Lagrangian equation

Applying the Lagrangian equations to this form for L gives the equations of motion of the qj coordinates ... 

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 Lagrangian equations contain nothing more than the original Newtonian equations, but have the advantage that the coordinates may be of any kind whatever. This is of particular importance when analyzing phenomena in which the motion of material particles is not observed directly, such... 

The Lagrangian equations can be turned into another useful form involving generalized coordinates and momenta and by defining the Hamiltonian function... 

To incorporate the constraints in our least squares problem, we consider the Lagrangian equation... 

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. 

As a next step we combine the Lagrangian Equation (24-1) of an in-situ production/ consumption process, R(Ct), with the Eulerian view (Eq. 24-4) of transport and reaction. There are two ways to analyze the resulting situation, a more intuitive method and a formal approach. We elaborate on the former and explain the latter in Box 24.1. 

From the Lagrangian equation we calculate the temporal evolution of the concentration in a fixed water parcel, Ct. Given the production/consumption function P(Ct), Ct can be calculated from the (analytical or numerical) solution of the Lagrangian equation, dCt / dt - R(Ct), with initial concentration Ct((0), where t0 marks the time when the water volume has passed by a given cross section at x0 (see Fig. 24.2). For instance, if R(Ct) is a linear function ... 

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. 

An example of the alternative differential method is the principle of Jean Le Rond d Alemert (1717-1783). Perspicuous descriptions of the d Alemert principle and the derivation of the Lagrangian equations are, for example, given by Greiner [37] and Panat [73]. 

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