lagrange brachets,
\[
\frac{\partial[u, v]}{\partial t e}+\frac{\partial f v, w\}}{\partial u}+\frac{\partial(w, u)}{\partial v}=0
\]
where $14,1,$ and $w$ are three functions in tems of which the $(q, p)$ set can be specified.

Angela Guo

Numerade Educator

17:42

Problem 20

(a) Verify that the components of the two-dimensicand matrix A defined by Eq $(9.141),$ are constants of the motion for the two-dimensional isotropic harmicnic oscillator problem.
(b) Verify that the quantities $S_{i}, i=1,2,3,$ defined by Eqs. (9.144),(9.145),(9.146) have the properties stated in Eqs. (9.147) and (9.148)

Prachita Kush

Numerade Educator

04:31

Problem 21

(a) For a one-dimensional system with the Hamiltonim
\[
H=\frac{p^{2}}{2}-\frac{1}{2 q^{2}}
\]
show that there is a constant of the motion
\[
D=\frac{p q}{2}-H t
\]
(b) As a generaliration of part (a), for motion in a plane with the Hamiltonian
\[
H=|\mathbf{p}|^{n}-a r^{-n}
\]
where $\mathbf{p}$ is the vector of the mornenta conjugate to the Cartesian cocratinates. show that there is a constant of the motion
\[
D=\frac{\mathbf{p} \cdot \mathbf{r}}{n}-H t
\]
(c) The transformation $Q=\lambda q, p=\lambda P$ is obviously canonical. However, the sume transformation with $t$ time dilatation, $Q=\lambda q, p=\lambda P, t^{\prime}=\lambda^{2} t,$ is not. Show that, however, the equations of motion for $q$ and $p$ for the Hamiltonian in part (a) are invariant under this transformation. The constant of the motion $D$ is sad to he associated with this imvariance.

James Kiss

Numerade Educator

01:15

Problem 22

For the point transformation in a system of two deyrees of freedom,
\[
Q_{1}=q_{1}^{2}, \quad Q_{2}=q_{1}+q_{2}
\]
find the most gencral trinsformation cquations for $P_{1}$ and $P_{2}$ consistent with the over. all transformation bcing canonical. Show that with a perticular choice for $P_{1}$ and $P_{2}$ the Hamiltonian
\[
H=\left(\frac{p_{1}-p_{2}}{2 q_{1}}\right)^{2}+p_{2}+\left(q_{1}+q_{2}\right)^{2}
\]
can be transformed to one in which both $Q_{1}$ and $Q_{2}$ are ignorable. By this means solve the problem and obtain expressions for $q_{1}, 42, p_{1},$ and $p_{2}$ as functions of time and their initial values.

James Kiss

Numerade Educator

03:43

Problem 23

By any method you choose, show that the following transformation is canonical:
\[
\begin{array}{ll}
x=\frac{1}{\alpha}\left(\sqrt{2 P_{1}} \sin Q_{1}+P_{2}\right), & p_{2}=\frac{\alpha}{2}\left(\sqrt{2 P_{1}} \cos Q_{1}-Q_{2}\right) \\
y=\frac{1}{\alpha}\left(\sqrt{2 P_{1}} \cos Q_{1}+Q_{2}\right), & p_{2}=\frac{\omega}{2}\left(\sqrt{2 P_{1}} \sin Q_{1}-P_{2}\right)
\end{array}
\]
where $c$ is sorne fixed parameter. Apply this transformation to the problem of a particle of charge $q$ moving in a plane that is perpendicular to a constant magnetic field B. Express the Hamiltonian for this problem in the $\left(Q_{1}, P_{1}\right)$ coordinates letting the parameter $\alpha$ take the form
\[
\alpha^{2}=\frac{q B}{c}
\]
From this Hamiltonian, obtain the motion of the particle as a function of time.

Mahnoor Amin

Numerade Educator

05:23

Problem 24

(a) Show that the transformation
\[
Q=p+\operatorname{taq}, \quad P=\frac{P-i a q}{2 i a}
\]
is cancnical and find a generating function.
(b) Use the transformation to solve the linear harmonic oscillator problem.

Mehdi Hatefipour

Numerade Educator

03:22

Problem 25

(a) The Hamiltorian for a systcm has the form
\[
H=\frac{1}{2}\left(\frac{1}{q^{2}}+p^{2} q^{4}\right)
\]
Find the equation of motion for $q$
(b) Find a canonical transformation that reduces $H$ to the form of unmonic oscillator. Show that the solution for the transformed variables is such that the equation of motion found in part (n) is satisfied.

Deepak Kohli

Numerade Educator

02:05

Problem 26

A system of $n$ particles moves in a plane under the influence of interaction forces derived from potential terms depending conly upon the scalar distances between particles.
(a) Using plane polar conrdinates for cach particlo (relative to a common crigin). idcntify the form of the Hamiltonian for the system.
(b) Find a gencrating function for the canonical trinsformation that corresponds to a transformation to coordinates rotating in the plane counterclockwise with a uniform angular rate $\omega$ (the same for all particles). What are the transformation equations for the momenta?
(c) What is the new Hamiltonian? What physical significance can you give to the difference botween the old and the new Hamiltorians?

Chai Santi

Numerade Educator

01:10

Problem 27

(a) In the problem of small oscillations about stesdy motion, show that at the point of steady motion all the Hamiltcatian variables $\eta$ are constant. If the values for steady motion are $\eta_{0}$ so that $\eta-\eta_{0}+\xi$. show that to the lowest nonvanishing approximation the effective Hamiltontan for small oscillation can be expressed as
\[
H\left(\boldsymbol{\eta}_{0} \cdot \boldsymbol{\zeta}\right)=\frac{1}{2} \boldsymbol{\xi} \boldsymbol{S} \boldsymbol{\xi}
\]
where $\mathrm{S}$ is a square matrix with components that are functions of $\eta_{0}$ only.
(b) Assuming all frequencies of small oscillation are distinct, Ict M be a square $2 n \times$ $2 n$ matrix formed by the components of a possible sct of cigchvectons (for both positive and negative frequencies). Only the directions of the eigenvectors are fixed, not their magnitudes. Show that it is possible to apply conditions to the cigchvectors (in cficct fixing their magnitudes) that make $\mathrm{M}$ the Jacobian matrix of a canonical transformation.
(c) Show that the cancrical transformation so found transforms the effective Hamiltomian to the form
\[
H=i \omega_{j} q_{j} p_{j}
\]
where $\omega$, is the magnitude of the normal froqucncics. What are the cquations of motion in this sct of canonical coordinates?
(d) Finally, show that
\[
F_{2}=q_{j} P_{j}+\frac{i}{2} \frac{P_{j}^{2}}{\omega_{j}}-\frac{i}{4} \omega_{j} q_{j}^{2}
\]
Leads to a canonical transformation that decomposes $H$ into the Homilionians for a set of uncoupled linear harmonic oscillators that cscillate in the normal modes.

Dominador Tan

Numerade Educator

04:04

Problem 28

A charged particle moves in space with a constant magnetic ficid $\mathbf{B}$ such that the vector potential, A. is
\[
\mathbf{A}=\frac{1}{2}(\mathbf{B} \times \mathbf{r})
\]
(a) If $w$, are the Cartesian components of the velocity of the particle, evaluate the Potsson brackets
\[
\left[w_{i}, v_{j}\right], \quad f \neq j=1,2,3
\]
(b) If $p_{i}$ is the canonical momentum conjugate to $4,$ also evaluate the Poisson hrackets
\[
\begin{array}{l}
{\left[x_{i}, v_{j}\right], \quad\left[p_{i}, v_{j}\right]} \\
{\left[x_{i}, \dot{p}_{j}\right]_{.} \quad\left[p_{i}, \dot{p}_{j}\right]}
\end{array}
\]

Zhaojie Xu

Numerade Educator

01:48

Problem 29

The scrinimajor aris a of the elliptical Kepler orbit and the eccentricity $e$ are furictions of first integrais of the motion, and thercfore of the canonical variables. Stmilarly, the mean anomaly
\[
\phi=\omega(t-T)=\psi-e \sin \psi
\]
is a function of $r, \theta$, and the conjugate momenta. Here $T$ is the time of periapsis passage and is a constant of the motion. Evaluate the Poisson brickets that can be formed of $a, e, \phi, \omega,$ and $T .$ There are in fact only nine nonvanishing distinct Poiscon brackets out of these quantities.

Amrita Bhasin

Numerade Educator

01:28

Problem 30

(a) Prove that the Poison bracket of two constants of the motion is itself a constant of the motion cuen when the constants depend upon time cxplicitly.
(b) Show that if the Hamiltonian and a quantity $F$ are constants of the motion, then the $n$ the partial derivative of $F$ with respect to $t$ must also be a constant of the motion.
(c) As an illustration of this result. consider the uniform motion of a free particle of mass $m$. The Hamiltonian is certainly conserved. and there exists a constant of the motion
\[
F=x-\frac{p t}{m}
\]
Show by direct computation that the partial derivative of $F$ with $t$. which is a constant of the thotion, agrees with $[H, F]$

Penny Riley

Numerade Educator

02:28

Problem 31

Show by the use of Poisson brackets that for a onc-dimensional harmonic oscillator there is a constant of the motion $u$ defined as
\[
u(q \cdot p, t)=\ln (p+i m \operatorname{cov})-i \omega t, \quad \omega=\sqrt{\frac{k}{m}}
\]

Adriano Chikande

Numerade Educator

01:59

Problem 31

Show by the use of Poisson brackets that for a onc-dimensional harmonic oscillator there is a constant of the moun $u$ defined as
\[
u(q, p, t)=\ln (p+i m \cos p)-i \omega t, \quad \omega=\sqrt{\frac{k}{m}}
\]
What is the physical significance of this constant of the motion?

Surendra Kumar

Numerade Educator

04:31

Problem 32

A systcm of two degrecs of frecdom is described by the Harniltonian
\[
H=q_{1} p_{1}-q_{2} p_{2}-a q_{1}^{2}+b q_{2}^{2}
\]
Show that
\[
F_{1}=\frac{p_{1}-a q_{1}}{q_{2}} \text { and } F_{2}=q_{1} q_{2}
\]
are constants of the motica. Are there any other independent algchraic constants of the motion? Can any be constructed from Jacobi's identity?

James Kiss

Numerade Educator

00:59

Problem 33

Set up the magnetic monopole described in Exercise 28 (Chapter 3 ) in Hamiltonian formulation (you may want to use spherical polar cocrulinates). By means of the Poisson bracket formulation, show that the quantity $D$ defined in that exercise is conserved.

Mayukh Banik

Numerade Educator

02:10

Problem 34

Obtain the motion in time of a linear harmonic ascillator by means of the formal solution for the Poisson bracket version of the equation of motion as derived from Eq. $(9.116) .$ Assume that at time $t=0$ the initial values are $x_{0}$ and $p 0$

Penny Riley

Numerade Educator

02:10

Problem 35

A particle moves in one dimension under a potential
\[
V=\frac{m k}{x^{2}}
\]
Find $x$ as a function of time. by using the symbolic solution of the Poisson bracket form for the cquation of motion for the quantity $y=x^{2}$. Initial conditions are that at
\[
t-0, x=x_{0} \text { . and } v=0
\]

Penny Riley

Numerade Educator

01:13

Problem 36

(a) Using the theorem concerning Poisson brackets of vector functions and compo. nents of 'the angular momentum. show that if $\mathbf{F}$ and $\mathbf{G}$ are two vector functions of the coordinates and momenta only, then
\[
\left.[\mathbf{F} \cdot \mathbf{L}, \mathbf{G} \cdot \mathbf{L}]=\mathbf{L} \cdot(\mathbf{G} \times \mathbf{F})+L_{i} L_{j} | F_{i}, G_{j}\right]
\]
(b) Let $\mathbf{L}$ be the total angular momentum of a rigid body with one point fixcd and Iet $L_{n}$ be its component along a set of Cartesian axes fixed in the rigid body. By means of part (a) find a gencral expansion for
\[
\left[L_{u}, L_{v}\right], \quad \mu, v=1.2,3
\]
(Hint: Choose for $\mathbf{F}$ and $\mathbf{G}$ unit vectors along the $\mu$ and $\vee$ axes.)
(c) From the Poisson bracket cquations of motion for $L_{14}$ derive Euler's equations of motion for a rigid body.

Narayan Hari

Numerade Educator

19:29

Problem 37

Set up the problem of the sphencal pendulum in the Hamiltonian formulation. using spherical polar coordinates for the 4 ). Evaluate directly in terms of these canonical variables the following Poissen brickets:
\[
\left.| L_{x}, L_{y}\right], \quad\left|L_{y}, L_{z}\right|, \quad\left[L_{z}, L_{x}\right]
\]
showing that they have the values predicted by $\mathrm{Fa}$. $(9.128) .$ Why is it that $p_{4}$ and $p_{y}$ can be used as canonical momenta, although they are perpendicular components of the angular momentum?

Hafiz Shahzaib

Numerade Educator

01:29

Problem 38

In Section 9.7 , it is shown that if any two components of the angular mornenturn are conserved, then the total angular momentum is conserved. If two of the components are identically zero, the third must be conserved. From this it would appear to follow that in any motion confined to a plane, so that the components of the angular momentum in the plane are $x$ ero, the total angular momentum is constant. There appent to be a number of obvious contradictions to this prediction: for example, the angular momentum of an oscillating spring in a watch, or the angular momentum of a plane disk rolling down an inclincd plane all in the same vertical plane. Discuss the force of these objections and whether the statement of the theorem requires any restrictions.

Adriano Chikande

Numerade Educator

02:57

Problem 39

(a) Show from the Poisson bracket condition for conserved quantities that the Laplace Runge-Lenz vector $\boldsymbol{\Lambda}$
\[
\mathbf{A}=\mathbf{p} \times \mathbf{L}-\frac{m k \mathbf{r}}{r}
\]
is a constant of the motion for the Kepicr problem.
(b) Venfy the Poisson bracket relations for the components of $\mathbf{A}$ as given by
\[
\mathrm{Eq} \cdot(9,131)
\]

Eduard Sanchez

Numerade Educator

02:55

Problem 40

Consider a system that consists of a rigid body in three-space with one point fixed. Using cylindrical coordinates find the cancrical transformation corresponding to new axes rotating about the $z$ -axis with an arbitrary time-dependent angular velocity. Ver. ify that your proposed solution is canonical.

Nick Johnson

Numerade Educator

02:28

Problem 41

We start with a time independent Hamiltonian $H_{o}(q, p)$ and impose an extcreal oscillating ficld making the Hamiltonian
\[
H=H_{0}(q, p)-\varepsilon \sin \omega t
\]
where $z$ and $\omega$ are given constants.
(a) How are the canonical cquations modificd?
(b) Find a canomical transformation that restores the canonical form of the equations of motion and determine the "new" Hamiltonian.
(c) Give a possible physical interpretation of the imposed ficld.

Penny Riley

Numerade Educator