In computational physics it is common to formulate Hamilton's principle in a semi-discrete way, where space is continuous but time is discrete: in other words the Lagrangian $$L(q, \dot q, t): \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{Z}\to \mathbb{R}$$ becomes a function of two real and one integer variable. Imposing Hamilton's principle of extremal action leads to particular-nice discrete time evolution equations that automatically obey a discrete Noether's theorem, preserve symplectic structure, etc.

Is there any sensible way of pushing the above idea to a setting where $L$ is purely discrete? I.e., $$L:\mathbb{Z}^3\times\mathbb{Z}^3\times \mathbb{Z}\to\mathbb{Z}$$ where position, mass, velocity, time, and potential energy are all integer quantities?

What will the Euler-Lagrange equations look like? First-order optimality of the action looks very different, since there is no longer a continuous derivative to set equal to zero, but I think one can write down systems of inequalities that encode the fact that the action is (discretely) extremized. Do you get any kind of sensible time evolution from these? Is there some equivalent to Noether's theorem in this setting?


Comments to the question (v2):

  1. It often is possible to formulate (discrete) time-evolution equations/equations of motion (eoms) in a discretized theory. This is of course useful in computational physics. However OP asks for a variational action principle for a fully discretized theory. Hence we will not discuss further the case where the eoms (without a variational principle) constitute the first principle of the theory.

  2. In a less ambitious scheme we are just supposed to check that when some eoms are satisfied, then the variation of some action is zero. (In other words, the action has a stationary point.) This is possible in some case, but it runs against the spirit of an off-shell formulation, and we shall not discuss it further in this answer.

  3. In the most ambitious scheme we are supposed to derive a discrete version of Euler-Lagrange (EL) equations (eoms) from a variational action principle. This is what we will be interested in here.

  4. Horizontal discretization (e.g. discretization of the time variable $t$ in point mechanics and spacetime variables $x^{\mu}$ in field theory) is not a problem, as OP mentions, see e.g. this Phys.SE post and links therein.

  5. The problem is vertical discretization, i.e. discretization in the target space for the dynamical active variables of theory (say $q$ in point mechanics). We will only discuss this latter case from now on.

  6. One can still postulate a principle of least action, but it not clear how to obtain (conditions on) a variational derivative, if the $q$ variable only takes discrete values.

  7. In some cases, there will be a natural candidate for the replacement of Euler-Lagrange equation, cf. pt. 2, but it is unclear how that could be derived from the above principle of least action alone in the spirit of pt. 3. We stop short of declaring a no-go theorem, but it certainly does not look promising.

  8. Noether's theorem for discrete symmetries was discussed e.g. in this Phys.SE post, which also elucidate some of the difficulties in formulating a variational principle with vertical discretization.


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