Symmetry Reduced Dynamics of Charged Molecular Strands |
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Authors: | David C P Ellis François Gay-Balmaz Darryl D Holm Vakhtang Putkaradze Tudor S Ratiu |
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Institution: | 1. Mathematics Department, Imperial College London, London, SW7 2AZ, UK 2. Laboratoire de Météorologie Dynamique, école Normale Supérieure/CNRS, Paris, France 3. Control and Dynamical Systems, California Institute of Technology, Pasadena, CA, 91125, USA 4. Institute for Mathematical Sciences, Imperial College London, London, SW7 2PG, UK 6. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523-1874, USA 7. Section de Mathématiques and Bernoulli Center, école Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland
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Abstract: | The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible
continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are
nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal
dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that
illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods.
In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion
equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry
group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids.
On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic
structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant
formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation
of the nonlocal equations of molecular strand motion is transformed into quaternionic form. |
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