Lagrangian dynamics

A further advantage of using Lagrangian dynamics is that we can easily impose boundary conditions and constraints by applying the method of Lagrangian multipliers. This is particularly important for the dynamics of the electronic degrees of freedom, as we will have to impose that the one-electron wavefunctions remain orthonormal during their time evolution. The Lex of our extended system can then be written as ... 

Meneveau, C., T. S. Lund, and W. Cabot (1996). A Lagrangian dynamic subgrid-scale model of turbulence. Journal of Fluid Mechanics 319, 353-385. 

This section will be broken into a number of discussions. The first will be on a naive 5(7(2) x 5(7(2) extended standard model, followed by a more general chiral theory and a discussion on the lack of Lagrangian dynamics associated with the B3 field. This will be followed by an examination of non-Abelian QED at nonrelativistic energies and then at relativistic energies. It will conclude with a discussion of a putative 5(9(10) gauge unification that includes the strong interactions. 

To overcome some of the limitations just mentioned that are associated with purely empirical models, simulations that include various aspects of the inhaled aerosol dynamics have been developed. The simplest of these belong to a class of models we refer to as Lagrangian dynamical models (LDMs), meaning that the model simulates some of the dynamical behavior of the aerosol in a frame of reference that travels with the aerosol (i.e., a Lagrangian viewpoint ). 

Lagrangian dynamics provides two approaches for dealing with systems with general holonomic constraints ... 

The first approach of Lagrangian dynamics consists of transforming to a set of independent generalized coordinates and making use of Lagrange s equations of the first kind, which do not involve the forces of constraint. The equations of constraint are implicit in the transformation to independent gen-... 

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. 

Wells, D. A., Theory and Problems of Lagrangian Dynamics, McGraw-Hill, New York, 1967. 

Compaction, consolidation, and subsidence. A formal approach to modeling compaction, consolidation, and subsidence requires the use of well-defined constitutive equations that describe both fluid and solid phases of matter. At the same time, these would be applied to a general Lagrangian dynamical formulation written to host the deforming meshes, whose exact time histories must be determined as part of the overall solution. These nonlinear deformations are often plastic in nature, and not elastic, as in linear analyses usually employed in structural mechanics. This finite deformation approach, usually adopted in more rigorous academic researches into compressible porous media, is well known in soil mechanics and civil engineering. However, it is computationally intensive and not practical for routine use. This is particularly true when order-of-magnitude effects and qualitative trends only are examined. 

Book W.J., Recursive Lagrangian Dynamics of Flexible Manipulator Arms, Int. J. of Robotics Research 3,3 (1984), 87-101. 

The key to these more efficient treatments is a natural canonical formulation of the rigid body dynamics in terms of rotation matrices. The orientational term of the Lagrangian in these variables can be written simply as... 

Arbitrary-Lagrangian-Eulerian (ALE) codes dynamically position the mesh to optimize some feature of the solution. An ALE code has tremendous flexibility. It can treat part of the mesh in a Lagrangian fashion (mesh velocity equation to particle velocity), part of the mesh in an Eulerian fashion (mesh velocity equal to zero), and part in an intermediate fashion (arbitrary mesh velocity). All these techniques can be applied to different parts of the mesh at the same time as shown in Fig. 9.18. In particular, an element can be Lagrangian until the element distortion exceeds some criteria when the nodes are repositioned to minimize the distortion. 

M.J. Frits, Two-Dimensional Lagrangian Fluid Dynamics Using Triangular Grids, in Finite-Difference Techniques for Vectorized Fluid Dynamics Calculations (edited by D.L. Book), Springer-Verlag, New York, 1981. 

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]. 

Although constrained dynamics is usually discussed in the context of the geometrically constrained system described above, the same techniques can have many other applications. For instance, constant-pressure and constant-temperature dynamics can be imposed by using constraint methods [33,34]. Car and Parrinello [35] describe the use of the extended Lagrangian to maintain constraints in the context of their ab initio MD method. (For more details on the Car-Parrinello method, refer to the excellent review by Gain and Pasquarrello [36].)... 

We recall, from elementary classical mechanics, that symmetry properties of the Lagrangian (or Hamiltonian) generally imply the existence of conserved quantities. If the Lagrangian is invariant under time displacement, for example, then the energy is conserved similarly, translation invariance implies momentum conservation. More generally, Noether s Theorem states that for each continuous N-dimensional group of transformations that commutes with the dynamics, there exist N conserved quantities. 

The equality in the preceding constraint is in the sense of the Ti(Y) norm for fixed t. The actual Lagrangian controlling the dynamics then is defined as... 

In preparation for this, the equations of gas dynamics will reproduce the conservation laws of impuls, mass and energy that can be written in a number of different ways with respect to Eulerian (x,t) or Lagrangian (s,f) variables, where x is the coordinate of a particle and s is the initial coordinate of a particle or the quantity... 

Equations of gas dynamics with heat conductivity. We are now interested in a complex problem in which the gas flow is moving under the heat conduction condition. In conformity with (l)-(7), the system of differential equations for the ideal gas in Lagrangian variables acquires the form... 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

An important alternative to SCF is to extend the Lagrangian of the system to consider dipoles as additional dynamical degrees of freedom as discussed above for the induced dipole model. In the Drude model the additional degrees of freedom are the positions of the moving Drude particles. All Drude particles are assigned a small mass mo,i, taken from the atomic masses, m, of their parent atoms and both the motions of atoms and Drude particles (at positions r, and rdj = r, + d, ) are propagated... 

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