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Classical Mechanics and Field Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
The only requirements on ω in a general coordinate system is that it is anti-symmetric, has zero exterior derivative, and is non-degenerate, i.e., that ω(X) = 0 only if X = 0, such that the inverse Ω exists. A two-form that satisfies these conditions is called a symplecticform and a manifold along with the specification of such a two-form is called a symplectic manifold. Many of the properties and theorems of Hamiltonian mechanics can be described more succinctly in terms of the corresponding statements for symplectic manifolds and, just as a system in Lagrangian mechanics is specified by the configuration space and a function on its tangent bundle, a general system in Hamiltonian mechanics is described by a symplectic manifold and a function on it.
Hamiltonian Mechanics and Hamilton-Jacobi Theory
Published in M.D.S. Aliyu, Nonlinear H∞-Control, Hamiltonian Systems and Hamilton-Jacobi Equations, 2017
Hamiltonian mechanics is a transformation theory that is an off-shoot of Lagrangian mechanics. It concerns itself with a systematic search for coordinate transformations which exhibit specific advantages for certain types of problems, notably in celestial and quantum mechanics. As such, the Hamiltonian approach to the analysis of a dynamical system, as it stands, does not represent an overwhelming development over the Lagrangian method. One ends up with practically the same number of equations as the Lagrangian approach. However, the real advantage of this approach lies in the fact that the transformed equations of motion in terms of a new set of position and momentum variables are easily integrated for specific problems, and also the deeper insight it provides into the formal structure of mechanics. The equal status accorded to coordinates and momenta as independent variables provides a new representation and greater freedom in selecting more relevant coordinate systems for different types of problems.
Introduction to Nanosensors
Published in Vinod Kumar Khanna, Nanosensors, 2021
which equals the result of the Hamilton-Jacobi method, a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newton’s laws of motion, Lagrangian mechanics, and Hamiltonian mechanics.
Decision-Oriented Two-Parameter Fisher Information Sensitivity Using Symplectic Decomposition
Published in Technometrics, 2023
In summary, the use of the Fisher information as a sensitivity measure has wide applications across science and engineering. Nevertheless, practical issues can hinder the translation of sensitivity information into actionable decisions. In this article, we propose a new approach using the symplectic decomposition to extract the Fisher sensitivity information. The symplectic decomposition uses Williamson’s theorem (Williamson 1936; Nicacio 2021) which is a key theorem in Gaussian quantum information theory (Pereira, Banchi, and Pirandola 2021). Originating from Hamiltonian mechanics, the symplectic transformations preserve Hamilton’s equations in phase space (Arnol’d 1989). In analogy to the conjugate coordinates for the phase space, that is, position and momentum, we regard the input parameters as conjugate pairs and use a symplectic matrix for the decomposition of the FIM. The resulting symplectic eigenvalues of large magnitudes, and the corresponding symplectic eigenvectors of the FIM, then reveal the most sensitive two-parameter pairs.