Imagine that we have point interacting particles, that moving according law like x(i, t+dt)=x(i,t)+f(i,j,t)*dt. Or also velocities can be included. In the limit dt -> 0 we can get equations for well known continuous case with Hamiltonian. Can we have Hamiltonian for small dt?
Depends what you mean by discrete mechanics. The old quantum theory sometimes named semiclassical a la Bohr, has a discrete number of wave lengths in an orbit.
usually formulated as
closed line integral of p dx = nh
but uses the usuall hamiltonians.
On the other hand you can calculate orbits discretly using
Classical mechanics in small steps, but numerical stability is not a minor issue.
I mean: what dynamics we will get if will change time derivative to difference.
I see the problem in following: if you choose time step, then when particle will come to potential peak, it can jump over it. How can we fix this?
There do exist systems, evolving in discrete time, that do have a Hamiltonian structure: This http://www1.maths.leeds.ac.uk/~frank/math5492/Lectures.pdf
might be a good place to start.
See, also, Article Integrable Discrete-Time Systems and Difference Operators
and
https://pdfs.semanticscholar.org/f62a/b7106de256a593076cab39ddef0a2c8147bd.pdf
Dear Stam Nicolis ,
Thank you for links. It looks like we only can talk about approximately conserving algorithm, up to some definite order. I just was playing with quadratic Hamiltonians and got exactly energy conserving simple algorithm. But find no more complex cases.
The main problem I see, is that there is an approximation
involved, and that the iteration must be stable numerically, not runaway.
There are in fact some clever solutions.
Otherwise you must admit purely discrete dynamics. Hard to imagine a real system doing that.
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