Robert, thank you for your links.

From the point of view of "high theory" discretization of time breaks symmetry with respect to time shift. Just this symmetry  related with energy conservation, due to Noether theorem.

But we can still have discrete time symmetry  - shift for delta t. What is conserving in that case?

Remi, thank you for the link Now I got an idea of integrator and can choose a right one for my calculations :) My question will be closed now.

 I am trying but what Leimkuhler and Reach, have no any idea..................plz?

@Ram,

The correct names of the authors of the book to which Robert refers are Leimkuhler and Reich. A very readable book indeed!

To use the Noether theorem in classical mechanics you have to differentiate with respect symmetry parameter, so discreet symmetry does not produce conservation law in classics.

In quantum mechanics any symmetry is continuous, and any symmetry produces a conservation law. In quantum mechanics symmetry with respect to discreet time shift correspond to conservation of energy by modulo \delta E=(2\pi\hbar)/(\delta t). I.e. in discreet time one could not distinguish E and E+\delta E.

Robert, you are welcome!

When the system was considered discrete, continuous flow of energy can be converted to chain blocks. Confusion or turbulence between the energy block chain, the more freedom for their components and energy package apparently was missing ... just like defragmenting order entry and exit traffic on a winding highway in Tehran....

@Eugene,

although you declared the question closed, I would like to hint to the Chapter 8 of the following article of mine, which illustrates an interesting energy quasi-conservation in 'geometric' integrators. This gives a concrete example of what Robert described in his main contribution. 

Article Polyspherical grains and their dynamics

I think if we look at the Hamiltonian and the basis for energy conservation with this argument (i.e., the Hamiltonian is explicitly time independent)

Dear Mikhail,

I never heard about consistent quantum theory with discrete time.

As for discrete symmetry in classical theories, there definitely exist many discrete symmetries in crystals. We also can have crystal in 3+1 space and we will have time symmetry too.

@Ulrich, this is very interesting article. But some general questions stops me to like any knows algorithm for dynamics simulations:

1. When particle goes close to potential wall, there will be situations, when new position will be inside wall. How can I overcome this situation?

2. Even after reading your article I can't understand how to manage situation with 3 or more particles, coming close to each other? Sometimes situation prevent any next move, like for central particle under interaction with 5-6 external particles, moving into direction of central one.

Making time step too small may not solve such problems. Looks like God has very powerful cluster to calculate with infinite precision, or he just use really random numbers to resolve conflicts.

 see the  Hamiltonian Mechanics

Dear Ram, that is nice, that you know science "Hamiltonian Mechanics", but it has nothing to do with my question. Plz, in the future try to give advises with more sense.

@Eugene

To your questions:

1. Walls are best modelled as potential walls so that it are always forces (and not geometrical constrains) which hold the particles in their container. Potentials ~ d^4 or ~d^6 (d distance of a particle outside the geometrical container from the container surface) worked well in my applications. I once compared a cubical container with geometrical treatment to a spherical container with force based treatment and found energy conservation much better in the spherical case. In trying to cure the deficiency of the cubic case many possible directions opened up. I expected that it would take days of testing to find out which way to take. In the spherical case the implementation was unique and no need for optimization arose. Of course, other people may have other experiences, maybe based on a better insight.  

2. Even contact forces between solid grains have to be treated as 'soft body forces' where forces are determined by overlap (section of sets) of a particle with arbitrarily many neighbor particles. Here particles are assumed to have a fixed geometrical shape, but shapes are allowed to overlap and the repusive forces are computed from the overlaps. For the case that the fixed shape is a set union of overlapping spheres the forces and torques are worked out in my article (as are the resulting equations of motion in integrator form). The good news about this approach is that one has simply to implement the discretized version of equations of motions and they will run autonomously without ever facing conflicts. I found it absolutely necessary to control total energy during any simulation. If time steps are too large (say that a particle-particle collision or a particle-wall colision extends over less than 10 time steps) you get large energy fluctuations which spoil the simulation. For an efficient implementation it is important to have an efficient algorithms which finds out for a pair of particles if it could overlap and thus could give rise to contact forces. If your particles are spheres without rotational degrees of freedom one has to compute only forces and no torques and things become simpler. I found it, however, surprising how smoothly object oriented programming allows to treat point particles and rigid bodies on equal footing.

If you would disclose details of your system, I possibly could provide some more specific hints.

On the usefulness or uselesness of multiple precision arithmetics in this context there is also an article:

Article Precision-dependent symmetry breaking in simulated motion of...

Dear Ulrich, thank for your article. I'll come back to you with my system details.