Conservation of Momentum Law can be considered as a Non-Holonomic Constraint for example for space-robots in free-floating mode of operation. In developing the equations of motion using Lagrangian approach one should consider especial terms for Non-Holonomic constraints. My question is that is it necessary to consider this constraint when the equations of motion are generated through Lagrangian approach or this constraint inherently satisfied in Lagrangian approach?
This constraint is inherently satisfied. In other words, it is a motion invariant. Conservation of angular momentum is a manifestation of a symmetry in the system. It is a consequence of Lagrangian being invariant with respect to certain rotational variables.
Of course this second order nonholonomic constraint is inherently satisfied in the Euler-Lagrange equations of motion, however, if the mechanical system has first order (kinematic) constraints e.g. due to the presence of wheels, which implies that the instantaneous mobility of the system is restricted, then these kinematic constraints (which are actually geometric distributions on the configuration space) should be incorporated into the Euler-Lagrange equations. This incorporation can be done by adding a force, called the constraint force (a force that does no work on admissible velocities) to the equations of motion.
Using the valuable keywords presented by Dr Reyhanoglu I found "Noether Theorem" as a mathematical proof to support this statement: "Momentum conservation is inherently satisfied in Lagrangian approach".
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