Lagrange equation of motion

This sort of equation, known as a Lagrange equation of motion, is generally valid for systems with unconstrained degrees of freedom. If a system has n degrees of freedom, n second-order Lagrange equations will exist (functions of qj, qj, and time quadratic in qj). 

Using this particular form of L in the Euler-Lagrange equations of motion (3.1.6) we get... 

Consequently, 3N — 6 of the Lagrange equations of motion Avill be of the form... 

The Lagrange equations of motion allow another useful alternative. For instance, if the freely-jointed chain is described in Cartesian coordinates, the monomer motions are constrained by the constant bond lengths Bk- For a polymer chain of Nk+ 1 segments and bonds there is a set of constraint equations. 

The Lagrange equation of motion for this one-dimensional system is... 

ADAMS form either the Newton-Euler or the Lagrange equations of motion, a mixture of algebraic and differential equations. The results are given as the numerically integrated values for given initial conditions. 

The Lagrange equations of motion for the multibody system can be written as M q + = g... 

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. 

There are various ways to obtain the solutions to this problem. The most straightforward method is to solve the full problem by first computing the Lagrange multipliers from the time-differentiated constraint equations and then using the values obtained to solve the equations of motion [7,8,37]. This method, however, is not computationally cheap because it requires a matrix inversion at every iteration. In practice, therefore, the problem is solved by a simple iterative scheme to satisfy the constraints. This scheme is called SHAKE [6,14] (see Section V.B). Note that the computational advantage has to be balanced against the additional work required to solve the constraint equations. This approach allows a modest increase in speed by a factor of 2 or 3 if all bonds are constrained. 

The Hamiltonian for a charged particle in an electromagnetic field can be obtained from Hamilton s principle and Lagrange s equations of motion (Section 3.3) ... 

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 ... 

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... 

In the framework of the Euler-Lagrange formalism, we write the equation of motion for the displacements of the atoms as ... 

Here, Pp = mpR is a Cartesian bead momentum, U is the internal potential energy of the system of interest, Xi,.. .,Xk are a set of AT Lagrange multiplier constraint fields, which must be chosen so as to satisfy the K constraints, and is the rapidly fluctuating force exerted on bead p by interactions with surrounding solvent molecules. The corresponding Hamiltonian equation of motion is... 

Equation (4.9) is the equation of motion of the Lagrange multiplier that restricts the solution to satisfy the Schrodinger equation it is to be solved subject to the final-state condition (4.10). Equation (4.11) is the Schrodinger equation for our system it is to be solved subject to the initial condition (4.12). The field that results from these calculations is given by... 

The equations of motion of the held at the Higgs minimum (the minimum potential energy of the vacuum) are the Euler-Lagrange equations... 

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... 

Generalized momenta are defined by pk = and generalized forces are defined by Qk = f - When applied using t as the independent variable, Euler s variational equation for the action integral / takes the form of Lagrange s equations of motion... 

The equations of motion, generalized to include holonomic or nonholonomic constraints with Lagrange multipliers, are... 

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