differential equations

[gusl]

Does anyone ever use ODE models in which the derivative is not wrt time?

Comments

Re: differential equations

[http://users.livejournal.com/cunctator_/]

are u kiddin? Well, yes. Say, auction theory, google some lecture notes on vuong perrigne paper.

[elsumis.livejournal.com]

one thing that comes to my mind are computations of normal modes in 1D physical systems.

[gustavolacerda.livejournal.com]

Aren't the normal modes just the eigenvectors of the Hamiltonian?

Re: differential equations

[gustavolacerda.livejournal.com]

so what are the derivatives wrt, in these cases? price?

[elsumis.livejournal.com]

yup, eigenmodes of the Schrodinger equation for instance (which may be represented as eigenvectors of a Hamiltonian matrix), but also normal modes of acoustic/photonic cavities, or even the linear heat/diffusion equation. these all do have in common, though, that they are 'steady-state' solutions of a temporal problem, so they aren't totally unrelated to time.

tangentially

[gustavolacerda.livejournal.com]

ah, since they describe steady-states, I guess I should say "the principal eigenvector of the Hamiltonian". (Or it is eigenfunction?)

Is the Hamiltonian matrix a dynamics matrix, which tells you how to iterate the system (i.e. like the transition matrix for a Markov Chain)? If so, that would imply that the system can be described as a finite-dimensional vector.

Re: tangentially

[elsumis.livejournal.com]

Hmm, not quite a transition matrix. The equation for quantum wave evolution is the schrodinger equation. (See http://en.wikipedia.org/wiki/Schrodinger_equation under "General Quantum System", the first equation).

To convert the Hamiltonian (H) into a time-iteration matrix (call that M) for a time step dt, you'd have M = exp(-i*dt*H/hbar) (http://en.wikipedia.org/wiki/Matrix_exponentiation#Linear_differential_equations) which is approximately 1-i*dt*H/hbar for short timesteps dt. All the eigenvalues of M are complex numbers with magnitude 1 (M is a unitary matrix)... so they are all principal eigenvalues.

Usually, Hamiltonians are inifinite-dimensional since they model (for example) a particle which can occupy an infinite number of positions. Physicists seem to get away with using infinite dimensional matrices due to some math details I never learned (Hilbert spaces and the spectral theorem and such).

Re: tangentially

[gustavolacerda.livejournal.com]

Thanks.

After some confusion, I now see how you can have many principal eigenvectors if the eigenvalues aren't restricted to the reals.

Re: differential equations

[http://users.livejournal.com/cunctator_/]

Type - valuation of the object being auctioned. Like on page 28 here: http://books.google.ru/books?id=fIPSYiNYvK4C
Simplest setup of the Maximum principle gives you an ODE so there might be a lot of examples.