Why are so many things squared in Physics?

Question

I don't know much about Physics, and I wasn't sure whether to ask here or Mathematics, but I have always wanted to know why when it comes to the physical world/universe, there are so many descriptions and problems and solutions that are squared.

I'm sure there are other constructs out there, but it seems everywhere I look when studying I see something squared, like meters per second squared, time with the t^2, the spring, bouncing ball, etc.

Does the universe just happen to be that way?

Why is there all this usage of squares?

If you are just starting out in physics then you are being carefully protected from the full complexity of the discipline. — dmckee --- ex-moderator kitten, May 11 '14 at 03:19
@dmckee I'm not in school or anything. I always just wanted to know. — johnny, May 11 '14 at 03:22
I think squares come in because of the symmetry of most things happening in space and time. If not squares, the square root of squares ( as with the 1/r potential of gravity). In a Tailor expansion for any function the first power changes sign and is useless for most physical descriptions. Same with the third power. The fourth will have a much smaller coefficient ( in useful functions). — anna v, May 11 '14 at 03:49
Relevant: God loves the 2-norm - http://www.scottaaronson.com/blog/?p=1345 — Johannes, May 11 '14 at 04:46
In mechanics, it's because Newton's laws are second order differential equations and the easiest approximation after linear is quadratic. — jinawee, May 11 '14 at 08:39
Out of curiosity, how can this be opinion base? Is there no scientific answer? — johnny, May 11 '14 at 17:52

score 2 · Answer 1 · answered May 11 '14 at 03:37

One way to think about such nonlinear behavior is as a feedback loop.

When you stretch a spring, you change its immediate stretchiness so it's a bit harder to stretch further.
When you push an object into motion, a moving object takes more work to push than a stationary one, so increasing its speed takes incrementally more energy.

The lazy answer is that things get squared because that's just what happens mathematically when you isolate a particular variable. Any exponent might pop out, and sine waves (or other transcendental functions) occur when an exponent would go to infinity. This occurs if you close a feedback loop, such as by putting a mass on a spring, and it's called a harmonic oscillator.

If it makes you feel better, exponents and nonlinearity are perpetually annoying in computational physics, because linear equations are easier to solve. In computer simulation, linear approximations are common and even nonlinear systems get "linearized" one of the final steps.

Why are so many things squared in Physics?

1 Answers1