Abstract
The relation of the Eckart axis conditions for polyatomic vibrating molecules to the problem of optimal superposition of molecular structures has been pointed out recently [J. Chem. Phys. 122, 224105 (2005)]. Here, it is shown that both problems are intimately related to Gauss' principle of least constraint, for which a concise derivation is presented. In the context of this article, Gauss' principle leads to a rotational superposition problem of the unconstrained atomic displacements and the corresponding displacements due to a molecular rigid-body motion. The Eckart axis conditions appear here as necessary conditions for a minimum of the constraint function. The importance of Eckart's problem for extracting the internal motions of macromolecules from simulated molecular dynamics trajectories is pointed out, and it is shown how the case of coarse-grained sampled trajectories can be treated.
Highlights
Classical point mechanics can be considered as the oldest branch of theoretical physics, there are still applications to discover that concern less well known but very useful formulations of mechanical problems
The basic problem here is to separate the internal motions of the molecule from the global ones, which include rigid-body translations and rotations
It was shown that Eckart’s problem can be solved in cases where the consecutive atomic positions are not separated by differentials of positions, but by finite differences
Summary
Classical point mechanics can be considered as the oldest branch of theoretical physics, there are still applications to discover that concern less well known but very useful formulations of mechanical problems. An example is Gauss’ principle of least constraint, an elegant reformulation of D’Alembert’s principle that allows one to establish the equations of motion of systems of point particles under constraints in a very intuitive and general way.. An example is Gauss’ principle of least constraint, an elegant reformulation of D’Alembert’s principle that allows one to establish the equations of motion of systems of point particles under constraints in a very intuitive and general way.1 Despite this fact, it has not found its way into many standard textbooks on classical mechanics. The method leads immediately to the Eckart conditions for the intramolecular atomic displacements and, in particular, to a superposition problem of molecular structures This has been recently pointed out by Kudin and Dymarsky.. This has been recently pointed out by Kudin and Dymarsky. In this paper, I will briefly comment on the relation of Gauss’ principle with the more familiar formulation of mechanics by D’Alembert and on the justification of the latter from the point of view of modern linear algebra
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