Newton’s laws of motion in classical mechanics

Classical mechanics as a unifying principle of physics began with Newton s laws of motion. As discussed in Section 0.1, Newton s second law 

In classical mechanics the particle obeys Newton s second law of motion 

In a molecular dynamics calculation, equations of motion are integrated to determine the trajectories of all atoms in the molecule. The equations of motion can. in principle, be either classical (Newton s laws of motion) or quantum mechanical. But, in practice, due to the very laige number of atoms in a macromolecule. Newton s equations of motion are used. Quantum mechanical methods are too time consuming, complicated, and at this stage too inaccurate to be popular in the field of polymer chemistry. 

The laws of classical mechanics (in particular the Newton s second law of motion) are applicable to the motion of gaseous molecules. 

In a molecular dynamics experiment the equation of motions for each particle follows the laws of the classical mechanics, and most notably the Newton s law for each atom / in a system constituted by N atoms 

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, 

MD is a technique in which the time evolution of the molecular motions is simulated following the laws of classical mechanics. Therefore, the physical variable time must be considered explicitly. In this way, the dynamic evolution of coordinates and moments, i.e., the trajectory of the system, is calculated by numerically solving Newton s equations of motion. This trajectory, together with the associated energies and forces, leads to the static and dynamic thermodynamic properties of the studied system via statistical analysis methods. MD is also a powerful tool to understand dynamic processes at the atomistic level that involve fluids or materials [9]. 

Although some of the physical ideas of classical mechanics is older than written history, the basic mathematical concepts are based on Isaac Newton s axioms published in his book Philosophiae Naturalis Principia Mathematica or principia that appeared in 1687. Translating from the original Latin, the three axioms or the laws of motion can be approximately stated [7] (p. 13)  

Quantum mechanics provides the law of motion for microscopic particles. Experimentally, macroscopic objects obey classical mechanics. Hence for quantum mechanics to be a valid theory, it should reduce to classical mechanics as we make the transition from microscopic to macroscopic particles. Quantum effects are associated with the de Broglie wavelength A = h/mv. Since h is very small, the de Broglie wavelength of macroscopic objects is essentially zero. Thus, in the limit A —> 0, we expect the time-dependent Schrddinger equation to reduce to Newton s second law. We can prove this to be so (see Problem 7.56). 

Schrodinger s work in quantnm theory resulted in the creation of a new scientific discipline— wave mechanics, which has as its centerpiece the Schrodinger wave equation, explained in a series of four papers published in 1926. This equation and the later relativistic versions are considered by many scientists to have the same central importance to molecular quantum mechanics as Newton s laws of motion have to large-scale classical mechanics. 

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