<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
    NO2, and (c) a net
    rotation of 42 degrees in 105 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

    [1] F. J. Lin and J. E. Marsden, Symplectic reduction and
    topology for applications in classical molecular dynamics,
    Math. Phys.
    , Vol. 33, 1281 – 1294 (1992).

    [2] F. J. Lin, Hamiltonian dynamics of atom-diatomic molecule
    complexes and collisions, Discrete and Continuous Dynamical
    Systems, Supplement
    , Vol. 2007, to appear (2007).

    [3] J. E. Marsden, R. Montgomery, and T. Ratiu, Reduction,
    symmetry, and phases in mechanics, Mem. Amer. Math. Soc.,
    88, No. 436
    (American Mathematical Society, Providence,

    * This material was presented during the time set aside for Second
    Chances on Thursday, July 26, at 4:30 – 4:40 p.m.