  The origins of thermodynamics from the microscopic properties of matter have
not been satisfactorily accounted for. This work presents a formulation that
connects Lagrangian mechanics to thermodynamics. By using such a formulation
and following similar steps to those performed in the laboratory, the heat
capacities, energetic and mechanical response coefficients, absorbed and
emitted heats, entropy changes, and thermodynamic energies of a prototype water
system, employed as an illustrative example, are found to be close to the
experimental results on water bulk. The present formulation is realistic. After
connecting Lagrangian mechanics to Langevin stochastic dynamics, the first law
of thermodynamics is derived. The formulation not only exhibits the
transformation of time-reversible equations onto time-irreversible equations,
and the molecular origins of heat as well, but also reveals new pathways to
investigate the thermodynamics of systems of nanometric sizes.
