University of Minnesota
School of Physics & Astronomy

Phys 5011.001

Classical Physics I

Problem Set 5 (Due Oct. 27)
modified 18-Oct-2017 at 5:13PM 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- .

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

Final Exam
posted 12-Oct-2017 at 11:33AM by Robert Lysak

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

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 19-Sep-2017 at 2:12PM by Robert Lysak

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

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

modified 20-Oct-2017 at 2:00PM 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