Rigid bodies

Fig. 5. Rigid-body analysis of citrate synthase, using two X-ray structures (after Hayward and Berendsen, Proteins 30 (1998) 144). The decomposition of the protein into two domains (dark gray and white) and two interconnecting regions (light gray) is shown, together with the hinge axis for the closing/opening motion between them.

Hayward et al. 1994] Hayward, S., Kitao, A., Go, N. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Prot. Sci. 3 (1994) 936-943 [Head-Gordon and Brooks 1991] Head-Gordon, T., Brooks, C.L. Virtual rigid body dynamics. Biopol. 31 (1991) 77-100... 

Compsirison of Geometric Integrators for Rigid Body Simulation... 

The key to these more efficient treatments is a natural canonical formulation of the rigid body dynamics in terms of rotation matrices. The orientational term of the Lagrangian in these variables can be written simply as... 

We now consider the formulation of the equations of motion for a rigid body pinned at its center of mass and acted on by a (possibly nonlinear) potential field. The Lagrangian in this case is... 

Geometric Integrators for Rigid Body Simulation 355 where I = diag(/i,/2,/a) is the (diagonalized) inertial tensor,... 

A Hamiltonian version of the quaternionic description is also possible by viewing the quaternions as a set of generalized coordinates, introducing those variables into the rigid body Lagrangian (1), and finally determining the canonical momenta through the formula... 

Rotation matrices may be viewed as an alternative to particles. This approach is based directly on the orientational Lagrangian (1). Viewing the elements of the rotation matrix as the coordinates of the body, we directly enforce the constraint Q Q = E. Introducing the canonical momenta P in the usual manner, there results a constrained Hamiltonian formulation which is again treatable by SHAKE/RATTLE [25, 27, 20]. For a single rigid body we arrive at equations for the orientation of the form[25, 27]... 

It has been observed by [27, 24] that the equations of motion of a free rigid body are subject to reduction. (For a detailed discussion of this interesting topic, see [23].) This leads to an unconstrained Lie-Poisson system which is directly solvable by splitting, i.e. the Euler equations in the angular momenta ... 

Fig. 1. Motion of a material point on the body over time (left, short time interval right, long interval). The rigid body swings repeatedly toward the plane where it is repelled by the strong short-range force.

This represents an attractive (Coulombic) potential coupled with a repulsive soft wall, relative to a plane situated just below the rigid body. The rigid body is repeated drawn toward the plane, then repelled sharply from the wall. 

These experiments confirm observations in the recent articles [20] and [11] symplectic methods easily outperform more traditional quaternionic integration methods in long term rigid body simulations. 

S. Reich, Symplectic integrators for systems of rigid bodies. Fields Institute Communications, 10, 181-191 (1996). 

Ferro D R and J Hermans 1977. A Different Best Rigid-body Molecular Fit Routine. Acta Crystallographica A33 345-347. 

With respeet to this prineipal axis point of view, the rotation of the moleeule is deseribed in terms of three angles (it takes three angles to speeify the orientation of sueh a rigid body) that detail the movement of the a, b, and e axes relative to the lab-fixed X, Y,... 

Returning now to the rigid-body rotational Hamiltonian shown above, there are two special cases for which exact eigenfunctions and energy levels can be found using the general properties of angular momentum operators. 

---