PHYS 207

You are familiar with the basic Newtonian mechanics concepts in this chapter from your introductory mechanics course. We will review these concepts and use more advanced mathematics, such as ordinary differential equations to solve more sophisticated problems.

• Newton’s law of motion: historical introduction
• Rectilinear motion: uniform acceleration under a constant force
• Forces that depend on position: the concept of kinetic and potential energy
• Velocity-dependent forces: fluid resistance and terminal velocity

We will analyze the motion of a damped harmonic oscillator under the influence of an external driving force. We will use of complex quantities in dealing with the mathematics involved here (solving ordinary second-order linear differential equations).

• Introduction
• Linear restoring force: harmonic motion
• Energy considerations in harmonic motion
• Damped harmonic motion
• Forced harmonic motion: resonance

We will expand the application of Newton’s laws to three dimensions and use the mathematical concepts of gradient and line integral to relate conservative forces and potential energies; these mathematical concepts are revisited in PHYS 216 in more detail.

• Introduction: general principles
• The potential energy function in three-dimensional motion: the del operator
• Forces of the separable type: projectile motion
• The harmonic oscillator in two and three dimensions

We will reformulate Newton’s second law using polar coordinate coordinates to analyze the motion of a particle under the influence of a central force in general and under an inverse-square force in particular. Planetary orbits and Kepler’s laws will be derived.

• Velocity and acceleration in plane polar coordinates (F&C, Sect. 1.11)
• Kepler’s laws of planetary motion:
  first law (the law of ellipses), second law (equal areas), third law (the harmonic law)
• Angular momentum in a central field
• Orbital equation in a central field, Kepler’s orbit
• Relation between energy E and orbital parameters in an inverse-square field
• Limits of the radial motion: effective potential

We will develop the general tools for dealing with a system of particles when it is subjected to internal and external forces and torques. We will introduce the concept of center of mass and examine the momentum and angular momentum of a system of particles in both inertial and center-of-mass reference frames. We will formulate Newton’s second law in both linear and angular forms for a system of particles.

• Introduction: center of mass and linear momentum of a system
• Angular momentum and kinetic energy of a system
• Motion of two interacting bodies: the reduced mass
• Motion of a body with variable mass: rocket motion

We will first examine how angular momentum and angular velocity are related to each other in general when a system of particles or a rigid body is undergoing a rotation about an instantaneous axis. We will then focus on the analysis of planar motion of rigid bodies.

• Center of mass of a rigid body
• The inertia tensor (Sect. 9.1 of F&C)
• Rotation of a rigid body about a fixed axis: moment of inertia
• Calculation of the moment of inertia
• The angular momentum of a rigid body in laminar motion
• Examples of the laminar motion of a rigid body

We will derive the differential wave equation in 1D and discuss its solution in the forms of traveling and standing waves. We will then revisit driven, damped oscillators. But unlike in Chapter 3, we do not impose linear approximation. We will see that even though the differential equation is still perfectly deterministic, the solution may no longer be periodic and chaos can set in. We will introduce some basic concepts and terminology encountered in the study of chaos.

• Waves: an introduction (Sect. 11.6 of F&C)
• The nonlinear oscillator: chaotic motion (Sect. 3.8 of F&C)