University of Minnesota
School of Physics & Astronomy

Phys 5011.001

Classical Physics I

Final Exam
modified 7-Dec-2017 at 11:48AM by Robert Lysak

The final exam for Phys 5011 will be on December 15, 8:30-11:30 am, in Tate 110.

Suggested Problems for Final: Jackson Chapter 4, Problems 7, 9, 13.

Solutions to Suggested Problems | Download posted 7-Dec-2017 at 11:09AM
Final Exam Formulas | Download posted 7-Dec-2017 at 11:40AM
Sample Final | Download posted 7-Dec-2017 at 11:46AM
Sample Final Solutions | Download posted 7-Dec-2017 at 11:48AM

Problem Set 9 (Due Dec. 8)
posted 1-Dec-2017 at 11:14AM by Robert Lysak

Jackson Chapter 3, Problems 1, 6, 12.

modified 5-Dec-2017 at 10:04AM by Robert Lysak

In this section, I will post expansions on the notes at various times.

Bessel Functions
modified 5-Dec-2017 at 10:04AM by Robert Lysak
Bessel Functions | Download posted 5-Dec-2017 at 10:04AM

Sturm-Liouville Equations
modified 1-Dec-2017 at 9:54AM by Robert Lysak
Sturm-Liouville Equations | Download posted 1-Dec-2017 at 9:54AM

Addition Theorem for Spherical Harmonics
modified 1-Dec-2017 at 9:52AM by Robert Lysak
Addition Theorem | Download posted 1-Dec-2017 at 9:52AM

Kepler Problem in Action-Angle Variables
modified 1-Nov-2017 at 11:20AM by Robert Lysak
Kepler Action-Angle | Download posted 1-Nov-2017 at 11:20AM

Motion of Charged Particles in Magnetic and Electric Fields
modified 24-Oct-2017 at 11:31AM by Robert Lysak
Charged Particle Motion | Download posted 24-Oct-2017 at 11:31AM

Closed Orbits: Bertrand's Theorem
modified 19-Sep-2017 at 2:12PM by Robert Lysak

9/19: These notes give the full derivation of Bertrand's theorem for finite size perturbations. The first page refers to small deviations from circular motion, following section 3.6 of Goldstein and discussed in class. The second page gives the expansion to 3rd order that determines that only the inverse square law and the harmonic oscillator can give closed orbits.

Notes on Closed Orbits | Download posted 19-Sep-2017 at 2:12PM

Motivation for the Lagrangian
modified 8-Sep-2017 at 2:08PM by Robert Lysak

9/8: Here is a description of the rather complicated motivation for using the Lagrangian for mechanics. I say "motivation" rather than "derivation" since it is not really a rigorous derivation. However, the proof is in the results: this formulation gives the correct equations of motion in a wide range of circumstances.

Lagrange Equations | Download posted 8-Sep-2017 at 2:08PM

Potential Energy for a System of Particles
modified 6-Sep-2017 at 1:18PM by Robert Lysak

9/6: The discussion of potential energy seemed to be confusing. Here is a clearer version of these notes.

Potential energy for a system of particles | Download posted 6-Sep-2017 at 1:18PM

Problem Set 8 (Due December 1)
modified 6-Dec-2017 at 11:19AM by Robert Lysak

Jackson Chapter 2: Problems 2,5,7.

Problem Set 8 Solutions | Download posted 6-Dec-2017 at 11:19AM

No Wednesday Office Hours
posted 20-Nov-2017 at 9:55AM by Robert Lysak

Due to the Thanksgiving holiday, I will not have office hours on Wednesday, Nov. 22.

Problem Set 7 (Due Nov. 22)
modified 1-Dec-2017 at 11:14AM by Robert Lysak

Jackson Chapter 1: Problems 2,6,7,9.

Problem Set 7 Solutions | Download posted 1-Dec-2017 at 11:14AM

Quiz 2 (Nov. 9)
modified 13-Nov-2017 at 12:46PM by Robert Lysak

Quiz statistics: Mean 71.6; Median 73; Top Quartile 84; Bottom Quartile 60.

Covers Chapters 6, 8-10, and some parts of Chapter 11 (as in class).

Practice problems: 10.19, 11.1 and 11.5

Hint for 10.19: The integral in r can be done by complex integration in a similar way as in the Kepler problem. In this case, there will be a 3rd order pole, where the residue is lim_u->0 (1/2)(d^2/du^2) (u^3 f(u)).

Quiz 2 Solutions | Download posted 13-Nov-2017 at 12:45PM
Quiz 2 Practice Problem Solutions | Download posted 3-Nov-2017 at 3:59PM
Sample Quiz 2 | Download posted 3-Nov-2017 at 4:19PM
Sample Quiz 2 Solutions | Download posted 3-Nov-2017 at 4:20PM
Quiz 2 Formulas | Download posted 3-Nov-2017 at 5:15PM

Problem Set 6 (Due Nov. 3)
modified 9-Nov-2017 at 10:48AM by Robert Lysak

Chapter 10, Problems 8, 13, 18. For problem 10.18, you can assume that the total distance between the walls is d = 3a, i.e., all of the springs are at their unstretched length in equilibrium.

Hint for Problem 10.8: In order to separate the Hamilton-Jacobi equation, try writing the principal function as S = f(t)+x*g(t).

Problem Set 6 Solution | Download posted 9-Nov-2017 at 10:48AM

Problem Set 5 (Due Oct. 27)
modified 30-Oct-2017 at 12:07PM by Robert Lysak

Chapter 8, Problems 20, 33; Chapter 9, Problem 22.

Note for problem 8.33, "Investigate the small oscillations..." means find the eigenfrequencies and the relative amplitudes of each body in each mode. For this part, you can assume m1 = m2 = m and 3g / r0 > > 2k / m . You can also assume that the unstretched length of the string is 0.

I have attached a new version of the solution of 8.33 which does not assume the unstretched length is 0 and considers both limits on the ratio of 3g/r0 to 2k/m.

Problem Set 5 Solutions | Download posted 30-Oct-2017 at 12:07PM

Problem Set 4 (Due October 20)
modified 20-Oct-2017 at 2:01PM by Robert Lysak

Do problems 6.4, 6.12 and the problem in the attached file.

Problem Set 4, Problem 3 | Download posted 12-Oct-2017 at 11:48AM
Problem Set 4 Solutions | Download posted 20-Oct-2017 at 2:01PM

Quiz 1
modified 8-Oct-2017 at 3:43PM by Robert Lysak

Quiz 1 will be on Friday, Oct. 6 in class at 10:10 am in B65. The exam will cover Chapters 1-5 of Goldstein. The exam is closed book, and I will provide a sheet giving important formulas. This sheet will be posted in advance early next week. The following problems from Chapters 4 and 5 will not be graded, but you will be responsible for this material on the quiz:

Chapter 4, Problem 21; Chapter 5, Problems 6, 15, 21.

Quiz 1 Solutions | Download posted 8-Oct-2017 at 3:43PM
Solutions to Quiz 1 Practice Problems | Download posted 2-Oct-2017 at 3:08PM
Quiz 1 Formulas | Download posted 2-Oct-2017 at 3:01PM
Sample Quiz 1 | Download posted 2-Oct-2017 at 3:06PM
Sample Quiz 1 Solutions | Download posted 2-Oct-2017 at 4:42PM

TA office hours
posted 26-Sep-2017 at 5:20PM by Robert Lysak

In my absence, Ragnar Stefansson will have office hours 3:30-5:00 Wednesday in PAN 434.

Lysak out of town
posted 21-Sep-2017 at 10:58AM by Robert Lysak

I will be out of town the week of September 25 to attend a conference. During that time we will have guest lecturers:

Sept. 25: Rafael Fernandes
Sept. 26: Dan Cronin-Hennessey
Sept. 27: Joe Kapusta
Sept. 29: Tom Jones

Problem Set 3 (Due Sept. 29)
modified 29-Sep-2017 at 1:59PM by Robert Lysak

1. NASA’s Polar satellite is in an orbit around Earth with an apogee of 9.0 RE and a perigee of 1.8 RE, measured from the center of the Earth (1 RE = 1 Earth radius = 6380 km). Determine its semi-major axis, eccentricity and orbital period. Find the energy and angular momentum per unit mass for the satellite. What are its maximum and minimum orbital velocities?

2. Do Problem 3.15

3. Do Problem 3.21

Problem Set 3 Solutions | Download posted 29-Sep-2017 at 1:59PM

Problem Set 2 (Due Sept. 22)
modified 22-Sep-2017 at 1:59PM by Robert Lysak
Problem Set 2 | Download posted 14-Sep-2017 at 2:00PM
Problem Set 2 Solutions | Download posted 22-Sep-2017 at 1:59PM

Problem Set 1 (Due Sept. 15)
modified 18-Sep-2017 at 1:54PM by Robert Lysak

Goldstein Chapter 1, Problems 5, 21, and the following problem:

PS1 Problem 3: Consider the double pendulum of Figure 1.4, with each body of mass m and each rod having length L. This system is placed in an external, constant gravitational force. Write down the kinetic and potential energy of the system in terms of the angles and their derivatives and determine the Lagrangian. Find the Lagrangian equations of motion for this system.

Problem Set 1 Solutions | Download posted 18-Sep-2017 at 1:54PM

modified 1-Dec-2017 at 11:13AM by Robert Lysak

Sections refer to Goldstein
9/5: Kinematics 1.1
9/6: Many particle systems; Constraints 1.2-3
9/8: Lagrangian equations 1.4-6

9/12: Hamilton’s principle; calculus of variations 2.1-3
9/13: Lagrange multipliers; Conservation of momentum 2.4-6
9/14: Conservation of angular momentum and Energy 2.7
9/16: Two-body problem 3.1-3

9/19: Closed orbits; Kepler problem; Virial Theorm 3.4, 6-7
9/20: Hyperbolic orbits; Scattering 3.10
9/21: Transformation to lab frame; Solar system example 3.11
9/23: Rigid Bodies; Body Coordinates 4.1-2

9/25: Orthogonal Transformations; Euler angles 4.3-6
9/26: Derivatives of Vectors; Coriolis effect 4.7-10
9/27: Inertia Tensor; Euler equations 5.1-5
9/29: Torque-free motion; Lagrangian formulation of Rigid Body Motion 5.6-7

10/2: Lagrangian formulation of Rigid Body Motion 5.7
10/3: Motion of top with fixed point 5.7
10/4: Precession of Earth and satellites 5.8
10/6: QUIZ 1

10/9: Systems of oscillators 6.1-2
10/10: Normal mode coordinates 6.3-4
10/11: Damped, driven pendulum 6.5-6
10/13: Hamiltonians 8.1

10/16: Conservation Laws; Poisson Brackets; Phase Space 8.2, 9.5
10/17: Hamilton’s Principle; Least Action; Canonical Transformations 8.5-6; 9.1
10/18: Harmonic Oscillator; Symplectic Form; 9.2-4
10/20: Infinitesimal Canonical Transformations; Liouville’s Theorem 9.5-6, 9

10/23: Angular momentum; Hamilton-Jacobi theory 9.7, 10.1-3
10/24: Harmonic Oscillator; Action-angle variables; Pendulum 10.6
10/25: Separation of Variables; Charged particle in B 10.4, 7
10/27: Charged particle in B and E; Hall effect 10.7

10/30: Guiding Center Drifts; Adiabatic Invariants 12.2,5
10/31: Completely Separable system; Kepler problem 10.5,8
11/1: Kepler Problem 10.8
11/3: Nonlinear oscillators; KAM theorem 11.1-2

11/6: Fixed points; Limit Cycles; Strange Attractors 11.3-4
11/7: Henon-Heiles Hamiltonian; Damped driven oscillator 11.5-7
11/8: Logistic equation; Fractals 11.8-9
11/10: QUIZ 2

From here on, section numbers refer to Jackson, except for numbers preceded by LL which are from Landau and Lifschitz, Electrodynamics of Continuous Media.

11/13: Fundamentals of E&M, Gauss and Coulomb’s Laws 1.1-4
11/14: Electric Potential, Poisson’s Equation 1.5-8
11/15: Green’s Functions; Electrostatic Energy 1.9-11
11/17: Physics of Conductors; Energy and Forces 1.11; LL 1, 5, 20

11/20: Numerical Methods; Method of Images 1.12-13; 2.1
11/21: Method of Images for a Sphere; Green’s functions 2.2-4
11/22: Green’s function of a Sphere 2.5-7
11/24: No Class: Thanksgiving weekend

11/27: Fourier series; Spherical Coordinates 2.9-3.1
11/28: Legendre Polynomials; BVP with Azimuthal Symmetry 3.2-3
11/29: Spherical Harmonics; Green’s function 3.5, 9
12/1: Addition Theorem; Sturm-Liouville Systems 3.6, 9-11

12/4: Cylindrical Coordinates; Bessel functions 3.7-8
12/5: Green’s function on cylinder 3.11
12/6: Multipole expansion 4.1-2
12/8: Polarization; Macroscopic equations 4.3, 5-6