Since the late 1700s, it has been implicitly agreed among researchers of analytic mechanics that the unity of mass is implied in the Lagrangian equations of motion. Their simultaneous system of mixed partial and exact differential equations

[math]\frac{d}{dt}\frac{\partial L}{\partial \dot{q_k}}=\frac{\partial L}{\partial q_k}[/math], where k=1,2,3,…,n

And L is the energy function. If the left term is the absolute acceleration then the right term is a force. Conforming to Newton’s 2nd law of motion, the left term is equivalent to acceleration if and only if a factor of unity of mass exists.

What is the unity of mass? Is the value determined already? Is it the quantum of mass? Can this value be arbitrarily chosen? Is this value the same as the Planck mass?