1,187 karma · joined May 22, 2013
The Hamiltonian approach in contrast is breadth-first: you single out a time axis, start with some initial state, and consider all possibilities that a particle (or field in QFT) could evolve forwards in time just a tiny bit (this is what the Hamiltonian operator does). Then you add up all these possibilities to find the next state, and so you move forwards through time by keeping track of all possible evolutions all at once. This massive superposition of everything that is possible (with corresponding amplitudes) is what you call a state (or wavefunction) and the space that it lives in is the Hilbert (or Fock) space.
So Lagrangian/path-integral: follow full trajectories, then add up all possible choices. depth-first
Hamiltonian/time-evolution: add up all choices for a tiny step in time, then simply do more steps: breadth-first
I imagine it a bit like a scanline algorithm calculating an image as it moves down the screen (Hamiltonian) vs something like a stochastic raytracer that can start with an empty image and refine it pixel by pixel by shooting more rays (Lagrangian)
This is my layman explanation anyways...hopefully it helps, even though i can't say much about their relationship in classical physics.
I highly appreciate this approach: "As i have ranted about before, linear algebra is done wrong by the extensive use of matrices to obscure the structure of a linear map. Similar problems occcur with multivariable calculus, so here I would like to set the record straight"
Math education and textbooks are doing an awesome job obscuring simple ideas by focusing on weird details and bad notation. Always good to see people trying to counter this :)
The pdp-1 "demoscene" has been rather inactive for ~6 decades now :) Our hope is to breathe some new life into it.
A real tape reader would be great and is something we are considering for the future, but without a punch it's not very satisfying. There will be a paper tape visualization though (currently writing the code for that).
The photos don't show it but you can also have your PiDP-1 in white instead of blue. It will come with both sets of front panels.