The motion of a particle is described by the Lagrange’s equations :


               (3.1)

   L – Lagrangian of a system, qk – its generalised coordinates.
   We want to build a theory in which equations (3.1) remain valid in the case of crypto-informational interaction. And only the Lagrangian of the system should undergo a change.
   The Lagrangian L of particle, if it moves in accordingly with open information, is a famous physical Lagrangian Lphys. But how the Lagrangian will be changed in the case of crypto-informational interaction (when the particle receives encrypted information) ?
   Information is an additive quantity. In addition, in the absence of crypto-informational interactions, the full Lagrangian Lfull must go into the famous physical Lagrangian Lphys. So, we proposed that the full Lagrangian Lfull is sum of physical Lagrangian Lphys and Lagrangian of crypto-information interaction Lс-inf :

   Lfull = Lphys + Lc-inf .                 (3.2)

   In this case, Lc-inf must go to zero in the absence of crypto-information interaction. So, our task to find the Lagrangian Lс-inf.


   3.3.2. Non-relativistic case.

   For non-relativistic case the physical Lagrangian ([Landau, 1976], §4) :


,               (3.3)


   where :
   min – inertial mass;
   v – velocity of particle relative to the inertial reference frame;
   |v| – magnitude of the velocity.

   For crypto-information interaction we can expected that :
   1. Lc-inf - must be an additive to the Lagrangian Lphys(see. (3.2)).
   2. Lc-inf - should look like a Lagrangian Lphys (3.3).

   According to the provisions of the crypto-information concept (see 3.2), space and time are only a form of presentation of open information. There is an unambiguous correspondence between the open information received by the body and its coordinates. Therefore, we can differentiate open information by coordinates and time. Encrypted information cannot be represented in form of space and time. For this reason, there is not one-to-one correspondence between the received encrypted information and coordinates of space and time. And amount encrypted information cannot be differentiated by coordinates and time. Encrypted information is transferred by open information. That is, a message that carries encrypted information always carries open information too. Therefore, the Lagrangian of classified information Lc-inf will contain both public information and encrypted information. Therefore, the Lc-inf should contain terms that can be differentiated by time and space, and that cannot be differentiated by time and space. In the simplest case, the Lc-inf for a free particle has the form:

               Lc-inf = |v|`∙ (ac-inf ∙ |v|) .                 (3.4)

   Lc-inf is the same, as lagrangian for open information (3.3), but in it only the first factor describing the open information |v|`- undergoes differentiation in time and space (we marked this fact with the symbol ` with this factor). The second factor (ac-inf ∙|v|) describe encrypted information cannot be differentiated in time and space.
   This strange differentiation rule is explained by the fact that information in the case of crypto-informational interaction is transmitted using the three-pass Shamir’s protocol. A mathematically correct explanation of this rule can be given only if we know the algorithm that nature uses. For the proposed phenomenological model, we formulated this rule based on general considerations.
   Note, the function (3.4) is independent of the direction of v, and is a function only of its magnitude as (3.3).
   Since the information cannot be negative, we assume that ac-inf > 0.
            So, the full Lagrangian Lfull of a free particle is addition of Lagrangian of open information (3.3) and Lagrangian of encrypted information (3.4) :


               (3.5)

   We believe that the, ac-inf coefficient should be proportional to the inertial mass min :

ac-inf = a0 c-inf ∙ min ,                (3.6)

   where a0c-inf - is dimensionless constant of crypto-information interaction, the same for all particles of the system.
   If there is not crypto-information interaction a0c-inf = 0 and Lfull = Lphys.