**An Integrated, Quantitative
Introduction to the Natural Sciences**

**Part 1: Dynamical Models**

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**0. Introduction**

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** **0.1 A physicistÕs point
of view (including some more general introductory material, and notes to
students)

0.2
A chemistÕs point of view

0.3
A biologistÕs point of view

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**1. NewtonÕs laws, chemical kinetics, É**

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 ÒcreatedÓ 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.

1.2
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 ÒstableÓ or
ÒunstableÓ 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

2.7 Resonance in the cell membrane

**3. We are not the center of the universe**

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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 weÕll 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

3.4 KeplerÕs laws

3.5 Biological counterpoint