Lagrangian Oscillator Dynamics

The Lagrangian L — T — V consists of the total kinetic energy T and the total potential energy V. denotes the generalized non-conservative forces. Here, it is the compression force The time-dependent x(r), here, it is l and mh. represents the generalized variables, denoting the displacements of O atoms from their 

dco = lo + hq is the distance between the adjacent oxygen ions at equilibrium without contribution of Coulomb repulsion. The dc = di + dn denotes the distance at quasi-equilibrium with involvement of the repulsion. The displacement of mc = ml -f Al — mh + Ah is the change in distance between the adjacent oxygen ions at quasi-equilibrium. Ax is the dislocation caused by repulsion. The ml and mh take opposite signs because of the 0 H, and H-O dislocates in the same direction [9]. A harmonic approximation of the potentials at each quasi-equilibrium site by omitting the higher-order terms in their Taylor s series yields 

In the Taylor series, the = 1 term in the potentials equals zero for the L and the H segment at ideal equilibrium. At quasi-equilibrium, the criterion of V x( x) + Vcidc) = 0, or V x Mx + V c-uc — 0. must meet Here, V denotes the first-order derivative at the quasi-equilibrium position, i.e., dVc/dr l. Terms of n = 2, or the curvatures of the respective potentials, denote the force constants, i.e., kx = V = d Vx/dr ljj for harmonic oscillators. Terms of n 3 are insignificant and omitted. 

Substituting Eqs. (37.2) and (37.4) into (37.1) yields the Lagrangian equations for the 0 H-0 coupled oscillators. 

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Lagrangian, respectively. At first we note that the total energy is conserved in both the dynamics, with oscillations orders of magnitude smaller than the oscillations of the potential energy. The latter presents on the other hand a behavior that is quite different in the two cases. For the case in which the charges are equilibrated at each step, the oscillations are quite large, of the order of 3.5 x 10-3 au, and they last for the whole trajectory. On the other hand, for the extended Lagrangian approach, after an initial period... 

Many MD studies on the kinetic processes of proteins have revealed the existence of a specific pathway of energy flow [6-10]. In this chapter, for the purpose of studying anharmonic dynamics, we propose two models in which protein motions are assumed to be described by perturbed harmonic oscillators [11-13]. These models are based on the success of the harmonic/quasiharmonic model in the description of the equilibrium properties of proteins. Firstly, we consider vibrational energy transfer between normal modes, based on the Lagrangian [11,12] ... 

We first assume that, within a sufficiently short period of time on the trajectory, the system can be considered to approximately behave as a set of harmonic oscillators. Under this assumption, it is possible to define normal mode coordinates characterizing the short-term dynamics at any instant of time on the trajectory. It is, however, expected that due to the influence of anharmonicity, any two sets of coordinate systems, each derived from different portions of the trajectory, would not be identical to each other, but rather would have different origins and orientations. Based on this idea, we introduced the time-dependent Lagrangian L(t) in Equation 5.2, which we refer to as moving normal mode coordinates. ... 

Lemkul, J. A., Roux, B., van der Spoel, D and MacKerell, A. D., Jr., Implementation of Extended Lagrangian Dynamics in GROMACS for Polarizable Simulations Using the Classical Drude Oscillator Model, In Press./ Comput. Chem., 2015. 36,1480-1486. 

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