Article records
https://feeds.library.caltech.edu/people/Koon-Wang-Sang/article.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenThu, 30 Nov 2023 18:09:27 +0000The Hamiltonian and Lagrangian Approaches to the Dynamics of
Nonholonomic Systems
https://resolver.caltech.edu/CaltechAUTHORS:20100820-092142844
Authors: Koon, Wang Sang; Marsden, Jerrold E.
Year: 1997
DOI: 10.1016/S0034-4877(97)85617-0
This paper compares the Hamiltonian approach to systems with nonholonomic constraints
(see Weber [1982], Arnold [1988], and Bates and Sniatycki [1993], van der Schaft and Maschke
[1994] and references therein) with the Lagrangian approach (see Koiller [1992], Ostrowski [1996]
and Bloch, Krishnaprasad, Marsden and Murray [1996]). There are many differences in the
approaches and each has its own advantages; some structures have been discovered on one side
and their analogues on the other side are interesting to clarify. For example, the momentum
equation and the reconstruction equation were first found on the Lagrangian side and are useful
for the control theory of these systems, while the failure of the reduced two form to be closed
(i.e., the failure of the Poisson bracket to satisfy the Jacobi identity) was first noticed on the
Hamiltonian side. Clarifying the relation between these approaches is important for the future
development of the control theory and stability and bifurcation theory for such systems. In
addition to this work, we treat, in this unified framework, a simplified model of the bicycle (see
Getz [1994] and Getz and Marsden [1995]), which is an important underactuated (nonminimum
phase) control system.https://authors.library.caltech.edu/records/bkhjg-r8q26Poisson reduction for nonholonomic mechanical systems with symmetry
https://resolver.caltech.edu/CaltechAUTHORS:20100820-093224864
Authors: Koon, Wang Sang; Marsden, Jerrold E.
Year: 1998
This paper continues the work of Koon and Marsden [1997b] that began the
comparison of the Hamiltonian and Lagrangian formulations of nonholonomic
systems. Because of the necessary replacement of conservation laws with the
momentum equation, it is natural to let the value of momentum be a variable
and for this reason it is natural to take a Poisson viewpoint. Some of this
theory has been started in van der Schaft and Maschke [1994]. We build on
their work, further develop the theory of nonholonomic Poisson reduction, and
tie this theory to other work in the area. We use this reduction procedure
to organize nonholonomic dynamics into a reconstruction equation, a nonholonomic
momentum equation and the reduced Lagrange d'Alembert equations in
Hamiltonian form. We also show that these equations are equivalent to those
given by the Lagrangian reduction methods of Bloch, Krishnaprasad, Marsden
and Murray [1996]. Because of the results of Koon and Marsden [1997b],
this is also equivalent to the results of Bates and Sniatycki [1993], obtained by
nonholonomic symplectic reduction.
Two interesting complications make this effort especially interesting. First
of all, as we have mentioned, symmetry need not lead to conservation laws
but rather to a momentum equation. Second, the natural Poisson bracket fails
to satisfy the Jacobi identity. In fact, the so-called Jacobiizer (the cyclic sum
that vanishes when the Jacobi identity holds), or equivalently, the Schouten
bracket, is an interesting expression involving the curvature of the underlying
distribution describing the nonholonomic constraints.
The Poisson reduction results in this paper are important for the future
development of the stability theory for nonholonomic mechanical systems with
symmetry, as begun by Zenkov, Bloch and Marsden [1997]. In particular, they
should be useful for the development of the powerful block diagonalization
properties of the energy-momentum method developed by Simo, Lewis and
Marsden [1991].https://authors.library.caltech.edu/records/0tj8z-gj274Mass-Related Dynamical Barriers in Triatomic Reactions
https://resolver.caltech.edu/CaltechAUTHORS:20101006-083122185
Authors: Yanao, T.; Koon, W. S.; Marsden, J. E.
Year: 2006
DOI: 10.1007/s00601-005-0138-7
A methodology is given to determine the effect of different mass distributions for triatomic reactions using the geometry of shape space. Atomic masses are incorporated into the non-Euclidean shape space metric after the separation of rotations. Using the equations of motion in this non-Euclidean shape space, an averaged field of velocity-dependent fictitious forces is determined. This force field, as opposed to the force arising from the potential, dominates branching ratios of isomerization dynamics of a triatomic molecule. This methodology may be useful for qualitative prediction of branching ratios in general triatomic reactions.https://authors.library.caltech.edu/records/t19f0-rc936Application of Tube Dynamics to Non-Statistical Reaction Processes
https://resolver.caltech.edu/CaltechAUTHORS:20101001-154821335
Authors: Gabern, F.; Koon, W. S.; Marsden, Jerrold E.; Ross, S. D.; Yanao, T.
Year: 2006
DOI: 10.1007/s00601-005-0136-9
A technique based on dynamical systems theory is introduced for the computation of lifetime distributions and rates of chemical reactions and scattering phenomena, even in systems that exhibit non-statistical behavior. In particular, we merge invariant manifold tube dynamics with Monte Carlo volume
determination for accurate rate calculations. This methodology is applied
to a three-degree-of-freedom model problem and some ideas on how it might
be extended to higher-degree-of-freedom systems are presented.https://authors.library.caltech.edu/records/9x0ee-1mc81Intramolecular energy transfer and the driving mechanisms for large-amplitude collective motions of clusters
https://resolver.caltech.edu/CaltechAUTHORS:20090818-113933909
Authors: Yanao, Tomohiro; Koon, Wang Sang; Marsden, Jerrold E.
Year: 2009
DOI: 10.1063/1.3098141
This paper uncovers novel and specific dynamical mechanisms that initiate large-amplitude collective motions in polyatomic molecules. These mechanisms are understood in terms of intramolecular energy transfer between modes and driving forces. Structural transition dynamics of a six-atom cluster between a symmetric and an elongated isomer is highlighted as an illustrative example of what is a general message. First, we introduce a general method of hyperspherical mode analysis to analyze the energy transfer among internal modes of polyatomic molecules. In this method, the (3n−6) internal modes of an n-atom molecule are classified generally into three coarse level gyration-radius modes, three fine level twisting modes, and (3n−12) fine level shearing modes. We show that a large amount of kinetic energy flows into the gyration-radius modes when the cluster undergoes structural transitions by changing its mass distribution. Based on this fact, we construct a reactive mode as a linear combination of the three gyration-radius modes. It is shown that before the reactive mode acquires a large amount of kinetic energy, activation or inactivation of the twisting modes, depending on the geometry of the isomer, plays crucial roles for the onset of a structural transition. Specifically, in a symmetric isomer with a spherical mass distribution, activation of specific twisting modes drives the structural transition into an elongated isomer by inducing a strong internal centrifugal force, which has the effect of elongating the mass distribution of the system. On the other hand, in an elongated isomer, inactivation of specific twisting modes initiates the structural transition into a symmetric isomer with lower potential energy by suppressing the elongation effect of the internal centrifugal force and making the effects of the potential force dominant. This driving mechanism for reactions as well as the present method of hyperspherical mode analysis should be widely applicable to molecular reactions in which a system changes its overall mass distribution in a significant way.https://authors.library.caltech.edu/records/6frba-5bc14Intramolecular Energy Flow and the Mechanisms for Dissociation of Atomic Clusters
https://resolver.caltech.edu/CaltechAUTHORS:20131101-100222294
Authors: Yanao, Tomohiro; Oka, Yurie; Koon, Wang Sang
Year: 2013
DOI: 10.1299/jtst.8.423
This paper explores the mechanisms for dissociation of atomic clusters in terms of internal energy flow and driving forces. We employ the hyperspherical coordinates to investigate internal dynamics of atomic clusters. The hyperspherical coordinates consist of three gyration radii and 3n-9 hyperangular degrees of freedom that parameterize the shape of an n-atom system in the three-dimensional physical space. The latter 3n-9 hyperangular degrees of freedom are further classified into three twisting modes and 3n-12 shearing modes. We numerically characterize the patterns of energy flow among the internal degrees of freedom leading to dissociations. It is shown that a large amount of kinetic energy tends to accumulate in the largest gyration radius upon dissociations of the cluster. We also identify some of the twisting and shearing modes that are active right at the instant of dissociation. These modes may be regarded as the triggers that drive dissociation of the cluster by pumping energy into the largest gyration radius. Physically, this pumping of energy is mediated by the internal centrifugal forces that originate from twisting and shearing motions of the system. These results are consistent with theoretical expectations from the equations of motion for gyration radii, and could be an initial step towards the control of large-amplitude collective motions of complex molecular systems.https://authors.library.caltech.edu/records/t4xhd-ntd43Control of a model of DNA division via parametric resonance
https://resolver.caltech.edu/CaltechAUTHORS:20130502-133827036
Authors: Koon, Wang Sang; Owhadi, Houman; Tao, Molei; Yanao, Tomohiro
Year: 2013
DOI: 10.1063/1.4790835
We study the internal resonance, energy transfer, activation mechanism, and control of a model of DNA division via parametric resonance. While the system is robust to noise, this study shows that it is sensitive to specific fine scale modes and frequencies that could be targeted by low intensity electro-magnetic fields for triggering and controlling the division. The DNA model is a chain of pendula in a Morse potential. While the (possibly parametrically excited) system has a large number of degrees of freedom and a large number of intrinsic time scales, global and slow variables can be identified by (1) first reducing its dynamic to two modes exchanging energy between each other and (2) averaging the dynamic of the reduced system with respect to the phase of the fastest mode. Surprisingly, the global and slow dynamic of the system remains Hamiltonian (despite the parametric excitation) and the study of its associated effective potential shows how parametric excitation can turn the unstable open state into a stable one. Numerical experiments support the accuracy of the time-averaged reduced Hamiltonian in capturing the global and slow dynamic of the full system.https://authors.library.caltech.edu/records/vcgte-p9413Roles of dynamical symmetry breaking in driving oblate-prolate transitions of atomic clusters
https://resolver.caltech.edu/CaltechAUTHORS:20150507-103425333
Authors: Oka, Yurie; Yanao, Tomohiro; Koon, Wang Sang
Year: 2015
DOI: 10.1063/1.4915928
This paper explores the driving mechanisms for structural transitions of atomic clusters between oblate and prolate isomers. We employ the hyperspherical coordinates to investigate structural dynamics of a seven-atom cluster at a coarse-grained level in terms of the dynamics of three gyration radii and three principal axes, which characterize overall mass distributions of the cluster. Dynamics of gyration radii is governed by two kinds of forces. One is the potential force originating from the interactions between atoms. The other is the dynamical forces called the internal centrifugal forces, which originate from twisting and shearing motions of the system. The internal centrifugal force arising from twisting motions has an effect of breaking the symmetry between two gyration radii. As a result, in an oblate isomer, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two largest gyration radii is crucial in triggering structural transitions into prolate isomers. In a prolate isomer, on the other hand, activation of the internal centrifugal force that has the effect of breaking the symmetry between the two smallest gyration radii is crucial in triggering structural transitions into oblate isomers. Activation of a twisting motion that switches the movement patterns of three principal axes is also important for the onset of structural transitions between oblate and prolate isomers. Based on these trigger mechanisms, we finally show that selective activations of specific gyration radii and twisting motions, depending on the isomer of the cluster, can effectively induce structural transitions of the cluster. The results presented here could provide further insights into the control of molecular reactions.https://authors.library.caltech.edu/records/6j3y1-cjr25