Everything in the real world is described fully by non-linear analysis. In order to make such systems simpler, we can linearize (differentiate) them, and use a first or second order approximation, and in order to represent them on a computer, we can discretize them, which turns analytic techniques into algebraic ones. Therefore we’ve turned our non-linear analysis into linear algebra.
They use a row of spinning fins mid-way through their rockets to indirectly steer missiles by creating turbulent vortices which interact with the tail-fins and add an extra oomfph to the steering mechanism. The exact algorithm is classified, for obvious reasons.
This is cool I never heard of this. There are many other exceptions of course. Particularly with “turning things on” (car starting, computer starting, etc)
Compare also the central conceit of QM /Koopmania.
Take a classical nonlinear finite-dimensional system X described by a say a PDE. This is a dynamical system with evolution operator X → X.
Now look at the space H(X) of C/R-valued functions on the phase space of X. After completion we obtain an Hilbert space H.
Now the evolution operator on X induces a map on H= H(X).
We have now turned a finite-dimensional nonlinear problem into an infinite-dimensional linear problem.
There is a mystery which many applied mathematicians have asked themselves: Why is linear algebra so over-powered?
An answer I like was given in Lloyd Trefethen’s book An Applied Mathematician’s Apology, in which he writes (my summary):
Everything in the real world is described fully by non-linear analysis. In order to make such systems simpler, we can linearize (differentiate) them, and use a first or second order approximation, and in order to represent them on a computer, we can discretize them, which turns analytic techniques into algebraic ones. Therefore we’ve turned our non-linear analysis into linear algebra.
Seems like every field of engineering is like:
step 1: put the system in a happy state where everything is linear or maybe quadratic if you must
step 2: work out the diameter of the gas tube or whatever
step 3: cover everything in cement to make sure you never ever leave the happy state
if you found an efficiency improvement that uses an exponential then go sit in time out and come back when you can act like an adult
That description is distinctly reminiscent of the rise of containerization in software.
Not quite, helpful video, summary:
They use a row of spinning fins mid-way through their rockets to indirectly steer missiles by creating turbulent vortices which interact with the tail-fins and add an extra oomfph to the steering mechanism. The exact algorithm is classified, for obvious reasons.
This is cool I never heard of this. There are many other exceptions of course. Particularly with “turning things on” (car starting, computer starting, etc)
Compare also the central conceit of QM /Koopmania. Take a classical nonlinear finite-dimensional system X described by a say a PDE. This is a dynamical system with evolution operator X → X. Now look at the space H(X) of C/R-valued functions on the phase space of X. After completion we obtain an Hilbert space H. Now the evolution operator on X induces a map on H= H(X). We have now turned a finite-dimensional nonlinear problem into an infinite-dimensional linear problem.