Classical mechanics Lagrangian formalism

Classical Mechanics of Constrained Systems within Lagrangian and Hamiltonian Formalisms... 

An alternative approach to describe steady-state thermodynamics for shear flow was formulated by Taniguchi and Morriss.192 Their method involves the development of a canonical distribution for shear flow by a Lagrangian formalism of classical mechanics. They then derive the Evans-Hanley thermodynamics, i.e. 

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. 

All aspects of Newtonian mechanics can equally well be formulated within the more general Lagrangian framework based on a single scalar function, the Lagrangian. These formal developments are essential prerequisites for the later discussion of relativistic mechanics and relativistic quantum field theories. As a matter of fact the importance of the Lagrangian formalism for contemporary physics cannot be overestimated as it has strongly contributed to the development of every branch of modem theoretical physics. We will thus briefly discuss its most central formal aspects within the framework of classical Newtonian mechanics. 

The third equivalent formulation of classical mechanics to be briefly discussed here is the Hamiltonian formalism. Its main practical importance especially for molecular simulations lies in the solution of practical problems for processes that can be adequately described by classical mechanics despite their intrinsically quantum mechanical character (such as protein folding processes). However, more important for our purposes here is that it can serve as a useful starting point for the transition to quantum theory. The basic idea of the Hamiltonian formalism is to eliminate the / generalized velocities in favor of the canonical momenta defined by Eq. (2.54). This is achieved by a Legendre transformation of the Lagrangian with respect to the velocities. 

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... 

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]. 

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