How do physicists use differential equations to describe the world?

[aychamo]

Hey guys

I don't understand how physicists arrive at equations that describe things.

Even Newton's equations, which look very simple, I don't see how he came up with them.

I can't imagine dropping a ball, observing it and then making a formula. I assume that took a lot of measurements, etc, and I know the units probably helped. When I am working physics problems and forget my equations I can write a unit style equation and help me remember it, but he had to damn invent the units too (ie, force.)

And those were so simple. I saw some program on Discovery channel and this physics guy was looking through an old math book and saw a formula and he realized that formula perfectly described whatever he was looking to describe. How in the hell? :)

An enamored biologist,
Aychamo

 

[HallsofIvy]

How do physicist do it?


With electricity!

 

[Mr. Robin Parsons]

Mathematics, plotting points, calculating numerical values, and figuring out how they act in space...formula translation...math is a descriptive language, it describes shapes, Circle/triangle/square and from that can be used to describe many other phenomenon...

...except "emptiness" that one they haven't quite finished counting yet...

(that ones a joke )

 

[Nereid]

That's a really good question aychamo!

The approach you describe isn't all that inaccurate, as a description of what 'experimental' physicists (and astronomers) do. In astronomy there are lots of examples of deep results which began as a diligent astronomer plotting some data on a chart and noticing they fell (more or less) on a straight line. Of course, there are many ways to plot data, but kneading and pummelling it with simple tools can often make a pattern show up as a straight line, and the rest, as they say, becomes history.

Theoreticians approach things rather differently. Some have a cool idea, and can take that idea and write the most beautiful equations from it (the experimentalists then take delight in seeing if it resembles anything remotely testable). Some take deep physics from the giants and seek to find ways to describe - with equations - what that deep physics means in this situation or that (e.g. the space between galaxy clusters, ordinary matter squeezed into a ball with the density of atomic nuclei, ...). And some have yet other ways of making their magic.

Let's see how others respond to your post; my two paragraphs are very short, and in many ways do much violence to what really takes place.

 

[Integral]

One key is being able to recognize what is changing, frequently observations revel how one thing changes with respect to something else. The relationship of these changes can be expressed mathematically as a differential equation.

For example, you can say that the CHANGE in population is proportional to the number of people so if P is the population you can write

$$\frac {dP} {dt} = kP$$

Now by solving the differential equation you arrive at a time dependent model of the population (This leads to Malthus's Model, it is an over simplification!)