Coordinate systems Lagrangian

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

While the Eulerian system has intuitive appeal, it is the Lagrangian coordinate system that is more convenient mathematically and in many practical applications. In this system, the coordinate is fixed to the material and moves with it. It is sometimes called the material coordinate system. In Fig. 2.2, the boxcars can be numbered, so the position of a car in this system never changes. By convention, the Lagrangian coordinate (h) is chosen so that it is equal to the Eulerian coordinate (x) at some time t = 0. Figure 2.10(b) illustrates a Lagrangian h-t diagram of the same system as shown in Fig. 2.10(a) with the Eulerian system. Because the flow is independent of the coordinate system chosen to describe it, both systems must lead to the same results. 

The form of the jump conditions also depends on the coordinate system. Substituting (2.40) into the general Eulerian form of the momentum jump condition (Table 2.1) yields the Lagrangian jump condition... 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

Lagrangian coordinates The coordinate system in whieh the material position (h) and time (t) are the independent variables. The dependent variables are deseribed as funetions of a partiele position within the material which had coordinate x = hat time t = 0. Also known as material coordinates. 

In quantum mechanics, the spatial variables are constituted by generalized coordinates (, ), which replace the individual Cartesian coordinates of all single particles in the set. The Lagrangian equations of motion are the Newtonian equations transposed to the generalized coordinate system. 

Recent efforts to distinguish between the terms burning velocity and flame speed on the basis of Eulerian and Lagrangian coordinate systems appear to introduce confusion. Therefore, the terms are used interchangeably here, as synonyms for such terms as deflagration velocity, wave speed, and propagation velocity. They all refer to velocities measured with respect to the gas ahead of the wave. 

There is one major technical problem that must be overcome In a liquid crystal most properties are best expressed relative a to a director based coordinate system. This is not a problem in a macroscopic system where the reorientation rate of the director is virtually zero but it can be a problem in a small system such as a simulation cell where the director is constantly diffusing on the unit sphere. When NEMD methods are applied the fictitious mechanical field exerts torques that twist the director and might make it impossible to reach a steady state. This problem has been solved by devising a Lagrangian constraint algorithm that fixes the director in space so that a director based coordinate system becomes an inertial frame. 

Other regional transport models, such as the Regional Lagrangian Model of Air Pollution (RELMAP Eder et al., 1986), use a different computational scheme than Eulerian models. In a Lagrangian model, the coordinate system moves with a parcel of air and mass balance of pollutant concentrations is computed on a parcel as it moves through space. 

Since the above derivation could be carried out for any value of j, there are 3n such equations, one for each coordinate <7,. They are called the equations of motion in the Lagrangian form and are of great importance. The method by which they were derived shows that they are independent of the coordinate system. 

The starting point of the theoretical treatment is the Lagrangian for an ensemble of charged particles in an exterior magnetic field II. Under the neglect of all intramolecular magnetic interactions this Lagrangian, if referred to the space fixed laboratory coordinate system reduces to D... 

As the second step in our derivation we will introduce a molecule fixed coordinate system and we will rewrite the Lagrangian using the corresponding generalized coordinates. 

This Lagrangian should be thought of as dependent on 3Ne + 6 generalized coordinates, qt, and velocities, g, respectively. These are the 3Ng coordinates be, Ce which describe the relative positions of the Ne electrons with respect to the nuclear frame three coordinates Xo, Vo and Zo which describe the position of the molecular center of mass as referred to the laboratory coordinate system, and three Eulerian angles 6, and x which describe the instantaneous orientation of the molecular coordinate system with respect to the space fixed X-, Y- and Z-axes. There are numerous ways of specifying Eulerian angles. Because of later reference we will follow the choice used by Wilson et where and 6 are the ordinary polar coordinates of the molecular c-axis O d n 0 < < 2n) and x is the angle between the nodal line N and the positive b axis as is illustrated in Fig. IV.2. x is positive for clockwise rotation about the c axis. 

In order to remove the explicit dependence on the position of the molecular coordinate system, Ro — ro), the Lagrangian given in Eq. (IV. 13) may be modified by subtracting the total difierential of an appropriate scalar function with respect to time ... 

Subtracting this result from the Lagrangian in Eq. (IV. 13) shows that the (Ro — Wo)-dependence is indeed removed and the field independent contributions which also depend on the velocity of the origin of the molecular coordinate system, Vq — [ o + (ft) X Wo)], add up to the potential energy of the molecular electric dipole moment within a virtual electric Stark-field... 

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