The N-Body Problem

Wisdom, J., Holman, M. Symplectic Maps for the n-Body Problem Stability Analysis. Astron. J. 104 (1992) 2022-2029... 

To test various stochastic assumptions for molecular motion that would simplify the N-body problem if they were valid. Molecular dynamics is far superior to experiment for this purpose since it provides much more detailed information on molecular motion than is provided by any experiment or group of experiments. 

Littlejohn, R G. and Reinsch, M. Gauge fields in the separation of rotations and internal motions in the n-body problem, Rev.Mod.Phys., 69 (1997) 213-275... 

A final word concerns the n-body problem with n > 3. Here the main problem is the rate of convergence of the EHF and corr series expansions, which we have discussed in Section IV.C. Although we may consider the knowledge of the potential-energy surface in its full dimensionality to be of fundamental importance in the case of four or five atoms, we suspect that the same may not be true for systems with a larger number of atoms, since the main role in the system chemical reactivity may then be attributed to three, four, or five atoms which define the active molecular center. Currently under way are studies for the H03 and 04 systems, and we hope, by using the DMBE... 

All well and good - the H atom can be said to be understood. How does this help us with the other atoms as the n-body problem cannot be solved explicitly Here is where the other solutions of the H atom problem come in handy. They provide the model the chemist uses every day as a fundamental part of his or her chemical language. To review, each solution is denoted by a primary quantum number n (n = 1 for the ground state,... 

K. R. Meyer and Hall, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Springer, Berlin, 1992. 

D. S. Vlachos and T. E. Simos, Partitioned Linear Multistep Method for Long Term Integration of the N-Body Problem, Appl. Num. Anal. Comp. Math., 2004,1(2), 540-546. 

Difficulties arise, however, when one considers more than two bodies. The motions in a system of three interacting bodies, defining (sensibly enough) the three-body problem, cannot fully be treated analytically, meaning that one cannot derive on paper a single equation to predict the positions and velocities of the three bodies in the system at some arbitrary time in the future. An analytic solution is likewise impossible for a larger system of some number N of objects, and this seemingly intractable situation is called the N-body problem. 

The N-body problem is one ideally suited to a computer—a machine that can repeat a prescribed string of commands with blazing speed. So the task now seems simple plug N vton s law of gravitation into a gigahertz computer, specify the initial positions and velocities of the objects under study, hit return, and sit back with a cool drink to watdi the fun. Of course, nothing is ever that easy. 

It seems we have matured to the realization that any influential effects must imply involvement through some sort of force fields between catalyst and reaction partners we acknowledge these forces to be electronic and therefore chemical in nature, and thus we imply the existence of at least temporary chemical complex or bond formation with the catalyst. Clearly then, the n-body problem of chemistry (e.g., of A and B) becomes at least an (2n- - l)-body problem (A, B X AX, BX) of catalytic chemistry (even before we worry about such strictly additional problems as energy heterogeneity of sites, polyfunctional catalysis, side reactions, etc.). 

The relations (8) and (9) defining the generalized semi-major axis a and the generalized semi-latus rectum p remain the same, with the integrals of motion c and h of the n-body problem (in the axes of the center of mass). 

As equation (3) caimot be solved analytically for the n-body problem ( >2) numerical methods must be utilized to solve n-body systems. In practice, MD calculations are carried out in time steps of the order of 10" s and are repeated several thousands or even millions of times. 

The primary aim of molecular dynamics is to numerically solve the N-body problem of classical mechanics. Molecular dynamics methods are used for used for simulating molecular-scale models of matter in order to relate collective dynamics to single-particle dynamics. Typical situations for its application are self-assembly of structures, such as micelles and vesicles. 

Greengard, L., The numerical solution of the N-body problem, Comput. Phys., 142-152, 1990. 

Let / be an analytic first integral of the n-body problem on the energy level [H = h). Then there exists an analytic first integral / of the geodesic flow on T M such that f — f ox onT U,... 

C. K. McCord, Planar Central Configuration Estimates in the N-Body Problem, preprint, 1994. 

---