Note on the Interpretation of Proper Mass as a Constant Lagrange Multiplier

The timelike world line C₀ of a free particle is a maximizing curve for the integral I = ∫ds in the class Γ of neighboring admissible timelike curves joining the events A, B, and satisfying the side-condition imposed on the 4-velocity $\upvarphi ={g}_{ij}\ {\overset{\cdot }{x}}^i{\overset{\cdot }{x}}^j=1\left({\overset{\cdot }{x}}^i={dx}^i/ ds\right)$. Considering the problem of extremizing integral I as a time optimal problem, we show that the multiplier λ (s) associated with the equation φ = 1 is constant along C₀ and may be identified with the proper mass m of the free particle. The constancy of m can thus be regarded as a consequence of the path dependence of proper time.