Analytical Mechanics HT-19 Tutorial 1

Problem 1. The Hamiltonian of a relativistic particle of mass m has the form p 2 2 2H = m c4 + p c ,

where p is the 3-dimensional momentum and c is the speed of light. Find a Lagrangian corresponding to this Hamiltonian.

Problem 2. A dynamical systems with the following Lagrangians

5 12 2(a) L = q1 + q2 + q1q2 cos(q1 − q2) + 3 cos q1 + cos q2,2 2 h i1 2(b) L = (q1 − q2)

2 + aq1 t2 − a cos q2.

2

Find a Hamiltonian corresponding to the above Lagrangians.

Problem 3. Find a Lagrangian corresponding to the following Hamiltonian

H = p1p2 + q1q2.

Problem 4. The Hamiltonian for a system with one degree of freedom is given by

2p ab 2 −αt � α + be−αt

� kq2

H = − bqpe−αt + q e + ,2a 2 2

where a, b, α and k are constants.

(a) Find a Lagrangian corresponding to this Hamiltonian.

(b) Find an equivalent Lagrangian that does not depend explicitly on time.

(c) Find the Hamiltonian corresponding to the second Lagrangian. What is the relation between this and the original Hamiltonian?

Problem 5. Find the Hamiltonian corresponding to a Lagrangian of the form

TL (q, q , t) = L0 (q, t) + q a +1 q TT q ,

2 ⎛ ⎞ ⎛ ⎞ q1 a1 ⎜ . ⎟ ⎜ . ⎟where q = ⎝ . ⎠ , a = ⎝ . ⎠ and T is a symmetric n × n matrix.. . qn an

Analytical Mechanics HT-19 Tutorial 2

Problem 1. Find a Lagrangian corresponding to the following Hamiltonian � �2 2H = q1p2 − q2p1 + a p1 + p .2

Problem 2. Derive Hamilton’s canonical equations for the following Hamiltonian and find a general solution to these canonical equations.

1 � � � �2 2 2 2H = p p2 + q + q .1 2 12

Problem 3. A dynamical system has the following Lagrangian � � mR2 ϕ2

ϑ2L = + − mgR cos ϑ. 2 sin2 ϑ

(i) Find the corresponding Hamiltonian.

(ii) Find a cyclic variable and reduce the problem to a family of problems with one degree of freedom.

(iii) Draw phase portraits for the one dimensional system.

Problem 4. Consider a point particle of mass m moving in a 2 dimensional central potential

qV (r) = − .

r

(i) Find the Hamiltonian for the system using polar coordinates.

(ii) Find a cyclic variable and reduce the problem to a family of one dimensional problems.

(iii) Draw phase portraits.

Analytical Mechanics HT-19 Tutorial 3

Problem 1. The length l of a mathematical pendulum varies with time as l = l(t).

(i) Write a Hamiltonian and Lagrangian for the system.

(ii) Draw phase portraits.

Problem 2. Find a canonical transformation defined by the following generating functions

(i) F = ln(qt) eP ,

(ii) F = q ln P .

Calculate the Poisson brackets to check that these transformations are indeed canonical.

Problem 3. A dynamical system is described by the Hamiltonian

2 2 2H =1 q + at q 2 − 2tpq +

1 p ,

2 a

where a is a constant. Apply a canonical transformation defined by the generating function

F =1 atq2 − qP

2

and find the new Hamiltonian in terms of P and Q.

Analytical Mechanics HT-19 Tutorial 4

Problem 1. A free point particle is described by the following Hamiltonian

2pH = .

2m

(i) Find a time independent canonical transformation such that the new momentum P coincides with H.

(ii) Give a general solution of the equations of motion using this canonical transformation.

Problem 2. A change of variables in the phase space is defined by the following equations

P = p cos t + q sin t,

Q = q cos t − p sin t.

Show that this is a canonical transformation and find the generating function F (q, P, t).

Problem 3. A dynamical system has the Hamiltonian

3pqH = .

2t

Find the new Hamiltonian after the following canonical transformation � � 3 1/q2

P = pq 1 + te , � � 1 1 pq3

Q = − + ln . 2 q2 t

Problem 4. Find the Poisson bracket of the following function on phase space

1 � � Φ = p 2 + q 2 ,

2 p

Ψ = arctan . q

d 1 (Hint: arctan x =

2 .)dx 1 + x

Analytical Mechanics HT-19 Tutorial 5

Problem 1. Consider a canonical transformation given by the generating function

F (q, P ) = P 2 + ln(P + q).

(i) Find the new variables P and Q in terms of p and q.

(ii) Check that the new variables have the correct Poisson bracket.

Problem 2. A dynamical system is described by the Hamiltonian H = p1p2 + q1q2. Show that the following functions are conserved quantities.

2 2 2 2Φ1 = p1 + q2 , Φ2 = p2 + q1 .

Problem 3. Show that the following functions

Φ1 = p1

, Φ2 = (p2 − q2) e −t , q2

2are conserved quantities for the Hamiltonian H = p1q1 − p2q2 + q2 . Find one more constant of motion using Poisson brackets. Find the general solution of the equations of motion using the consvered quantities.

Problem 4. Using Poisson brackets, show that the components of the Laplace-Runge-Lenz vector

mk A = p × L − r,

r

are conserved quantities in the Kepler problem with the Hamiltonian

p2 k H = − .

2m r

Analytical Mechanics HT-19 Tutorial 6

Problem 1. The Hamiltonian of a system has the form � � 1 1

H = + p 2 q 4 . 2 q2

Find the equation of motion for q. Find a canonical transformation which reduces H to the form of a harmonic oscillator.

Problem 2. Find Hamilton’s principal function for a one dimensional free particle of mass m. Find the general solution to the equation of motion for p and q.

Problem 3. Consider a particle of mass m moving in a one dimensional potential which depends linearly on time, such that the Hamiltonian is given by

2pH = − mAtx,

2m

where A is a constant. Solve the dynamical problem by using Hamilton’s principal function. Use the initial conditions t = 0, x = 0, p = mv0 to fix the constants.

Problem 4. The Hamiltonian of a system has the form

t � 2 �

e pH = + Aq .

2 2m

Find the general solution for p and q using Hamilton’s principal function.

Analytical Mechanics HT-19 Tutorial 7

Problem 1. A system with two degrees of freedom has the following Lagrangian � �I ϑ2 ϕ2 2 ϑL = + ˙ cos .

2

Solve the Hamilton-Jacobi equation to find Hamilton’s principal function for this system in terms of one dimensional integrals (You do not need to evaluate the integrals).

Problem 2. Do the same for the following Lagrangian

2 2x y 2L = + − αx2 y . 2y2 2

Problem 3. A system with two degrees of freedom is defined by the following Hamiltonian

1 � � H = p1q2 + 2p1p2 + q 2 .12

(i) Solve the Hamilton-Jacobi equation.

(ii) Find general solutions for q1(t) and q2(t).

Problem 4. Do the same for the following Hamiltonian � � p2 + q2

H = e t . p1 + q1

Analytical Mechanics HT-19 Tutorial 8

Problem 1. A particle of mass m moves in the presence of two attracting centers placed at the points x = −a, y = 0 (with potential −A/r1) and x = a, y = 0 (with potential −B/r2) as shown in the figure.

Our goal is to find Hamilton’s principal function for this system. In order to separate variables, we have to use the following coordinates

1 1 ξ = (r1 + r2) , η = (r1 − r2) ,

2 2

since they do not separate in the coordinates r1, r2.

(i) Write x and y in terms of ξ and η.

(ii) Write the Lagrangian in the coordinates ξ, η.

y

x −a a

m

r1

r2

(iii) Write the Hamiltonian in ξ, η coordinates and solve the Hamilton-Jacobi equation to find Hamilton’s principal function.

Problem 2. A system with two degrees of freedom is given by the following Hamiltonian

2 2p q � �21 1 2 2H = + p2 + q .22 8

(i) Solve the Hamilton-Jacobi equation in terms of one dimensional integrals. (You do not need to evaluate the integrals).

(ii) Draw phase portraits on the phase plains (p1, q1) and (p2, q2).

(iii) Introduce action-angle variables and express the Hamiltonian as a function of the action variables J1 and J2. Find frequencies ν1 and ν2 as functions of the action variables.

Analytical Mechanics HT-19 Tutorial 9

Problem 1. An elastic ball oscillates between two very slowly moving walls as shown in the figure below.

L(t)

v m

(i) Find an action variable for the system.

(ii) How does the velocity of the ball depend on the distance L(t) between the walls?

Problem 2. A plane pendulum is constrained to move on an inclined plane as shown below.

l

m

α

How does the amplitude change when the inclination angle α of the plane is changed slowly?

Problem 3. To the lowest order in correction terms, the relativistic Hamiltonian for the one dimensional harmonic oscillator has the following form

p2 1 p4

H = + mω2 q 2 − .3 22m 2 8m c

Calculate the lowest order relativistic correction to the frequency of the harmonic oscillator.

Problem 4. Consider a plane isotropic harmonic oscillator that is perturbed by a change in the Hamiltonian of the following form

2ΔH = εp2 p .x y

Find the shifts in frequencies in first order in ε.

Problem 5. [Goldstein 12.2 ] A point mass m hangs at one end of a vertically hung spring (obeying Hooke’s law) of force constant k. The other end of the spring is oscillated up and down according to

z1 = a cos ω1t.

By treating a as a small quantity, obtain a first order solution to the motion of m in time using time dependent perturbation theory. What happens as ω1 approaches the unperturbed frequency ω0?