Lagrangian mode

The motion of a particle in the flow field can be described in the Lagrangian coordinate with the origin placed at the center of the moving particle. There are two modes of particle motion, translation and rotation. Interparticle collisions result in both the translational and the rotational movement, while the fluid hydrodynamic forces cause particle translation. Assuming that the force acting on a particle can be determined exclusively from its interaction with the surrounding liquid and gas, the motion of a single particle without collision with another particle can be described by Newton s second law as... 

For simulating computationally the spatial and temporal evolution of both physical and chemical processes in mixing devices operated in a turbulent singlephase mode, two essentially different approaches are available the Lagrangian approach and the Eulerian technique. These will be explained briefly. 

Where ip is an arbitrary and unimportant phase. Thus, we see that the normalization of the mode is given by the commutator (7.182) and the normalization of the Lagrangian. The justification of the prefactor of Eq. (7.168) comes from the fact that the Lagrangians for us and ut can be found by varying at second order the Lagrangians for the scalar field and of the gravitational field, respectively. The meaning of the amplitude Uk here is that it corresponds to the variance of the quantities us,ut-... 

The subsystem or atomic Lagrangian integral is defined by the standard mode of integration used in the definition of the subsystem functional [ F, iJ] for a stationary state and for the definition of subsystem properties. 

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] ... 

If these basis functions, (O (0. e (0. and Xq(0, are time-independent, then Equation 5.2 is simply a Lagrangian for a set of harmonic oscillators. In a protein system, however, strong anharmonicity makes the basis functions time-dependent, and thus we refer to the model as moving normal mode coordinates. ... 

We will explain energy transfer at near zero temperatures using the classical formulation of a model system composed of only two modes, the perturbed mode and the resonance mode 2- The Lagrangian is... 

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. ... 

Other modes of numerical differentiations are based on Lagrangian differentiation formulas, Thylor exptmsions, and Spline and Splaus functions. Also, differentiation is used after discrete Fourier transformation. In this case, noise can be eliminated by neglecting the higher terms of the polynomial, but often it is not etisy to rind the limit of noise frequencies and the higher frequencies of the effective signals. 

In the Euler—Lagrangian approach of two-phase flow (see, e.g., Crowe et al, 1996), the particles are treated as point particles the finite volume of the particles is not considered and the flow around the particles is not resolved. The motion of the particle is simulated by means of Newton s second law and that is why the fluid—particle interaction force is needed and the empirical correlations enter. Although the flow around the particles is not resolved, any empirical correlation does reflect the hydrodynamics of the canonical case involved. The use of the Euler—Lagrange approach, or point-particle method, is usually restricted to the more dilute gas—solid and liquid—soHd systems. Ignoring the mutual interaction of particles is therefore not too serious a simplification. The fluid-particle interaction can be treated in the simpler one-way mode or according to the more complicated two-way coupling mode in which the particles also affect the carrier phase flow field (Decker and Sommerfeld, 2000 Derksen, 2003 Derksen et al, 2008). 

The mode shapes, as presented previously for describing the deformations of the links, can be obtained from a finite element analysis of each of the bodies of the structure. The Lagrangian formalism enables a straightforward derivation of the equations of motion. The Lagrangian can be expressed in terms of nodal displacements and their time derivatives. Here it is formulated as a function of the joint angles 0, the joint deformation variables p and the link modal coordinates g, and their time derivatives, as follows L(0yp q p q) = ifJ5(, p,g,, p,g) -PE fP q), For a serial link manipulator with n joints and n links, the equations of motion take the form ... 

---