Equations of motion Lagrangian

The specific first Piola-Kirchhoff stress 11, specific second Piola-Kirchhoff stress and specific EuIct stress t are defined by 

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The first approach is based on introducing simple velocity or position rescaling into the standard Newtonian MD. The second approach has a dynamic origin and is based on a refonnulation of the Lagrangian equations of motion for the system (so-called extended Lagrangian formulation.) In this section, we discuss several of the most widely used constant-temperature or constant-pressure schemes. 

Another popular approach to the isothennal (canonical) MD method was shown by Nose [25]. This method for treating the dynamics of a system in contact with a thennal reservoir is to include a degree of freedom that represents that reservoir, so that one can perform deterministic MD at constant temperature by refonnulating the Lagrangian equations of motion for this extended system. We can describe the Nose approach as an illustration of an extended Lagrangian method. Energy is allowed to flow dynamically from the reservoir to the system and back the reservoir has a certain thermal inertia associated with it. However, it is now more common to use the Nose scheme in the implementation of Hoover [26]. 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

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. 

The particles position in the flow field is computed by solving the Lagrangian equations of motion for the particles with the inertial drag force, dependent on the density and size of the particles taken into account. 

In order to determine the normal frequencies, the general form of equation (14) is introduced into the Lagrangian equations of motion in terms of the g s. The result is... 

Following the logic of Euler, after integrating by parts to replace the term in Sq by one in Sq, this implies the Lagrangian equations of motion. 

The variational formalism makes it possible to postulate a relativistic Lagrangian that is Lorentz invariant and reduces to Newtonian mechanics in the classical limit. Introducing a parameter m, the proper mass of a particle, or mass as measured in its own instantaneous rest frame, the Lagrangian for a free particle can be postulated to have the invariant form A = mulxiilx = — mc2. The canonical momentum is pf, = iiiuj, and the Lagrangian equation of motion is... 

One situation in which solution of Hamilton s equations becomes trivial is when H is a constant of the motion and where the coordinates qk do not appear in the Lagrangian. Such coordinates are said to be cyclic or ignorable. In this special case the Lagrangian equation of motion reduces to... 

Other methods for performing constant-temperature molecular dynamics calculations have been proposed recently. Evans (72) has introduced an external damping force in addition to the usual intermolecular force in order to keep the temperature constant in the simulation of a dissipative fluid flow. In another method, Haile and Gupta 13) have imposed the constraint of constant kinetic energy on the lagrangian equations of motion to perform calculations al constant temperature. 

In terms of variables r and x in the plane of motion, the Lagrangian equations of motion are... 

Equations [24] are simply the equations of motion for the four-particle n-butane molecule, and for s = 0, Eq. [24] could have been obtained directly from the Lagrangian equations of motion Eq. [2]. This is the procedure by which it is obtained in Reference 49, and Eq. [24] is identical to Eq. [5] of that work. To be consistent with Reference 49, Eq. [24] is recast in the more compact form ... 

We now derive the Lagrangian equations of motion keeping in mind that Uj < k) and Wy(qA ) are treated as independent coordinates. 

By substituting from (2.20) into the Lagrangian equation of motion... 

Incremental Form of the Total Lagrangian Equation of Motion... 

The methods described above address the solution to Newton s equations of motion in the microcanonical NVE) ensemble. In practice, there is usually the need to perform MD simulations under specified conditions of temperature and/or pressure. Thus, in the literature there exist a variety of methodologies for performing MD simulations under isochoric or isothermal conditions [2,3]. Most of these constitute a reformulation of the Lagrangian equations of motion to include the constraints of constant T and/or P. The most widely used among them is the Nose-Hoover method. 

Many finite difference analogs of the Lagrangian equations of motion of a compressible fluid in two dimensions have been proposed. The Magee method was developed and used during the last 50 years at the Los Alamos National Laboratory by Kolsky, Orr, Browne and Hoyt, and others. 

The Eulerian equations of motion are more useful for numerical solution of highly distorted fluid flow than are Lagrangian equations of motion. Multicomponent Eulerian calculations require equations of state for mixed cells and methods for moving mass and its associated state values into and out of mixed cells. These complications are avoided by Lagrangian calculations. Harlow s particle-in-cell (PIC) method uses particles for the mass movement. The first reactive Eulerian hydrodynamic code EIC (Explosive-in-cell) used the PIC method, and it is described in reference 2. The discrete nature of the mass movement introduced pressure and temperature variations from cycle to cycle of the calculation that were unacceptable for many reactive fluid dynamic problems. A one-component continuous mass transport Eulerian code developed in 1966 proved useful for solving many one-component problems of interest in reactive fluid dynamics. The need for a multicomponent Eulerian code resulted in a second 2DE code, described in reference 4. Elastic-plastic flow and real viscosity were added in 1976. The technique was extended to three dimensions in the 1970 s and the resulting 3DE code is described in Appendix D. 

It can now be shown that the Lagrangian equations are equivalent to the more familiar Newton s second law of motion. If qi = r, the generaUzed coordinates are simply the Cartesian coordinates. Introducing this definition of the generalized coordinates and the corresponding Lagrangian function (2.6) into the Lagrangian equations of motion (2.14), we get ... 

The function L is known as the Lagrangian function for the system, and the set of equations represented by 2-22 as the Lagrangian equations of motion. 

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