CDS 280  Two Talks

Tomohiro Yanao (CDS, Caltech) and
Hiroaki Yoshimura (Waseda University, Japan)

1st Talk 12:30 PM  Tomohiro Yanao -
Gyration-Radius Dynamics of Atomic Clusters and Polymers:

Large-amplitude collective motions of clusters and biopolymers are of great interest in chemistry and biophysics. For the true understanding of these high-dimensional dynamics, it is crucially important to find a small number of coarse variables that essentially dominate the collective motions of the system. In this talk, it is shown that the gyration radii (or moments of inertia) of molecules are good collective variables both in coarse-graining the high-dimensional dynamics and in clarifying the mechanism of collective motions. Equations of motion for the gyration radii are investigated based on the reduction theory and the hyperspherical coordinates. By averaging these equations of motion, we introduce a mean force potential along a reaction coordinate defined in the space of gyration radii. This mean force potential explains phase changes and branching ratios of structural isomerization dynamics of clusters and a prototypal polymer.

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2nd Talk: 2:00 PM Hiroaki Yoshimura:

Dirac Structures and the Legendre Transformation for Implicit Lagrangian and Hamiltonian Systems
[ https://www.cds.caltech.edu/news/uploads/wiki/files/173/yoshimura_abstract.pdf ]

We investigate mechanical systems with degenerate Lagragians in the context of Dirac structures. We first show an induced Dirac structure $D_{\Delta_{Q}}$ can be defined on $T^{\ast}Q$  by a given constraint distribution $\Delta_{Q} \subset TQ$ and demonstrate how an implicit Lagrangian system $(L,\Delta_{Q},X)$ can be constructed in the case that a given Lagrangian $L$ is degenerate and with a vector field $X$ on $T^{\ast}Q$. Then, we introduce a generalized Legendre transformation to define a Hamiltonian $H_{P}$ on a constraint momentum space $P=\mathbb{F}L(\Delta_{Q})$ and also define a generalized Hamiltonian $H$ on the Pontryagin bundle $TQ \oplus T^{\ast}Q$ by incorporating primary constraints in the sense of Dirac. Thus, we develop an implicit Hamiltonian system $(H,\Delta_{Q},X)$ from the degenerate Lagrangian, that is, a Hamiltonian analogue of an implicit Lagrangian system. Further, we illustrate the equivalence between implicit Lagrangian and Hamiltonian systems in the context of the generalized Legendre transformation, where we also clarify the duality relation between the Legendre map and the primary constraints. We shall illustrate an example of L-C circuits, which is a typical physical system of a degenerate Lagrangian with constraints in both contexts of implicit Lagrangian and Hamiltonian systems.