Lagrangian of point particles.
Ruslan A. Sharipov.
In previous texts, we discussed gravity within the framework of a new non-Einsteinian theory called the "3D-brane universe model". But besides gravity, the universe also contains matter. We've already considered matter, but only in general terms, without specifying its forms. Now we'll begin to specify them. First, we'll consider point particles.
The action integral for a point particle is the time integral of its Lagrangian.
The Lagrangian itself is simply a function. This distinguishes it from the field Lagrangian, which is the integral over spatial variables of the Lagrangian density. In our case, we choose the Lagrangian as follows:
The index nb indicates that we have chosen a non-baryonic particle, i.e., a dark matter particle. In the case of a baryonic particle, the index nb is replaced by the index br. In this formula, m denotes the particle's rest mass, and v denotes its velocity. The quantity c with the index nb is the critical velocity of the non-baryonic particle. Note that the critical velocity in the new theory depends on the particle type: for baryonic particles, it is one value, while for non-baryonic particles, it may be different. Moreover, non-baryonic matter may contain particles of various types with different critical velocities.
From the formula for the Lagrangian of a pointed particle, formulas for its energy and the components of its momentum are derived:
These formulas are in good agreement with those in Einstein's theory. The only difference is the magnitude of the critical velocity. In Einstein's theory it is equal to the speed of light for all types of matter. In the new theory each type of matter has its own critical velocity.
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