Re: This Week's Finds in Mathematical Physics (Week 260)

John Baez wrote:
Also available as http://math.ucr.edu/home/baez/week260.html

8) James Nearing, Mathematical Tools for Physics, available at
http://www.physics.miami.edu/~nearing/mathmethods/

Unlike the usual dry and formal textbook, it reads like a friendly
uncle explaining things in plain English, trying to cut through the red
tape and tell you how to actually think about this stuff.

For example, on page 3 he introduces the hyperbolic trig functions:

Where do hyperbolic functions come from? If you have a mass
in equilibrium, the total force on it is zero. If it's in *stable*
equilibrium then if you push it a little to one side and release
it, the force will push it back to the center. If it is *unstable*
then when it's a bit to one side it will be pushed farther away
from the equilibrium point. In the first case, it will oscillate
about the equilibrium position and the function of time will be
a circular trigonometric function - the common sines or cosines of
time, A cos(wt). If the point is unstable, the motion will be
described by hyperbolic functions of time, sinh(wt) instead of
sin(wt). An ordinary ruler held at one end will swing back and
forth, but if you try to balance it at the other end it will fall
over. That's the difference between cos and cosh.

No, not at all.

The Jacobi am(w t, k) as general solution of the pendulum equation
1/2 phi'^2 + 1-cos(phi)=E
has the sin function for all times as a reasonable approximation for E~0
stable equilibrium, but hyperbolic functions at the instable equilibrium
1/2 phi'^2 + cos(phi)=E
are good for nothing at E~1.

Implicitely the author seems to assume the instable equilibrium angle
phi(t) is small for all times. Short time approximations up to second
order in time with complicated functions are not so much the goal of
solutions than the theme of the equation of motion itself.

--

Roland Franzius

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