Lagrangian equation classical mechanics

This means that all moving nuclei (atoms) are treated as classical particles which is a serious approximation, but which was found to work very well (60,61). Applying the Euler-Lagrange equation (Eq. 2) to the Lagrangian C (Eq. 1) leads to the same equations as the well-known Newton s second law (Eq. 3). Or in other words, in classical mechanics the derivative of the Lagrangian is taken with respect to the nuclear positions. 

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... 

For a system with N degrees of freedom, q, i = 1 to N, this equation is obtained for each of the N coordinates qi. These are Lagrange s equations of motion, the equations of motion for a system obeying classical mechanics. Thus, the Lagrangian, which minimizes the value of the action integral along the true trajectory between the times tj and fj, is also the function which yields the equations of motion when inserted into the Euler equation (8.50). 

It is possible to formulate the classical laws of motion in several ways. Newton s equations are taught in every basic course of classical mechanics. However, especially in the presence of constraint forces, the equations of motion can often be presented in a simpler form by using either Lagrangian or Hamiltonian formalism. In short, in the Newtonian approach, an /V-point particle system is described by specifying the position xa = xa(t) of each particle a as a function of time. The positions are found by solving the equations of motion,... 

Hamiltonian mechanics refers to a mathematical formalism in classical mechanics invented by the Irish mathematician William Rowan Hamilton (1805-1865) during the early 1830 s arising from Lagrangian mechanics which was introduced about 50 years earlier by Joseph-Louis Lagrange (1736-1813). The Hamiltonian equations can however be formulated on the basis of a variational principle without recourse to Lagrangian mechanics [95] [2j. 

From the definition of the Lagrangian function (2.6) it can be shown that the time coordinate is both homogeneous and isotropic meaning that its properties are the same in both directions [52]. For, if t is replaced by —t, the Lagrangian is unchanged, and therefore so are the equations of motion. In this sense all motions which obey the laws of classical mechanics are reversible. 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

The use of functionals and their derivatives is not limited to density-functional theory, or even to quantum mechanics. In classical mechanics, e.g., one expresses the Lagrangian C in terms of of generalized coordinates q(x,t) and their temporal derivatives q(x,t), and obtains the equations of motion from extremizing the action functional 4[g] = J C q, q t)dt. The resulting equations of motion are the well-known Euler-Lagrange equations 0 = = fy — > which are a special case of Eq. (14). 

For example, even in classical mechanics we have equations (like Newton s) and the variational principle of the least action. If we introduced something similar to varying constants into the Lagrangian we will change the equations. Similar... 

The force and velocity are vectors, whose direction and magnitude are both of importance. In complex problems it is often preferable to reformulate classical mechanics in terms of a scalar, such as the energy, which is characterized only by its magnitude. This gives rise to the Lagrangian and Hamiltonian equations of motion. The latter equations are of most interest here and are dqi d K dt dpi ... 

Molecular dynamics (MD) is an application of classical mechanics using computer simulations. Good introductions can be found in many textbooks, for example the excellent book by Tuckerman [9]. In order to carry out MD, equations describing the motion of molecules are needed. These equations of motion can be derived for example from the classical Lagrangian , a function of the kinetic (K) and the potential energy (U) ... 

All phenomena of classical nonrelativistic mechanics are solely based on Newton s laws of motion, which are valid in any inertial frame of reference. The natural symmetry operations of classical mechanics are the Galilean transformations, mediating the transition from one inertial coordinate system to another. The fundamental laws of classical mechanics can equally well be formulated applying the elegant Lagrangian and Hamiltonian descriptions based on Hamilton s action principle. Maxwell s equations for electric and magnetic fields are introduced as the basic laws of classical electrodynamics. 

The field operators (f) x) obey equations of motion that are derived from a Lagrangian L via a variational principle, in direct analogy to classical mechanics, and interactions between different fields are produced by adding to the free Lagrangian Lq an interaction term V containing products of the various field operators that are to influence each other. The equations of motion thereby become coupled equations relating the different fields to each other. Usually L is written as an integral over all... 

Contrary to Newton s second law the Lagrangian and Hamiltonian formulations of classical mechanics are form invariant under a change of coordinates, i.e. the Euler-Lagrange equations have... 

Newton s formulation is not the only way in which classical equations of motion can be formulated. Lagrange (Joseph Louis Lagrange, France, 1736-1813), Hamilton (William Rowan Hamilton, Ireland, 1805-1865), and others developed different means, and it is the formulation of Hamilton that has proven the most useful framework for developing the mechanics of quantum systems. It is important to realize that Newtonian, Lagrangian, and Hamiltonian mechanics offer equivalent descriptions of classical systems. 

The above considerations leads to the somewhat troubling question of whether (128) represents the true non-relativistic limit of the Dirac equation in the presence of external fields. Referring back to (110) we have certainly obtained the non-relativistic limit of the free-particle part Lm, but we have in fact retained the interaction term as well as the Lagrangian of the free field. In order to obtain the proper non-relativistic limit, we must consider what is the non-relativistic limit of classical electrodynamics. This task is not facilitated by the fact that, contrary to purely mechanical systems, the laws of electrodynamics appear in different unit systems in which the speed of light appears differently. In the Gaussian system Maxwell s laws are given as... 

A number of manipulations are possible, once this formalism has been established. There are useful analogies both with the Eulerian and Lagrangian pictures of incompressible fluid flow, and with the Heisenberg and Schrodinger pictures of quantum mechanics T, chapter 7], [M, chapter 11]. These analogies are particularly useful in formulating the equations of classical response theory [39], linking transport coefficients with both equilibrium and nonequilibrium simulations [35]. 

These are the 3N equations of motion in Lagrangian mechanics. They generate the same solution for the motion of classical particles as Newton s equations of motion. The benefit of Lagrangian mechanics is that Eqs. 3.24 are invariant to coordinate transformations. This simply means that the functional form of the equations of motion remains... 

---