Lagrangian equations value formulation

To obtain the corresponding Lagrangian equations of motion. Eg is initially treated as a constant and later expanded using Eq. (83). It appears that Eq. (78) has exactly the form of Eq. (18), i.e., the Nose-Hoover thermostat has one equation of motion in common with both the Woodcock/Hoover-Evans and the Berendsen thermostats. However, in contrast to these other thermostats where the value of y was uniquely determined by the instantaneous microstate of the system (compare Eq. (79) with Eqs. (45), (50), and (56)), y is here a dynamical variable which derivative (Eq. (79)) is determined by this instantaneous microstate. Accompanying the fluctuations of y, heat transfers occur between the system and a heat bath, which regulate the system temperature. Because y = s s = y = s (Eq. (77)), the variable y in the Nose-Hoover formulation plays the same role as s in the Nose formulation. When y (or s) is negative, heat flows from the heat bath into the system due to Eq. (78) (or Eq. (60)). When the system temperature increases above To, the time derivative of y (or s) becomes positive due to Eq. (79) (or Eq. (63)) and the heat flow is progressively reduced (feedback mechanism). Conversely, when y (or s) is positive, heat is removed from the system until the system temperature decreases below To and the heat transfer is slowed down. 

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