In non-relativistic classical mechanics a mechanical system can be characterised by a function called the Lagrangian, S(q, q) where q denotes the coordinates, and the motion of the system is such that the action S, defined by [Pg.68]

A Legendre transformation also connects the Lagrangian and Hamiltonian functions in classical mechanics. For a particle moving in one dimension, the Lagrangian L = T—V can be written as [Pg.197]

We may also note an analogy between mean field theory and classical mechanics, and treat the integrand of the Fb functional as the Lagrangian Then [Pg.13]

Following Hamilton s principle in classical mechanics, the required time dependence can be derived from a variational principle based on a seemingly artificial Lagrangian density, integrated over both space and time to define the functional [Pg.78]

A familiar example of Legendre transformation is the relationship that exists between the Lagrangian and Hamiltonian functions of classical mechanics [17]. In thermodynamics the simplest application is to the internal energy function for constant mole number U(S, V), with the differentials [Pg.419]

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 [Pg.37]

From classical mechanics (e.g., Goldstein 1950), we can show that the presence of a vector potential requires that the Hamiltonian function must be constructed using the kinetic momentum (or mechanical momentum), which is the momentum that is given in nonrelativistic theory by m. We must express this momentum in terms of the canonical momentum of Lagrangian mechanics, because it is the canonical momentum to which the quantization rule p —ihV applies. Here (and hereafter) we will use p for the canonical momentum and n for the kinetic momentum. The relation between the two is [Pg.36]

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. [Pg.239]

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) [Pg.112]

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, [Pg.272]

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