Note that sometimes lecture will be held in the discussion session period, and vice versa.
Shortcuts to particular weeks:
Week of January 8
January 13
January 20
January 27
Week of February 3
February 10
February 17
February 24
Week of March 2
March 9
March 16
March 23
March 30
Week of April 6
April 13
April 20
April 27
This class will be held in the discussion slot, 6-7:20 pm.
Readings: Taylor 1.1-1.5. If you have not yet taken linear algebra, you may want to spend extra time on this material.
After some logistical information, we'll review some properties of vectors and basic principles of Newtonian mechanics. We'll cover inertial reference frames, Newton's three laws, and conservation of momentum.
Readings: Taylor 1.6-1.7, 2.1-2.2
We'll continue review, and look at examples of solving equations of motion for Newton's laws. After a classic Cartesian coordinate example, we'll look at describing motion in polar coordinates, for which unit vector directions are not constant in time. In order to find velocity and acceleration in polar coordinates so that we can apply Newton's Laws, we need to differentiate these unit vectors. We'll do this, and apply the result to an example.
This lecture will be held in the discussion slot, 6-7:20 pm.
Readings: Taylor 2.1-2.4. We will skip the rest of Chapter 2.
We will discuss drag (air resistance) and set up the equations of motion for a proje ctile with drag. We'll analytically solve the case for horizontal and vertical motion with linear drag.
Readings: Taylor Chapter 3
This lecture will be a review of a number of concepts from introductory mechanics: conservation of momentum (which we saw before in Lecture 1), center of mass, angular momentum and torque, moment of inertia, and conservation of angular momentum. We'll do some examples as time permits.
This will be a Mathematica introduction section run by your TA. See here for information on how to install Mathematica and some starter information. Also, look on Sakai under "Assignments" for an introductory notebook.
Readings: Taylor 4.1-4.6
We'll define work and kinetic energy, and discuss the work-kinetic energy theorem. Then we'll discuss conservative forces and the potential energy associated with conservative forces. Next we'll consider total mechanical energy, and when this quantity is conserved. We'll look at how 1D potential energy diagrams can be useful in understanding the behavior of a system.
Materials are on Sakai under "Assignments." Please note that 2 out of 10 points for the first will be awarded for having completed the "Tutorial" portion by start of the session.
Readings: Taylor 4.7-4.10, 5.1-5.2
We'll finish up some material from Chapter 4: how to describe systems in spherical coordinates, central forces, and energy in multiparticle systems. For the latter, we'll skip the derivations, but please read the relevant text sections. Next, we'll start on discussion of oscillators, an important topic. We'll cover the 1D simple harmonic oscillator, define terms, and look at solutions, including description of solutions using complex exponentials.
This will be a discussion/software lab help class run by your TA.
Readings: Taylor 5.2-5.4. We will cover some material not in the text on phase space (this shows up in later chapters as "state space").
We will discuss energy for the simple harmonic oscillations. Next we'll consider the SHO in 2D, and the resulting elliptical motion solutions for different initial conditions, including Lissajous figures when frequencies are different in different directions (the "anisotropic" case). I'll introduce the concept of phase space, an abstract space helpful for describing motion. We'll then discuss damped oscillations, including the cases of underdamping, overdamping, and critical damping.
Readings: Taylor 5.5-5.7 (we won't cover 5.8 and 5.9 explicitly but they are useful to read).
After a bit of review and some questions from the material last class, we'll start to discuss driven, or forced, oscillations. We will cover solutions for the equation of motion for an applied sinusoidal force. The main behavior is characterized by transient and steady-state solutions. We'll look amplitude and phase shift as a function of frequency, and the resonance phenomenon.
Readings: Taylor 5.7, 5.8, 12.1. We'll cover some material not in the text.
We'll discuss Fourier series, and how to use a Fourier decomposition to solve a linear damped driven oscillator problem for an arbitrary driving function. Then we'll change gears and start to discuss nonlinear oscillators, and how to describe their solutions using potential energy diagrams and phase space.
Readings: Some of the material corresponds to Taylor 12.2-12.8, but we will not cover all of that material, and I'll also cover some examples not in the text.
We will go through some examples of nonlinear oscillators: the van der Pol oscillator (which doesn't have a good mechanical analog, but which exhibits some relevant properties of nonlinear differential equations) and the free plane pendulum. Then we will discuss in some detail the complex behavior of the nonlinear damped, driven plane pendulum. This undergoes period doubling bifurcations and eventually chaotic motion as the driving parameter (ratio of applied force to gravity) increases.
Readings: Taylor Chapter 6
We'll cover the basics of the calculus of variations: the method of minimizing the value of an integral over a path using the Euler-Lagrange equations.
This will be a review session for the midterm.
Readings: Taylor 7.1-7.5. We won't cover the proof in 7.4 in class, but please read it
We will define the Lagrangian and show how Newton's Laws in the form of the Euler-Lagrange equations imply Hamilton's principle, which states that the path followed by a particle is that minimizing the integral of the Lagrangian (the "action") over the path. We'll formulate this in terms of generalized coordinates and constraints, and look at some examples of using Lagrange's equations to determine equations of motion for systems with constraints.
Readings: Taylor 7.6, 7.7, 7.8, 7.10
We will consider generalized forces and momenta in the context of Lagrangian formalism, and look at how cyclic or ignorable coordinates correspond to conservation laws. More generally, Noether's Theorem states that invariances of the Lagrangian result in conserved quantities. We'll look in detail at two examples: how invariance under space translation results in conservation of momentum, and how invariance under time translation results in conservation of energy. In discussion of the specific conditions for the latter, I'll introduce the Hamiltonian quantity.
Readings: Taylor 7.10
We'll cover in detail the method of Lagrange multipliers to handle problems with constraints, and go through some examples (including some not in the textbook.)
This will be a problem-solving session.
Readings: Taylor 13.1-13.5
We will cover Hamilton's equations and the basics of Hamiltonian mechanics, an alternative formulation to Lagrangian mechanics.
Readings: Taylor 8.1-8.5
We'll consider in detail the problem of two-body central force motion. Such a system's motion can be reduced to motion in a 2D plane of the center of mass of the system, along with the relative motion, which can be described by the motion of a particle with the "reduced mass". We'll discuss the "effective potential", which is a convenient quantity for treating this problem.
Class times will be open for Zoom trial connection--- please see Sakai for connection information.
All subsequent classes, including discussions and software labs, will be by Zoom. Homework will need to be uploaded to Sakai. Information for connection is on Sakai. The lecture notes will be posted, and recordings of the lectures will be available on Sakai. Minute questionnaires will be done using Duke Qualtrics surveys.
Readings: Taylor 8.5-8.8. We didn't get to the section on changes of orbit in class (8.8) but please read it.
We'll derive the "equation of orbit", a change of variables that allows us to solve for radius as a function of angle in the two-body central-force problem. The Kepler problem is a special case of this problem, for which the force is an inverse square law. The solution applies to planetary orbits (i.e., gravitational force.) We'll look at solutions for Kepler orbits, which turn out to be conic sections.
Readings: Taylor 9.1-9.6. We won't cover 9.2 on tides in class, but please read it.
We will discuss non-inertial reference frames. We'll look at how to describe motion in a frame rotating with respect to an inertial one. In such a frame, Newton's laws are not valid, but we can introduce "fictitious" or "effective" forces to recover a description in which a relation like Newton's second law is valid. We'll look specifically at the example of motion in the Earth's rotating frame, and look in detail at the effective centrifugal force.
Readings: Taylor 9.7-9.8. We're not covering the Foucault pendulum in class (a Covid-19 casualty) but you might be interested to read section 9.9.
We'll continue to look at description of motion in the rotating reference frame of the Earth, and consider the centrigual and Coriolis forces. We'll look at some examples.
Readings: Taylor 10.1-10.3
We'll discuss how to describe rotation of a rigid body about any axis, in terms of the inertia tensor. We'll look at several examples.
This will be a virtual lab session led by your TA. The assignment is posted on Sakai.
Readings: Taylor 10.4-10.6
We'll first do some examples of calculating the tensor of inertia. We'll Then discuss the principal moments of inertia for a given object and coordinate system: how to find them, and their physical meaning. We'll then look at some examples as time permits.
This will be a review session.
Readings: Taylor 10.6-10.8 (optional: 10.9-10.10)
We'll look at precession of a spinning top under the influence of weak gravitational torque. Next we'll cover Euler's equations, and look at their implications for a couple of cases of zero torque, including the case of free precession.
We won't have time in the course to cover Euler angles and their use in understanding of more complex motions of the spinning top, but please read the sections in your text if you are interested.
Readings: Taylor 11.1-11.4
We will go through the example of two 1D coupled harmonic oscillators in some detail: we'll find the general solution, and then find the simpler solution in terms of carefully chosen coordinates that describe independent motion in each coordinate (the normal coordinates). We'll look also at the specific case of weak coupling between the oscillators.
Readings: Readings: Taylor 11.4-11.7
I'll go through some examples of coupled oscillator systems. We will look at the general case of system of coupled oscillations with n degrees of freedom, treated in a linear approximation.
This session will be used for office hours, 6-7:30 pm. Please see email for Zoom coordinates.
We'll do some review problems.