Hamiltonian Mechanics with Geometric Calculus

This much-needed monograph presents a systematic, step-by-step approach to the continuum modeling of flow phenomena exhibited within materials endowed with a complex internal microstructure, such as polymers and liquid crystals. By combining the principles of Hamiltonian mechanics with those of irreversible thermodynamics, Antony N. Beris and Brian J. Edwards, renowned authorities on the subject, expertly describe the complex interplay between conservative and dissipative processes. Throughout the book, the authors emphasize the evaluation of the free energy--largely based on ideas from statistical mechanics--and how to fit the values of the phenomenological parameters against those of microscopic models. With Thermodynamics of Flowing Systems in hand, mathematicians, engineers, and physicists involved with the theoretical study of flow behavior in structurally complex media now have a superb, self-contained theoretical framework on which to base their modeling efforts.

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Hamiltonian Mechanics

10.1093/oso/9780198743040.003.0007 ◽

2017 ◽

Author(s):

Jennifer Coopersmith

Keyword(s):

Angular Momentum ◽

Jacobi Equation ◽

Modern Theory ◽

Hamiltonian Mechanics ◽

Lagrangian Mechanics ◽

Canonical Transformations ◽

Canonical Variables ◽

Light Waves ◽

Hamilton Jacobi Equation ◽

Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

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Overcoming the dissipation obstacle with Bicomplex Port-Hamiltonian Mechanics

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2020.12.1569 ◽

2020 ◽

Vol 53 (2) ◽

pp. 5573-5578

Author(s):

Coen Hutters ◽

Max Mendel

Keyword(s):

Hamiltonian Mechanics

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Perturbation Analysis of Diffeomorphisms in Contact Dynamical System

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.529.264 ◽

2012 ◽

Vol 529 ◽

pp. 264-267

Author(s):

Da Wei Sun

Keyword(s):

Dynamical System ◽

Perturbation Analysis ◽

Contact System ◽

Hamiltonian Mechanics ◽

Perturbation Techniques ◽

Small Perturbations

This paper studies the perturbations of strictly contact diffeomorphisms in contact dynamical system. By constructing new lifting methods for contact system and using some perturbation techniques in Hamiltonian mechanics, this paper proves that there exists an arbitrary small perturbations such that the corresponding function of the strictly contact isotopy does not equal to a constant at any time.

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