An Integrated, Quantitative Introduction to the Natural Sciences

Part 1:  Dynamical Models

 

 

0.  Introduction

 

         0.1 A physicist’s point of view (including some more general introductory material, and notes to students)

         0.2 A chemist’s point of view

         0.3 A biologist’s point of view

 

1.  Newton’s laws, chemical kinetics, …

 

This is the start of several lectures on dynamical models that revolve around relatively simple differential equations.  We'll begin with mechanics, then see how similar equations arise in chemical kinetics, electric circuits and population growth.  Sometimes the simple equations have simple solutions, but even these have profound consequences, such as understanding that most of the chemical elements in our solar system were “created” at some definite moment several billion years ago.  In other cases simple equations have strikingly complex solutions, even generating seemingly random patterns.  This is just a first look at this whole range of phenomena.

 

         1.1 Starting with F=ma

         1.2 From boxes and arrows to differential equations

         1.3 Radioactivity and the age of the solar system              

         1.4 Using computers to solve differential equations

         1.5 Simple circuits and population dynamics

         1.6 Second order kinetics and the complexity of DNA

 

2.  Resonance and response

 

In this section of the course we will begin with a very simple system—a mass hanging from a spring—and see how some remarkable ideas emerge.  We will see, for example, that it is useful to use imaginary numbers to describe real things.  Most importantly, we will understand how to describe the way systems respond to small perturbations, and this turns out to be very general.  Following this path, our intuitive notions that something is “stable” or “unstable” can be given precise mathematical formulations.  We will take all of this far enough to see how the ideas can be used in describing complex biological phenomena, from the switches that control the expression of genes to the electrical impulses that carry information throughout the brain.

 

         2.1  Simple harmonic oscillators

         2.2. Magic with complex exponentials

         2.3  Damping, phases and all that

         2.4  Linearization and stability

         2.5  Stability in real biochemical circuits

         2.6  The driven oscillator

         2.7  Resonance in the cell membrane

 

 

3. We are not the center of the universe

 

Differential equations give a local description of dynamics, telling us how the state of a system changes in time from one moment to the next.  In contrast, some natural statements about the world (e.g., the planets go around the sun, and the orbits are elliptical) are global rather than local.  In this section of the course we’ll see how to connect global and local statements, with the added bonus that some things can be predicted without actually solving the equations (!).  The key link is a set of conservation laws, which are among the most profound organizing principles that we have for thinking about the world.  Surprisingly, these conservation laws are connected to the fact that the laws of physics must look to the same to any observer, no matter where they stand or which way they look.  In this sense, there is no privileged position from which to view the world, and our particular point of view is just one among many.    The idea that we are not the center of things is important not just in physics, but also in life.

 

         3.1  Conservation of energy

         3.2  Energy functions in complex systems

         3.3  Conservation of P and L

         3.4  Kepler’s laws

         3.5  Biological counterpoint