We present a reformulation of the coupled dynamics of a Bose-Einstein condensate and its thermal cloud as a two-field Lagrangian theory, with the condensate amplitude (x, t) and a dimensionless entropy field (x, t) as dynamical variables. All parameters are fixed by microscopic quantities with no free constants. The Euler–Lagrange equations conserve total momentum by construction and reduce to the ZNG hydrodynamic equations in the adiabatic limit. A collective-coordinate reduction yields Newton's second law with an entropy-gradient force that vanishes identically for decoherent classical systems, explaining the empirical success of Newtonian mechanics. For coherent quantum systems, spatial entropy gradients---engineered via applied temperature gradients---produce a measurable relative displacement between condensate and thermal cloud. In a ^87Rb BEC with T/ x = 100~K/m and = 2 100~Hz, we predict xₑ₄₋ 10~m with ^-1 scaling, providing an experimentally testable signature.
Lohith et al. (Sun,) studied this question.