Using Punch Cards by Bubbles Whiting (2016)
Source. Interview at the The Centre for Computing History.The key difference from Lagrangian mechanics is that the Hamiltonian approach groups variables into pairs of coordinates called the phase space coordinates:This leads to having two times more unknown functions than in the Lagrangian. However, it also leads to a system of partial differential equations with only first order derivatives, which is nicer. Notably, it can be more clearly seen in phase space.
- generalized coordinates, generally positions or angles
- their corresponding conjugate momenta, generally velocities, or angular velocities
There is a Turing machine that halts for every member of the language with the answer yes, but does not necessarily halt for non-members.
Video "Finite fields made easy by Randell Heyman (2015)" at youtu.be/z9bTzjy4SCg?t=159 shows how for order . Basically, for order , we take:For a worked out example, see: GF(4).
- each element is a polynomial in , , the polynomial ring over the finite field with degree smaller than . We've just seen how to construct for prime above, so we're good there.
- addition works element-wise modulo on
- multiplication is done modulo an irreducible polynomial of order
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