Numerical Methods and Spreadsheet Physics

[cj]

I understand that one can use numerical methods
to solve a derivative or integral that can't be
solved analytically.

What are some simple examples of physics Diff Eqs
and/or Integrals that can only be solved using
numerical methods?

 

[arildno]

Well, I would say that practically every diff eq in use has no analytical solution.
The class of Diff Eqs. that does have these remarkably simple solutions is such a tiny class as to be altogether negligible.
For starters, non-linear oscillators will typically have no analytical solutions.

 

[krab]

I've heard it said that by the time a person gets a bachelor's degree in physics, (s)he has solved learned all the problems in physics that can be solved analytically. I have no reason to doubt it. It's also unfortunate, since as arildno says, the vast majority of real-life problems must be solved numerically, so the result is that the graduate has an inflated impression of his/her abilities. My own experience agrees with this. All the tricks on how best to approximate an intractable problem, and how best to numerically solve it, I learned in graduate school.
An example of a problem that can be solved analytically is the two-body gravitational problem. The three-body problem can only be solved numerically. So the earth-moon system can be solved by pretending their are no other bodies. If the accuracy is insufficient, the sun is brought in as a perturbation. The technique for doing this is the real test of a physicist. Alternatively, one could put all the bodies into a Runge-Kutta solver. This requires little skill and is compute-inefficient and in the end is not very enlightening.

 

[cj]

If the accuracy is insufficient, the sun is brought in as a perturbation. The technique for doing this is the real test of a physicist. Alternatively, one could put all the bodies into a Runge-Kutta solver. This requires little skill and is compute-inefficient and in the end is not very enlightening.

Did you mean "sufficient" or "insufficient"?

This is very useful feedback -- thanks! I need to quickly
get fairly familiar with numerical methods. I specifically
need to find analytical solutions to probelms that lend themselves
to an analytical solution; I then need to figure-out how to alter
the problem so that it is only solve-able via numerical methods.

The 2- and 3-body graviational problem is a great suggestion --
any others?

Thanks again!

 

[JohnDubYa]

RE: "It's also unfortunate, since as arildno says, the vast majority of real-life problems must be solved numerically, so the result is that the graduate has an inflated impression of his/her abilities. My own experience agrees with this. All the tricks on how best to approximate an intractable problem, and how best to numerically solve it, I learned in graduate school."

If all differential equations were solvable, we would be out of a job and physics would be considered a dead field. The intractabilty of problems is precisely what makes physics so interesting.

Here is the problem with differential equations:

Consider a simple equation like $$\frac{dx}{dt} = f(x)$$. The solution to this equation is a function x(t) that can be plotted on an x-t graph. Therefore, the solution has a definite shape.

And the solution DOES exist. There IS such a shape. But why doesn't it have in general an analytical solution?

Divide both sides by $$f(x)$$ and you get

$$\frac{\frac{dx}{dt}}{f(x)} = 1$$

Think about what this equation is saying. It says that whatever shape of x(t) is, it must produce a constant horizontal line when, at every value of t, its slope is divided by the height above the x-axis of f(x).

There is such a shape, but what are the chances that the shape would be exactly described by a single mathematical function at every value of t?

The chances are slim, although some examples do exist.


The simplest example I know of an intractable problem is simple pendulum drawn back to a large initial angle and released. If you are not allowed to make the small-angle approximation, you end up with a differential equation that must be solved numerically.

Velocity dependent friction can also create intractable problems. The diamagnetic hydrogen problem is also a good example.

I would highly recommend downloading Numerical Receipes and reading it.

 

[cj]

This is very educational -- I'm seaching for "Numerical Recipes" now!

Thanks.

 

[Dr Transport]

www.nr.com

 

[Clausius2]

I wish you were never in front of one problem that does not have analitical solution. Believe me, it is the Hell.

 

[theFuture]

If you are looking for examples it might be useful to work with a problem that has an analytical solution and then solve it numerically. This will give some understanding how well your approximation works and which sorts of approximations works best in any given situation. Numerical Recipies is a great place to start, too.

 

[JohnDubYa]

RE: "I wish you were never in front of one problem that does not have analitical solution. Believe me, it is the Hell."

Analytical solutions are boring. The whole art of approximation is what makes physics interesting.

How much do you really have to know about a physical system if the solution to the differential equation is analytic?

 

[arildno]

RE: "I wish you were never in front of one problem that does not have analitical solution. Believe me, it is the Hell."

Analytical solutions are boring. The whole art of approximation is what makes physics interesting.

How much do you really have to know about a physical system if the solution to the differential equation is analytic?

I fully agree with you, JohnDubya, in that "the art of approximation", or the exercise of our abilities to attack and shed light upon intractable problems is what remains of interest once a certain level has been reached
(It's like being properly equipped to explore/survive new and unknown territories; what might we discover?)

However, I would like to point out (and I'm sure you'll agree with me) that analytical solutions do have a certain educational value:
In being simple, the dynamics of the physics described by the analytical solution becomes "clear", and hence, these experiences are eminently suited in developing a basic, physical "intuition", i.e, a basic understanding of how physical systems work, for example how the "restraint" of energy conservation shows itself.

(I do not regard (insights from) intuition as essentially different from (insights from) experience; rather, in my view intuition is experience in distilled from)

 

[JohnDubYa]

I agree. My point was that in order to SOLVE physical systems little intuition is required if the differential equation has an analytic solution. At that point physics is nothing more than applied mathematics.

 

[BobG]

Easy example: Find the square root of two without hitting the square root button on your calculator.

Medium examples: Kepler's equation - going from Mean Anomaly to True Anomaly.

Solving a quadratic equation - if you're good with spreadsheets and lazy, a numerical method is easier than the quadratic equation (of course, a slide rule solves quadratic equations even quicker and easier than the Newton-Raphson method, provided you can get by with only 3 significant digits).

All can be solved on a spread sheet using the Newton-Raphson method, an easy numerical method to use and understand.

Harder example: Stable equilibrium longitude for Earth's triaxiality (with spherical harmonics to the 3rd degree and order). The Newton-Raphson method converges so slowly, it's virtually worthless. (In fact, Newton-Raphson doesn't work that well with logarithms, either.)

Hence the 'better' numerical methods which I don't actually know - I just let somebody else's computer program solve those (or solve it with a good graphing calculator, in the example I gave).