Further study of the equivalent theorem of Hellinger-Reissner and Hu-Washizu variational principles

Abstract Mathematical models of finite deformation of a rolling viscoelastic cylinder in contact with a rough foundation are developed in preparation for a general model for rolling tires. Variational principles and finite element models are derived. Numerical results are obtained for a variety of cases, including that of a pure elastic rubber cylinder, a viscoelastic cylinder, the development of standing waves, and frictional effects.

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Accelerated Simulation of Kinetic Transport Using Variational Principles and Sparsity

10.2172/1417993 ◽

2017 ◽

Author(s):

Russel Caflisch

Keyword(s):

Variational Principles ◽

Accelerated Simulation

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The Physical World

10.1093/oso/9780198795933.001.0001 ◽

2017 ◽

Cited By ~ 1

Author(s):

Nicholas Manton ◽

Nicholas Mee

Keyword(s):

Quantum Mechanics ◽

Particle Physics ◽

Variational Principles ◽

Black Body ◽

Black Body Radiation ◽

Physical World ◽

Atomic Scale ◽

Solid State Physics ◽

Least Action ◽

Dimensional Spacetime

The book is an inspirational survey of fundamental physics, emphasizing the use of variational principles. Chapter 1 presents introductory ideas, including the principle of least action, vectors and partial differentiation. Chapter 2 covers Newtonian dynamics and the motion of mutually gravitating bodies. Chapter 3 is about electromagnetic fields as described by Maxwell’s equations. Chapter 4 is about special relativity, which unifies space and time into 4-dimensional spacetime. Chapter 5 introduces the mathematics of curved space, leading to Chapter 6 covering general relativity and its remarkable consequences, such as the existence of black holes. Chapters 7 and 8 present quantum mechanics, essential for understanding atomic-scale phenomena. Chapter 9 uses quantum mechanics to explain the fundamental principles of chemistry and solid state physics. Chapter 10 is about thermodynamics, which is built around the concepts of temperature and entropy. Various applications are discussed, including the analysis of black body radiation that led to the quantum revolution. Chapter 11 surveys the atomic nucleus, its properties and applications. Chapter 12 explores particle physics, the Standard Model and the Higgs mechanism, with a short introduction to quantum field theory. Chapter 13 is about the structure and evolution of stars and brings together material from many of the earlier chapters. Chapter 14 on cosmology describes the structure and evolution of the universe as a whole. Finally, Chapter 15 discusses remaining problems at the frontiers of physics, such as the interpretation of quantum mechanics, and the ultimate nature of particles. Some speculative ideas are explored, such as supersymmetry, solitons and string theory.

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Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

Journal of Applied Analysis ◽

10.1515/jaa-2018-0017 ◽

2018 ◽

Vol 24 (2) ◽

pp. 175-183

Author(s):

Jean-Claude Ndogmo

Keyword(s):

Explicit Form ◽

Nonlinear Equations ◽

Variational Principles ◽

Symmetry Algebra ◽

First Integrals ◽

Maximal Symmetry ◽

General Order ◽

Linear And Nonlinear ◽

Variational Symmetries ◽

Variational Symmetry

Abstract Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. A discussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

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