Symplectic methods for classical mechanics

The integration methods discussed above have been compared in terms of stability and accuracy however, no integration method with a finite time step is perfectly acciuate. Over long simulation periods, the discrepancy between the predicted and actual system behavior may be sigiuficant. 

This is of importance for applications such as celestial mechanics and molecular dynamics, in which we simulate the motion of a number of interacting particles of masses nia, positions ra, velocities Va, with a total potential energy function V(ri,r2,rji). The motion of each point mass is governed by Newton s second law of motion 

In the absence of an external potential or dissipation (frictional forces), the total system energy is constant. 

Due to integration errors, this property is generally not satisfied by the numerical trajectory of the system however, for a special class of integrators, total energy is conserved (approximately) even over long simulations. 

From Noether s theorem (Arnold, 1989), it is known that the conservation of energy is related to the invariance of Newton s equation of motion to time reversal. That is, if we follow a conservative system for some period of time and then reverse the direction of time, the system will exactly retrace, in reverse, its previous trajectory. It may be shown that integration rules that are symplectic (i.e., symmetric with respect to the direction of time) have favorable energy conservation properties that make them most suitable for simulating the classical mechanics of conservative systems. A more complete discussion of symplectic integrators is found in Frenkel Smit (2001). Here, we merely provide a popular symplectic 

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