ItemDual Pairs and Regularization of Kummer Shapes in Resonances(American Institute of Mathematical Sciences, 2019-06) Ohsawa, Tomoki; Ohsawa, TomokiWe present an account of dual pairs and the Kummer shapes for n : m resonances that provides an alternative to Holm and Vizman’s work. The advantages of our point of view are that the associated Poisson structure on su(2)* is the standard (+)-Lie–Poisson bracket independent of the values of (n, m) as well as that the Kummer shape is regularized to become a sphere without any pinches regardless of the values of (n, m). A similar result holds for n : −m resonance with a paraboloid and su(1, 1)* . The result also has a straightforward generalization to multidimensional resonances as well. ©2019 American Institute of Mathematical Sciences ItemThe Hagedorn–Hermite Correspondence(Birkhäuser Boston) Ohsawa, Tomoki; Ohsawa, TomokiWe investigate the relationship between the semiclassical wave packets of Hagedorn and the Hermite functions by establishing a relationship between their ladder operators. This Hagedorn–Hermite correspondence provides a unified view as well as simple proofs of some essential results on the Hagedorn wave packets. Particularly, we show that Hagedorn’s ladder operators are a natural set of ladder operators obtained from the position and momentum operators using the symplectic group. This construction reveals an algebraic structure of the Hagedorn wave packets, and explains the relative simplicity of Hagedorn’s parametrization compared to the rather intricate construction of the generalized squeezed states. We apply our formulation to show the existence of minimal uncertainty products for the Hagedorn wave packets, generalizing Hagedorn’s one-dimensional result to multi-dimensions. The Hagedorn–Hermite correspondence also leads to an alternative derivation of the generating function for the Hagedorn wave packets based on the generating function for the Hermite functions. This result, in turn, reveals the relationship between the Hagedorn polynomials and the Hermite polynomials. © 2018 Springer Science+Business Media, LLC, part of Springer Nature ItemOptimal Control Problems with Symmetry Breaking Cost Functions(SIAM Publications) Bloch, Anthony M.; Colombo, Leonardo J.; Gupta, Rohit; Ohsawa, Tomoki; Ohsawa, TomokiWe investigate symmetry reduction of optimal control problems for left-invariant control affine systems on Lie groups, with partial symmetry breaking cost functions. Our approach emphasizes the role of variational principles and considers a discrete-time setting as well as the standard continuous time formulation. Specifically, we recast the optimal control problem as a constrained variational problem with a partial symmetry breaking Lagrangian and obtain the Euler-Poincare equations from a variational principle. By using a Legendre transformation, we recover the Lie Poisson equations obtained by Borum and Bretl [IEEE Trans. Automat. Control, 62 (2017), pp. 3209-3224] in the same context. We also discretize the variational principle in time and obtain the discrete-time Lie Poisson equations. We illustrate the theory with some practical examples including a motion planning problem in the presence of an obstacle. ItemGeometry and Dynamics of Gaussian Wave Packets and Their Wigner Transforms(Amer Inst Physics, 2018-09-24) Ohsawa, Tomoki; Tronci, Cesare; 0000-0001-9406-132X (Ohsawa, T); Ohsawa, TomokiWe find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant's coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables. ItemSymmetry and Conservation Laws in Semiclassical Wave Packet Dynamics(American Institute of Physics Inc., 2015-03-18) Ohsawa, Tomoki; Ohsawa, TomokiWe formulate symmetries in semiclassical Gaussian wave packet dynamics and find the corresponding conserved quantities, particularly the semiclassical angular momentum, via Noether's theorem. We consider two slightly different formulations of Gaussian wave packet dynamics; one is based on earlier works of Heller and Hagedorn and the other based on the symplectic-geometric approach by Lubich and others. In either case, we reveal the symplectic and Hamiltonian nature of the dynamics and formulate natural symmetry group actions in the setting to derive the corresponding conserved quantities (momentum maps). The semiclassical angular momentum inherits the essential properties of the classical angular momentum as well as naturally corresponds to the quantum picture.