About Book

Mechanics is an area of science concerned with the behaviour of physical bodies when subjected to forces or displacements and the subsequent effects of the bodies on their environment. Classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. It is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. Classical mechanics was one of the first pillars of modern physics providing the first unification of the laws of motion and universal gravitation. Mechanics may be divided into three branches: statics, which deals with forces acting on and in a body at rest; kinematics, which describes the possible motions of a body or system of bodies; and kinetics, which attempts to explain or predict the motion that will occur in a given situation. This book is a collection of research, which the main purpose is to offer a broad glimpse of the advances in mechanics. It overviews modern mathematical tools and techniques used to solve and construct differential equations associated. It focuses on continuum mechanics, quantum mechanics, newtonian mechanics, non-euclidean superposition, complexified lagrangian mechanics, fluid mechanics and transversality condition. This book is a compilation of chapters that discuss the most vital concepts in the field of engineering mechanics. Most of the topics introduced in this book cover new techniques and applications of mechanics. It unfolds the innovative aspects of the field, which will be crucial for the holistic understanding of the subject matter. Those with an interest in engineering mechanics field would find this book helpful.

Preface............................................................................................................v

Chapter 1 Transversality Condition in Continuum Mechanics..........................1

1. Introduction............................................................................................1

2. Energy functional and transversality condition ...........................3

3. Adhesion of two micro-beams ..........................................................6

4. Collapsed morphology of a SWCNT ...............................................8

5. Conclusions .........................................................................................16

6. Acknowledgements ...........................................................................16

Chapter 2 Topology Optimization of Fluid Mechanics Problems...................19

1. Introduction..........................................................................................19

2. Topological optimization problem...................................................21

3. Topological sensitivity analysis........................................................23

4. Numerical experiments......................................................................27

5. Conclusion............................................................................................44

Chapter 3 Bohmian Trajectories and the Path Integral Paradigm – Complexified Lagrangian Mechanics.............................................................................48

1. Introduction..........................................................................................48

2. From classic realm to quantum ......................................................50

3. Interference pattern from an N-slit gratin ....................................53

4. Variational computations...................................................................67

5. Beyond the Bohm’s insight into QM .............................................74

6. Concluding remarks...........................................................................78

7. Acknowledgement...............................................................................79

Chapter 4 The Group Theory and Non-Euclidean Superposition Principle in

Quantum Mechanics................................................................................83

1. Introduction ........................................................................................83

2. Completeness of observables............................................................87

3. Non-Euclidean superposition principle..........................................97

4. Irreversibility in quantum mechanics .........................................108

5. Conclusion..........................................................................................112

Acknowledgement............................................................................118

Chapter 5 Quantum Mechanics and Statistical Description of Results of

Measurement............................................................................................120

1. Introduction........................................................................................120

2. Statistical description of results of measurement .....................123

3. Wave function....................................................................................124

4. Momentum operator .......................................................................126

5. Mean value of xp..............................................................................126

6. Fisher information............................................................................127

7. Kinetic energy....................................................................................129

8. Heisenberg uncertainty relations...................................................130

9. Two uncertainty relations................................................................131

10. Robertson–Schrödinger uncertainty relation...............................133

11. Free particle.......................................................................................134

12. Linear harmonic oscillator .............................................................136

13. Commutation relations and vector potential..............................138

14. Time.....................................................................................................139

15. Scalar potential..................................................................................140

16. Time–energy uncertainty relations................................................141

17. Equations of motion.........................................................................142

18. Conclusion ........................................................................................145

Chapter 6 Theory of Elementary Particles Based on Newtonian

Mechanics.................................................................................................148

1. Introduction........................................................................................148

2. Electrodynamics of physical vacuum ..........................................150

3. Elementary particles of a matter ..................................................157

4. Gravitation and gravitational waves.............................................170

5. Conclusion ........................................................................................171

Chapter 7 Quantum Mechanics on Surfaces.......................................................173

1. Introduction ......................................................................................173

2. Dimensional reduction.....................................................................174

3. The catenoid .....................................................................................177

4. Conformal deformations ................................................................183

5. Supersymmetric quantum mechanics ..........................................190

6. The Dirac equation ..........................................................................194

7. Conclusion .........................................................................................203

Chapter 8 Generalized Non-Relativistic Supersymmetric Quantum Mechanics.................................................................................................206

1. Introduction........................................................................................206

2. Introduction to supersymmetric quantum mechanics in one

dimension...........................................................................................208

3. Computational examples.................................................................213

4. Generalization to multi-dimensions..............................................221

5. Clusters of distinguishable particles.............................................228

6. Electronic structure of atoms: Hydrogen and helium...............242

7. Aufbau approach for excited states...............................................246

8. Conclusions and perspectives.........................................................248

Chapter 9 Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics..............257

1. Introduction ......................................................................................257

2. Generalized variational principle for dissipative

hydrodynamics .................................................................................259

3. Viscous terms in dissipative hydrodynamics..............................266

4. Conclusion .........................................................................................274

5. Acknowledgment..............................................................................275

Citation......................................................................................................279

Index...........................................................................................................281

About Author

Solomon Brown obtained his Ph.D. in Theoretical and Applied Mechanics from University of Science & Technology, Haifa. He has more than a decade of experience in the field of mechanics. He has contributed numerous articles and research papers on classical mechanics and quantum mechanics. He is a Research Associate at Public University, Liege. Brown is also working as an independent researcher, technical writer and editor in the fields of mathematics, physics, astronomy and astrophysics. He has attended numerous engineering conferences and symposiums and has presented 5 papers on fluid mechanics and continuum mechanics.