Lagrange coordinates

Xi are called material or Lagrange coordinates while x, are known as spatial or Euler coordinates. The motion of a body is a continuous sequence of configurations. Eqs. (3.1) and (3.2) as well as (3.F) and (3.2 ) represent, therefore, continuous functions of time. Sutmning up, it is presumed that the functions (3.1 ) and (3.2 ) are differentiable with respect to all variables as many times as needed. 

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

The constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. 

Ajt is the Lagrange multiplier and x represents one of the Cartesian coordinates two atoms. Applying Equation (7.58) to the above example, we would write dajdx = Xm and T y = Xdajdy = —X. If an atom is involved in a number of lints (because it is involved in more than one constrained bond) then the total lint force equals the sum of all such terms. The nature of the constraint for a bond in atoms i and j is ... 

Using this coordinate system the shape functions for the first two members of the tensor product Lagrange element family are expressed as... 

These difficulties have led to a revival of work on internal coordinate approaches, and to date several such techniques have been reported based on methods of rigid-body dynamics [8,19,34-37] and the Lagrange-Hamilton formalism [38-42]. These methods often have little in common in their analytical formulations, but they all may be reasonably referred to as internal coordinate molecular dynamics (ICMD) to underline their main distinction from conventional MD They all consider molecular motion in the space of generalized internal coordinates rather than in the usual Cartesian coordinate space. Their main goal is to compute long-duration macromolecular trajectories with acceptable accuracy but at a lower cost than Cartesian coordinate MD with bond length constraints. This task mrned out to be more complicated than it seemed initially. 

In the same way as described above, we can formulate the multidimensional theory without relying on the complex-valued Lagrange manifold that constitutes one of the main obstacles of the conventional multidimensional WKB theory [62,63,77,78]. Another crucial point is that the theory should not depend on any local coordinates, which gives a cumbersome problem in practical applications. Below, a general formulation is described, which is free from these difficuluties and applicable to vertually any multidimensional systems [30]. 

In Section 2.4 we have studied the behavior of the gravitational field of the spheroid outside of masses. Now let us focus our attention on the field of attraction inside masses. It may be proper to notice that the determination of the field caused by masses in the spheroid and, in general, by an ellipsoid, was a subject of classical works performed by Maclaurin, Lagrange, Laplace, Poisson, and others. As is well known, the equation of the ellipsoid, when the major axes are directed along coordinate lines is... 

For a family of trajectories all starting at the value X(to) and at t=t all arriving at X(t), there is one trajectory that renders the action stationary. The classical mechanical trajectory of a given dynamical system is the one for which 5S=0, i.e. the action becomes stationary. The equation of motion is obtained from this variational principle [59], The corresponding Euler-Lagrange equations are obtained d(3L/3vk)/dt = 9L/dXk. In Cartesian coordinates these equations become Newton s equations of motion for each nucleus of mass Mk ... 

In this approach, the potential energy V is a function of the reduced coordinates and the -matrix. For the kinetic energy, one would only be interested in the motion of the particle relative to the distorted geometry so that a suitable Lagrange function Lq for the system would read as follows ... 

Lagrange equation for relative movement of isolated system of two interacting material points with masses mi and m2 in coordinate x can be presented as follows ... 

The calculation of the torsional accelerations, i.e. the second time derivatives of the torsion angles, is the crucial point of a torsion angle dynamics algorithm. The equations of motion for a classical mechanical system with generalized coordinates are the Lagrange equations... 

Derivation of the Boltzmann distribution function is based on statistical mechanical considerations and requires use of Stirling s approximation and Lagrange s method of undetermined multipliers to arrive at the basic equation, (N,/No) = (g/go)exp[-A Ae/]. The exponential term /3 defines the temperature scale of the Boltzmann function and can be shown to equal t/ksT. In classical mechanics, this distribution is defined by giving values for the coordinates and momenta for each particle in three-coordinate space and the lin-... 

Here t,- = t,( ) is one of K Lagrange multiplier fields that are functions of the soft coordinates alone, which must be chosen so as to satisfy condition (2.107) for all of the hard coordinates. Equation (2.112) has the following properties ... 

Consider the two-dimensional motion of a particle in a central force field. The Lagrange function in Cartesian coordinates is... 

Euler s proof of the least action principle for a single particle (mass point in motion) was extended by Lagrange (c. 1760) to the general case of mutually interacting particles, appropriate to celestial mechanics. In Lagrange s derivation [436], action along a system path from initial coordinates P to final coordinates Q is defined by... 

If the end-points are fixed, the integrated term vanishes, and A is stationary if and only if the final integral vanishes. Since Sxa is arbitrary, the integrand must vanish, which is Newton s law of motion. Hence Lagrange s derivation proves that the principle of least action is equivalent to Newtonian mechanics if energy is conserved and end-point coordinates are specified. 

Newton s equations of motion, stated as force equals mass times acceleration , are strictly true only for mass points in Cartesian coordinates. Many problems of classical mechanics, such as the rotation of a solid, cannot easily be described in such terms. Lagrange extended Newtonian mechanics to an essentially complete nonrelativistic theory by introducing generalized coordinates q and generalized forces Q such that the work done in a dynamical process is Qkdqk [436], Since... 

On solving these equations for the coefficients A j, the solution of minimum norm is the interpolated gradient vector P, such that pi = P 2 0, at the interpolated coordinate vector Q. The Lagrange multiplier /x in this method provides an estimate of the residual error. 

Lagrange s equations follow by demanding that S is stationary under variations of the generalized coordinates q. Thus, for an arbitrary change in the field ... 

In Eq. 4.7, ij/j is the jth one-electron orbital accommodating the /rth electron with spatial coordinate and spin coordinate Likewise, the rth electron resides in the ith orbital denoted by (/r,. The Lagrange multiplier, Sj, guarantees that the solutions to this equation forms an orthonormal set and is the expectation value for the equation. Hence, it is the quantized one-electron orbital energy. The second term within the... 

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