PHYS 601 (Fall 2018): Theoretical Dynamics Homeworks. 1 Homework 1 (due Friday, September 7 in folder outside Room 3118 of PSC by 5 pm.

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1 PHYS 601 (Fall 2018): Theoretical Dynamics Homeworks Note: the 1st homework is simply signing the honor pledge (but still it is compulsory); the actual homework starts with #2. And, please sign the honor pledge on a separate sheet of paper from HW 2. 1 Homework 1 (due Friday, September 7 in folder outside Room 3118 of PSC by 5 pm.) Print your name and sign (with date) the Honor Pledge (which covers all assignments, including exams): I pledge on my honor that I will not give or receive any unauthorized assistance (including from other persons and online sources) on all examinations and homework assignments in this course. General guidelines for homework (1). Please read the problem number (when it is from the textbook by Goldstein, Poole and Safko, abbreviated as GPS) correctly. In particular, note that the number to left of decimal point refers to the relevant chapter and the one to the right is the actual problem number. Also, the problem (and in some cases, reference to section) numbers correspond to the third (domestic) edition of GPS: the numbering might be different in other editions. So, a brief description of the problem will be given (alongside the problem number) in order to help you identify the correct problem from your edition of GPS, in case it is not the third (domestic): as a double-check, you might then have to refer to a colleague s third, domestic edition. Finally, some some printings/editions of GPS have typos: for a list, see (2). Many of the following homework problems have multiple parts. So, it is your responsibility to read the full statement of the problem carefully. (3). Most (if not all!) of the problems here are not mathematical ones, i.e., are instead (almost purely) physics-based. So, unless it is (explicitly) stated otherwise, feel free to use any computer programs (such as mathematica) for this solving them or to look up (purely)

2 mathematical formulae in reference tables (including online). However, if you take such a step, please indicate what exactly you did here (this will help us grade appropriately). (4). You are welcome to ask for help (for example, hints) on homework from the instructor or TA. Also, limited discussion with other students is allowed (and encouraged): for example, just in order to get started, but the actual problem-solving part should be your own work. And, you should try not to get any other outside help. (5). In order to get full credit for homework (and exam) problems, you should show as many steps as you can. (6). Every homework assignment will (typically) have 5 problems, with each problem being worth (a maximum of) 10 points. (7). Homework will have a total weightage of 50% of the final grade, which will be divided equally among all the assignments. 2 Homework 2 (due Friday, September 7 in folder outside Room 3118 of PSC by 5 pm.) 2.1 Lagrangian with second time derivative of coordinate Exercise 2.12 of GPS. (The idea here is to analyze Lagrangians containing higher-than-first time-derivatives of coordinates, called generalized mechanics. It is an exercise in using the variational principle.) 2.2 Lagrangian for relativistic particle [The next two problems are from David Tong (DT) s University of Cambridge lecture notes.] The Lagrangian for a relativistic point particle of mass m (with coordinate r) is L = mc 2 1 (ṙ.ṙ) /c 2 V (r) (1) where c is the speed of light. Derive the equation of motion. Show that it reduces to Newtons equation in the limit ṙ c. Hint: it might be convenient to first compute Lagrange s equation for one of the Cartesian coordinates, say, x and then argue how the result can be generalized to the full position vector (r). (We will of course discuss the special theory of relativity in more detail later on in this course; for the above problem, just treat time and space differently, as in non-relativistic mechanics.)

3 2.3 Two different Lagrangians giving same equations of motion A particle moves in one-dimension with position x and potential V (x), governed by the Lagrangian: L = 1 12 m2 ẋ 4 + mẋ 2 V V 2 (2) Show that the resulting equation of motion is identical to that arising from the more traditional L = 1 2 mẋ2 V. (Note: be careful with taking time-derivative of V : for a similar subtlety, see incorporation of Lorentz force in Lagrangian formulation done in lecture.) 2.4 Change of coordinates/variables [This problem is from Phys601 taught by Thomas Cohen (TC).] Consider the Lagrangian for a simple 1-dimensional harmonic oscillator: where ω 0 is a parameter. (i) Find the equation of motion from Lagrange s equations. L(x, ẋ) = 1 2 mẋ2 1 2 mω2 0x 2 (3) Next, consider a change of variables to the generalized coordinate q = sinh 1 (x). (ii) Using the result of part (i), find the equation of motion for q. (iii) Find the Lagrangian for q [i.e., find L (q, q), using the notation from lecture]. (iv) Find the Lagrange s equation of motion for L (q, q). (v) How do the results of parts (ii) and (iv) compare? Why is this expected? 2.5 System with constraint Exercise 2.20 of GPS. (This problem involves a particle sliding on a wedge, which, in turn, moves on a horizontal surface.) Hint for constraint problems in general: it might be useful to solve using simply the Newtonian formalism, either to begin with or as a cross-check. Similar examples are worked out in Fig. 2.5 on page 50 of GPS and on top of page 47, just after Eq See also the case of simple pendulum discussed in lecture (and posted note). Note for constraint problems in general: the beginning step for solving problems where constraint is to be computed are standard, i.e., add terms to Lagrangian with Lagrange

4 multipliers multiplying the constraint equations, as in L at the top of the posted note on simple pendulum); then Lagrange s equations for the usual/actual coordinates will contain (in general) Lagrange multipliers (which, in turn, can be identified with constraint forces) as in Eqs. 1, 2, while Lagrange s equation for multiplier will simply enforce the constraint (just above Eq. 1). The above system of equations can (in principle at least) be solved for the coordinates and constraint forces as functions of time (as mentioned just below Eq. 2 of the posted note on simple pendulum). Having said this, there (unfortunately) might not be a fixed recipe to go about this 2nd step (to the best of my knowledge), i.e., it might have to be handled on a case-by-base basis. In particular, in some problems, one is asked to compute constraint force as function of coordinate instead, in which case clever manipulations can get you there without actually solving for quantities as functions of time (as in the discussion on simple pendulum: see Eq. 3). 3 Homework 3 (due Monday, September 17 in folder near Room 3118 of PSC by 5 pm.) 3.1 Normal reaction force using Lagrange multiplier Exercise 2.13 from GPS. Note: the vertical hoop (at the top of which the heavy particle is placed) is assumed to be fixed. 3.2 Canonical momentum; electromagnetic forces Derivation 2.2 from GPS. (This problem concerns Lagrangians containing velocity-dependent terms. For the 2nd part on electromagnetic forces, you can refer to lecture notes or section of DT or section 1.5 of GPS.) 3.3 Constant of motion for system with constraint Exercise 2.18 from GPS. Note: think of the point mass as a bead threaded on a hoop and moving along the (rotating) hoop without friction. It might be useful to sketch the setup first. Then, derive the Lagrangian for the single degree of freedom that this system has. 3.4 Conserved quantities for a two-dimensional harmonic oscillator, including one not associated with Noether s theorem (This problem is from Phys601 taught by TC.)

5 Consider a general two-dimensional harmonic oscillator: L (x, y; ẋ, ẏ) = 1m 2 (ẋ2 + ẏ 2 ) 1 2 mω2 (x 2 + αy 2 ), where α is a parameter specifying the degree of anisotropy. (i) Is energy, E = 1m 2 (ẋ2 + ẏ 2 ) mω2 (x 2 + αy 2 ) conserved? (ii) Is linear momentum conserved? (iii) What about angular momentum (especially for certain value of α)? Explain your answers in above parts. (iv) Use the equations of motion to show that there is another conserved quantity = 1 2 m (ẋ2 ẏ 2 ) mω2 (x 2 αy 2 ). (Think about and write briefly whether conservation of this quantity can be derived from a continuous coordinate transformation.) 3.5 Two masses connected to three springs Exercise 6.12 from GPS. Note: here is the (rough) general strategy for solving such problems (this should have been clear from the example of double pendulum done in detail in lecture: see also DT s sec ): (1).Obtain equations of motion (EOM s) as Lagrange s equations. (2).Determine the equilibrium configuration, i.e., with (net) force on each particle vanishing. (3).Taylor expand EOM s for small motion around equilibrium: you should be able to reduce it to the form in Eq. 1 of note posted at (which is Eq of DT), where η is a column-vector of small displacements and F is (n n) matrix. (4).Find eigenvalues/vectors of above F : general solution for small motion (we are interested in oscillations) is then given by these eigen-quantities as in Eq. 14 of above note (Eq of DT). 4 Homework 4 (due Monday, October 1 in folder near Room 3118 of PSC by 5 pm.) 4.1 Vibrations of a linear tri-atomic molecule The linear triatomic molecule consists of two identical outer atoms of mass m and a middle atom of mass M. It is a rough approximation to CO 2. The interactions between neighbouring atoms are governed by a complicated potential V (x i x i+1 ). If we restrict attention to motion in the x direction parallel to the molecule, the Lagrangian is L = 1 2 mẋ Mẋ mẋ2 3 V (x 1 x 2 ) V (x 2 x 3 ) (4) where x i is the position of the i th particle. Define the equilibrium separation r 0 = x i x i+1 of this system.

6 (i). Write down the equation describing small deviations from equilibrium in terms of the masses and the quantity k = 2 V (r) r 2 r=r0 (5) (ii). Show that the system has three normal modes and calculate the frequencies of oscillation of the system. (iii). What are the corresponding eigenvectors? Briefly describe (or better yet, draw rough pictures of) the motions in the above modes. In particular, one of these frequencies vanishes: what is the interpretation of this? 4.2 Two (electrically) charged masses connected to three springs Exercise 6.13 from GPS. Added part Compute the eigenfrequencies also. Hint: Will the equilibrium configuration in this case be the same as in HW 3.5 above? In more detail, let L denote an appropriate distance characterizing the equilibrium configuration. Setting up an equation for determining L (in terms of parameters of the potential) should be rather straightforward, but solving it might be non-trivial (there might not be an explicit/analytic solution) so that you are not expected to go through with this second step. Nonetheless, you should try to argue (either by hand or mathematica etc.) that a physically reasonable solution does exist. Your answer for the eigenfrequencies can be given in simply terms of L (i.e., without actually solving for it) and other parameters of the potential. 4.3 Power-law potentials With a maximum Consider a particle of mass m moving under the influence of the central force potential given by V (r) = a r 3 (6) which is discussed around Fig. 3.9 of GPS. (i) From the effective one-dimensional potential approach, for a given value of the angular momentum (l), find the value of the energy for which only a circular orbit is allowed. (ii) What are the kinetic and potential energies for this orbit? Does their ratio agree with the expectation based on the virial theorem?

7 4.3.2 Monotonic Suppose the central force potential is instead as follows: V (r) = b r 2 (7) Again, using the effective one-dimensional potential, (i) find the range of values of the angular momentum such that only unbounded motion is allowed? What range of energy does this motion correspond to? (ii) For other values of the angular momentum, determine the type(s) of motion allowed and the associated energies. 4.4 Collision of a comet with a planet moving in an elliptical orbit Exercise 3.10 from GPS. 4.5 Radial speed for elliptical orbit Exercise 3.24 from GPS. 5 Homework 5 (due Thursday, October 18 in folder near Room 3118 of PSC by 5 pm.) 5.1 Solving Kepler equation Derivation 3.2 from GPS Hint: you can start as you would for typical Fourier series problems, i.e., write a (formal) expression for those coefficients; then it is a series of manipulations (involving use of Kepler equation etc.). It should help that the answer is given: you might have to look up Bessel function formulae in some standard reference. 5.2 Yukawa potential: combination of power-law and exponential Exercise 3.19 from GPS: A particle moves in a force field described by the Yukawa potential: where k and a are positive. V (r) = k ( r exp r ) a (8) (a) Write the equations of motion and reduce them to the equivalent one-dimensional problem. Use the effective potential to discuss the qualitative nature of the orbits for different values of the energy and the angular momentum.

8 (b) Show that if the orbit is nearly circular, the apsides will advance approximately by π (ρ/a) 2 per revolution, where ρ is the radius of the circular orbit. Note: there is a typo in expression in part (b) above in some versions of GPS, where it says πρ/a instead. Also, you might need to assume ρ/a 1 here. Hints for part (b): Since the orbit is close to being circular, you can perturb around it, keeping only leading (non-trivial) powers of that deviation in the radial distance (r). Think about how you can determine the evolution of this deviation (again from the constant radius of a circular orbit), for example, either using Eq of GPS (this part was not done in lecture, but you can read it on your own) or using the original EOM. Also, the basic idea behind what is asked for here is as follows. Since the potential is neither r nor quite 1/r (although it s sort of a modified version of the latter), as per Bertrand s theorem (GPS sec. 3.6), the orbit will not be closed (even though it s close to being so, since it is, in turn, almost circular). Thus, between two consecutive closest (or farthest) approaches (apsidal distances), θ will not exactly change by 2π (as would be the case for a closed orbit, whether circular or elliptical), although (again) this shift in θ will be close to 2π, given the approximations made. That is what is meant by the statement in the problem about apsides advancing by the (small) amount shown above. 5.3 Differential cross-section for (repulsive) inverse cube law force law Exercise 3.31 from GPS. Note: as a possible (general) strategy for these (three) scattering problems (but no warranty is expressed or implied here!), you can use the general expression in Eq (or 3.97) of GPS (also done in lecture) for relating scattering angle (θ) to impact parameter (s). Recall that r m in Eq (the distance of closet approach: see Fig of GPS) is to be obtained by setting the total energy (E) equal to potential energy, plus angular kinetic energy, since radial part of kinetic energy ( ṙ 2 ) vanishes. Via Eq of GPS (for fixed E), r m is then a function of impact parameter (s). Finally, Eq of GPS gives cross-section once we know relation between s and θ as above. However, in some cases, you might know the trajectory (orbit equation) by other means so that you can by-pass actually using the above formula. Recall that this is how we solved Rutherford scattering, i.e., we knew orbit is hyperbola etc. (well, in this case, we had actually done the same integral already as part of Kelper problem studied earlier!). 5.4 Truncated repulsive Coulomb potential Exercise 3.34 from GPS. Note: you might wish to first determine (in terms of energy E of the particles and parameters of the potential) the impact parameter s 0 for which (as stated in the problem) periapsis (i.e., closet approach) occurs at r = a.

9 You might then have to consider the two cases s s 0 and s < s 0 separately. What will be distances of closet approach as compared to a for these two ranges of s? For one of these cases of s, can you then simply re-use formulae from Rutherford scattering that was done in lecture (and in GPS)? Also, you might find the discussion around Fig of GPS (this is a generic truncated repulsive potential and was also done in lecture) useful here. 5.5 Attractive potential (arising in nuclear physics) Exercise 3.32 from GPS. Note: you might find the discussion around Fig of GPS (which includes an attractive contribution to the potential) useful here. Also, just to be clear, this problem is in two dimensions (as usual, r is radial coordinate etc.) so that perhaps a more appropriate name would be circular well (instead of rectangular used in GPS). Hints: again, think about whether there is an easy way to figure out trajectory (even if it might be in a piece-wise fashion). Also, it should help that the answer is sort of given (i.e., you are told that scattering is like light rays). For the very last part, i.e., total cross-section, as a cross-check, you might wish to think whether you could have gotten the same answer simply based on geometry. 6 Homework 6 (due Friday, November 2 in folder near Room 3118 of PSC by 5 pm.) 6.1 (Recoil) Angle of target particle and energies of both Derivation 3.7 from GPS Note: you are supposed to assume that scattering is elastic (but the two masses need not be equal) in order to get the final result even in part (a) [just like explicitly instructed in part (b)]. Note: in part (b), probability distribution for energy just means differential cross-section, but with energy as variable (instead of θ s): so, you just have to (appropriately) re-express usual σ (θ) or σ (θ lab ) in terms of the new variable (i.e., energy), like what was done in going between 2 θ s themselves in Eq of GPS (also discussed in lecture): basically these cross-sections tell you what is the distribution of the corresponding variable. 6.2 Relating scattering angle in laboratory frame to energies of incident particle Derivation 3.8 from GPS

10 Note: a related derivation for the case of elastic scattering is given around Eq of GPS (also mentioned in lecture). (The next two problems are from DT s University of Cambridge lecture notes.) 6.3 Moments of inertia: specific solid (I) Consider a solid uniform cylinder of radius r, height 2 h and mass M. (i) Calculate the moments of inertia about its centre of mass (i.e., corresponding to the three principal axes). (ii) For what height-to-radius ratio does the cylinder spin like a sphere? (Hint: what is the relation between the moments of inertia about the three principal axes for a sphere?) 6.4 Moments of inertia: specific solid (II) Calculate the moments of inertia of a uniform ellipsoid of mass M, defined by x 2 a 2 + y2 b 2 + z2 c 2 R 2 (9) with respect to the (x, y, z) axes with origin at the centre of mass. (Hint: with a change of coordinates, you can reduce this problem to that of the solid sphere). 6.5 Angular velocity in terms of Euler angles Show that the angular velocity ω can be expressed in terms of Euler angles as [ ] [ [ ω = φ sin θ sin ψ + θ cos ψ e 1 + φ sin θ cos ψ θ sin ψ]e2 + ψ + φ ] cos θ e 3 (10) in the body frame {e a }. [Note that part of this derivation was already done in lecture and is in DT s notes, but in addition to filling-in the gaps therein, you should re-do (and show) all the steps given in the notes.] Show also that, alternatively, it can be written as [ ω = ψ sin θ sin φ + θ [ cos φ] ẽ 1 + ψ sin θ cos φ + θ ] [ ] sin φ ẽ 2 + φ + ψ cos θ ẽ 3 (11) in the space frame {ẽ a }. 7 Homework 7 (due Tuesday, November 13 during lecture or in folder near Room 3118 of PSC by 5 pm.) (The first two problems are from DT s University of Cambridge lecture notes.)

11 7.1 Moments of inertia: general solid Show that for any solid, the sum of any two principal moments of inertia is not less than the third. For what shapes is the sum of two equal to the third? 7.2 Free Euler s equations for a lamina A rigid lamina (i.e. a two dimensional object) has principal moments of inertia about the centre of mass given by, I 1 = ( µ 2 1 ), I 2 = (µ 2 + 1), I 3 = 2µ 2 (12) Write down Eulers equations for the lamina moving freely in space. Show that the component of the angular velocity in the plane of the lamina (i.e. ω ω 2 2) is constant in time. Choose the initial angular velocity to be ω = µne 1 + Ne 3. Define tan α = ω 2 /ω 1, which is the angle the component of ω in the plane of the lamina makes with e 1. Show that it satisfies d 2 α dt 2 + N 2 cos α sin α = 0 (13) and deduce that at time t, [ ] [ ] [ ] ω = µn (1/ cosh Nt) e 1 + µn tanh N t e 2 + N (1/ cosh Nt) e 3 (14) 7.3 A falling hinge, i.e., not quite a rigid body Exercise 5.13 from GPS. Hint: you can try using simply conservation of energy here. 7.4 Rolling cone Exercise 5.17 from GPS (You will need the moments of inertia of a cone about various axes: you can just look them up or better yet, derive them!) Hint: It might be simpler to decompose the angular velocity (and momentum) along bodyaxes, expressing these components in terms of Euler angles. 7.5 Euler s equations with torque Exercise 5.18 from GPS Eulers equations with torque are discussed briefly below Eq of DT and in Eq of GPS: in particular, N i in latter are obviously components of torque along the body-axes.

12 8 Homework 8 (due Tuesday, November 20 during lecture or in folder near Room 3118 of PSC by 5 pm.) 8.1 Using Euler s equations (with torque) for heavy, symmetric top Derivation 5.9 of GPS. The idea behind this problem (see also a related one done in lecture) is to (re-)derive some of the results (in terms of the Euler angles) for the heavy, symmetric top, but this time using Euler s equations of motion, i.e., those for the components of angular velocity along the body-frame axes (instead of the Lagrangian formalism done in lecture/dt s notes and GPS). This procedure can at times be a bit tedious, but the good news is that you already know the final answer. As a corollary, most (if not all!) the credit for this problem is for explicitly showing all the steps. As mentioned in HW 7.5 above, Eulers equations with torque are discussed briefly below Eq of DT and in Eq of GPS. (The next three problems are from DT s University of Cambridge lecture notes.) 8.2 Nutating heavy symmetric top As discussed in lecture, consider a heavy symmetric top of mass M, pinned at point P which is a distance l from the centre of mass. The principal moments of inertia about P are I 1, I 1 and I 3 and the Euler angles θ, φ and ψ are as usual. The top is spun with initial conditions φ = 0 and θ = θ 0. (i) Show that θ obeys the equation of motion, where I 1 d 2 θ dt 2 = V eff(θ) θ (15) V eff (θ) = I2 3ω3 2 (cos θ cos θ 0 ) 2 2I 1 sin 2 + M g l cos θ (16) θ (ii) Suppose that the top is spinning very fast so that I 3 ω 3 M g l I 1 (17) Show that θ 0 is close to the minimum of V eff (θ). Use this fact to deduce that the top nutates with frequency Ω ω 3I 3 I 1 (18) [Note: the initial parts of this problem were worked out in lecture, following DT s notes, but it will be nice if you re-derive (and show) it here.]

13 8.3 Hamiltonian for heavy symmetric top The Lagrangian for the heavy symmetric top is L = 1 ( 2 I θ2 1 + φ ) 2 sin 2 θ + 1 ( 2 I 3 ψ + φ 2 cos θ) M g l cos θ (19) Obtain the momenta p θ, p φ and p ψ and the Hamiltonian H (θ, φ, ψ, p θ, p φ, p ψ ). Derive Hamiltons equations. [In lecture (and in DT s notes), the Lagrangian approach is used, which of course should be related to the above.] 8.4 Lagrange s vs. Hamilton s equations A system with two degrees of freedom x and y has the Lagrangian, L = xẏ+yẋ 2 +ẋẏ. Derive Lagrange s equations. Obtain the Hamiltonian H(x, y, p x, p y ). Derive Hamilton s equations and show that they are equivalent to Lagranges equations. 8.5 More going back and forth between Lagrangian and Hamiltonian Exercise 8.16 from GPS. Hint: in part (a), you can suitably massage the usual relations, i.e., H = p q L and p = L in order to obtain (of course after some more work!) L (given H). q Hint: in part (b), for finding an equivalent Lagrangian, you might find useful part of the discussion done in the context of Noether s theorem: see around Eq. 2 on top of page 2 of posted note in particular, what happens to EOM when Lagrangian is shifted by a total time derivative? Note: some printings of 3rd edition of GPS seem to have a typo in the expression for Hamiltonian here: it should be p 2 /(2 a) in 1st term (and not α as the constant, which appears in exponent of next couple of terms). 9 Homework 9 (due Thursday, November 29 during lecture or in folder near Room 3118 of PSC by 5 pm.) 9.1 Conserved quantities using Hamiltonian Exercise 8.27 from GPS.

14 This problem involves figuring out the effect of a change of coordinates on conservation (or not) of Hamiltonian (you might wish to look at related discussion above and below Eq. 8.42/Fig. 8.1 on page of GPS). 9.2 Constants of motion from Poisson brackets for an abstract system, including their time-derivatives Exercise 9.30 from GPS. 9.3 Poisson bracket and constant of motion for harmonic oscillator Exercise 9.31 from GPS: Show by the use of Poisson brackets that for a one-dimensional harmonic oscillator there is a constant of motion u defined as u(q, p, t) = ln (p + imωq) iωt, ω = k m (20) What is the physical significance of this constant of motion? (The next two problems are from DT s University of Cambridge lecture notes. The first one is also a part of exercise 9.39 of GPS.) 9.4 Poisson bracket of Runge-Lenz vector with Hamiltonian A particle with mass m, position r and momentum p has angular momentum L = r p. Recall from discussion of Kepler problem that the Runge-Lenz vector is defined as A = p L mkˆr (21) where ˆr = r/ r and k is a constant. If the system is described by the Hamiltonian H = (p 2 /2m) k/r, prove using Poisson brackets that A is conserved. (The conservation of A was shown using EOM during the earlier discussion: see Eqs of GPS.) Hint: for computing the Poisson bracket above, you could try to express A in terms of H, plus other terms. Also, it suffices (and might be easier!) to just show that the Poisson bracket for one of the Cartesian components of A vanishes. 9.5 Canonical transformations Prove that the following transformations are canonical (using either Poisson brackets or Jacobian being symplectic): (a) P = 1 2 (p2 + q 2 ) and Q = tan 1 (q/p). (b) P = q 1 and Q = p q 2. (c) P = 2 q ( 1 + q cos p ) sin p and Q = log ( 1 + q cos p ).

15 10 Homework 10 (strictly due Tuesday, December 4 during lecture or in folder near Room 3118 of PSC before) 10.1 Using Poisson brackets to show that transformation is canonical (This problem is a part of exercise 9.15 from GPS) Using the Poisson brackets, find the values of α and β for which the equations: Q = q α cos βp, P = q α sin βp (22) represent a canonical transformation and find a generating function for it. The following two problems (first one being from DT) involve using canonical transformation in order to solve for the motion, i.e., determine the (original) coordinate as a function of time. You might wish to look at the example of simple harmonic oscillator done in lecture: see note posted at CT.pdf (based on section 9.3 of GPS and section of DT) Using canonical transformation to solve for constrained dynamics Prove that the following transformation is canonical for any constant λ: q 1 = Q 1 cos λ + P 2 sin λ, q 2 = Q 2 cos λ + P 1 sin λ p 1 = Q 2 sin λ + P 1 cos λ, p 2 = Q 1 sin λ + P 2 cos λ (23) If the original Hamiltonian is H (q i, p i ) = 1 2 (q2 1 + q p p 2 2), determine the new Hamiltonian H (Q i, P i ). Use this to solve for the dynamics under the constraint Q 2 = P 2 = Using canonical transformation to solve linear harmonic oscillator Exercise 9.24 from GPS: (a) Show that the transformation Q = p + iaq, P = p i a q 2 i a (24) is canonical and find a generating function. (b) Use the transformation to solve the linear harmonic oscillator problem.

16 The next two problems are solving for frequency of motion using action-angle variables: see section 10.6 of GPS for the case of simple harmonic oscillator (also done in lecture: see AA.pdf) Solving problem using action variable Exercise from GPS: A particle moves in periodic motion in one dimension under the influence of a potential V (x) = F x, where F is a constant. Using action-angle variables, find the period of the motion as a function of the particle s energy. Added part: cross-check your answer above by using the Newtonian formalism in order to compute the frequency Another problem-solving using action-angle variable Exercise from GPS. This is actually a two-dimensional problem, with r being the usual radial coordinate, i.e., perhaps a more appropriate name for it is a circular -well potential. You can study section of DT for an example of applying action-angle variables to a 2D system (i.e., Kepler problem, which is also done in sec of GPS); in particular, the net action variable can be taken to consist of two parts, one associated to the radial motion and the other to the angular motion. For a general discussion of action-angle variables in multi-dimensional systems, sec of GPS. Once again, you could cross-check your answer above by using the Newtonian formalism in order to compute the frequency. Note that a system has to be bounded and periodic (like simple harmonic oscillator done in lecture or the Kepler problem) in order to apply the method of action-angle variables. Hint: you can start with simple considerations such as what is trajectory of the particle when it is inside the well, what should happen upon the particle hitting boundary in order for the motion to be periodic etc. 11 Homework 11 (due Tuesday, December 11 in folder near Room 3118 of PSC by 5 pm.) 11.1 Hamilton s principal function for harmonic oscillator Exercise 10.5 For the following two problems, you can refer to solving for simple harmonic oscillator done in lecture: see HJ.pdf

17 11.2 Hamilton-Jacobi method for projectile motion Exercise Note that this is once again a two-dimensional problem: the idea here is to separate variables, as in the example of 2D harmonic oscillator worked out in GPS sec (Eq onwards). For a discussion of the general multi-dimensional case, see sec of GPS and beginning of 10.5 (later in sec. 10.5, i.e., Eq onwards, see the application to the Kepler problem) Another problem-solving using Hamilton-Jacobi method Assume that a particle moves in one dimension with a potential V = F x, where F is a constant; this is same as in HW 10.4 above or GPS exercise 10.13, where you already should have computed the period of the motion using action-angle variables (and checked that the Newtonian approach gives the same result). Here, use the Hamilton-Jacobi formalism instead in order to compute the coordinate of the particle as a function of time (once again, cross-check against the Newtonian formalism) Slowly varying potential Exercise (Note that Exercise referred to here was HW 10.4 above.) (The following problem is based on DT s University of Cambridge lecture notes.) 11.5 Computing period of motion from action variable, then including time dependence. Consider a system with Hamiltonian where λ is a positive constant and n is a positive integer. (i). by H(p, q) = p2 2m + λq2n (25) Show that the action variable (denoted by I in DT s notes) and energy E are related ( ) 2n/(n+1) ( ) n/(n+1) nπi 1 E = λ 1/(n+1) (26) J n 2m where J n = 1 0 (1 x)1/2 x (1 2n)/2n dx. (ii). If λ is fixed, determine the period of the motion, in particular, its dependence on λ and E. (iii). Conversely, if λ varies slowly with time, how does the particles total energy E depend on λ?