# <b>Second Chances (Benedict Leimkuhler, moderator)</b>

Thursday, July 26, 2007 - 4:30pm - 5:00pm

EE/CS 3-180

**Reduced phase spaces in molecular dynamics: Coupling of overall**

rotation and internal motions in a generalized Eckart frame *

Florence Lin (University of Southern California)

This presentation describes the internal” dynamics of

molecular N-body systems in an internal reduced phase space and

describes the coupling of the internal motions with the overall

rotation in the center-of-mass frame. Traditionally, molecular

motions have been separated into translational, rotational, and

internal motions. Symplectic reduction provides a systematic

method for obtaining reduced phase spaces and has been applied

to describe N-body molecular dynamics [1, 2]. This

presentation points out three examples of observations of net

rotation due to internal motions: (a) a net rotation in a

differential geometric study of a triatomic molecule, (b) a net

rotation of 20 degrees in the recoil angle of a departing O

atom in the experimental dissociation of

NO_{2}, and (c) a net

rotation of 42 degrees in 10^{5}reduced

time steps in a

computational study of protein dynamics. For the case of

Jacobi coordinates, the net rotation follows from (a) the

conservation of total rotational angular momentum and,

equivalently, (b) Hamilton’s equations in the center-of-mass

frame [2]. For Eckart generalized coordinates, the net

rotation follows from the conservation of total rotational

angular momentum. A phase space associated with the Eckart

frame is an example of a reduced phase space. Even after

reducing to the “internal” phase space, the reduced internal

dynamics are coupled to the overall rotation in the

center-of-mass frame. In terms of the net rotation, the

strength of the coupling is mass- and coordinate-dependent.

When the total angular momentum vanishes, the coordinates of

overall rotation and internal motions are separated when the

internal angular momentum vanishes, in agreement with a

condition for the separation of energies. Appendix A provides

further applications of the derived net rotation. Appendix B

provides a differential geometric [2, 3] description of the net

rotation.

[1] F. J. Lin and J. E. Marsden, Symplectic reduction and

topology for applications in classical molecular dynamics,*J.*, Vol. 33, 1281 – 1294 (1992).

Math. Phys.

[2] F. J. Lin, Hamiltonian dynamics of atom-diatomic molecule

complexes and collisions,*Discrete and Continuous Dynamical*, Vol. 2007, to appear (2007).

Systems, Supplement

[3] J. E. Marsden, R. Montgomery, and T. Ratiu, Reduction,

symmetry, and phases in mechanics,*Mem. Amer. Math. Soc.,*(American Mathematical Society, Providence,

Vol.

88, No. 436

RI,

1990).^{*}This material was presented during the time set aside for Second

Chances on Thursday, July 26, at4:30 – 4:40 p.m.