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El. knyga: Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds: A Geometric Approach to Modeling and Analysis

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This book provides an accessible introduction to the variational formulation of Lagrangian and Hamiltonian mechanics, with a novel emphasis on global descriptions of the dynamics, which is a significant conceptual departure from more traditional approaches based on the use of local coordinates on the configuration manifold. In particular, we introduce a general methodology for obtaining globally valid equations of motion on configuration manifolds that are Lie groups, homogeneous spaces, and embedded manifolds, thereby avoiding the difficulties associated with coordinate singularities.The material is presented in an approachable fashion by considering concrete configuration manifolds of increasing complexity, which then motivates and naturally leads to the more general formulation that follows. Understanding of the material is enhanced by numerous in-depth examples throughout the book, culminating in non-trivial applications involving multi-body systems.This book is written fo

r a general audience of mathematicians, engineers, and physicists with a basic knowledge of mechanics. Some basic background in differential geometry is helpful, but not essential, as the relevant concepts are introduced in the book, thereby making the material accessible to a broad audience, and suitable for either self-study or as the basis for a graduate course in applied mathematics, engineering, or physics. 

Mathematical Background.- Kinematics.- Classical Lagrangian and Hamiltonian Dynamics.- Langrangian and Hamiltonian Dynamics on (S1)n.- Lagrangian and Hamiltonian Dynamics on (S2)n.- Lagrangian and Hamiltonian Dynamics on SO(3).- Lagrangian and Hamiltonian Dynamics on SE(3).- Lagrangian and Hamiltonian Dynamics on Manifolds.- Rigid and Mult-body Systems.- Deformable Multi-body Systems.- Fundamental Lemmas of the Calculus of Variations.- Linearization as an Approximation to Lagrangian Dynamics on a Manifold.

Recenzijos

This book presents a monograph on foundational geometric principles of Lagrangian and Hamiltonian dynamics and their application in studying important physical systems. Throughout the book numerous examples of Lagrangian and Hamiltonian systems are included . It is very clearly written and it will be especially useful both for beginning researchers and for graduate students in applied mathematics, physics, or engineering. ( M. Eugenia Rosado Maria, Mathematical Reviews, May, 2018)

The starting point of this impressive textbook is the important fact that there are remarkable situations where the variables that describe a dynamical system do not lie in a vector space (i.e., a simple at algebraic structure) but rather lie in a geometrical setting allowing the differential calculus, namely a differential manifold. In conclusion, this book is extremely useful for each reader who wishes to develop a modern knowledge of analytical mechanics. (Mircea Crāmreanu, zbMATH 1381.70005, 2018)

This well-written and expansive book is ambitious in its scope in that it aims at sound and thorough pedagogy as far as its subject matter is concerned, and it also aims at preparing the reader for computational work . There are many good examples accompanying or even guiding the text, as well as extensive problem sets for the properly serious student. This book prove very valuable to its readership, be they mathematicians, engineers, or physicists. (Michael Berg, MAA Reviews, November, 2017)

1 Mathematical Background
1(52)
1.1 Vectors and Matrices
1(10)
1.1.1 Vector Spaces
2(2)
1.1.2 Symmetric and Skew-Symmetric Matrices
4(1)
1.1.3 Vector Operations in R2
5(1)
1.1.4 Vector Operations in R3
5(2)
1.1.5 Orthogonal Matrices on R3
7(1)
1.1.6 Homogeneous Matrices as Actions on R3
8(2)
1.1.7 Identities Involving Vectors, Orthogonal Matrices, and Skew-Symmetric Matrices
10(1)
1.1.8 Derivative Functions
11(1)
1.2 Manifold Concepts
11(16)
1.2.1 Manifolds
12(1)
1.2.2 Tangent Vectors, Tangent Spaces and Tangent Bundles
13(1)
1.2.3 Cotangent Vectors, Cotangent Spaces, and Cotangent Bundles
14(1)
1.2.4 Intersections and Products of Manifolds
15(1)
1.2.5 Examples of Manifolds, Tangent Bundles, and Cotangent Bundles
16(9)
1.2.6 Lie Groups and Lie Algebras
25(1)
1.2.7 Homogeneous Manifolds
26(1)
1.3 Vector Fields on a Manifold
27(16)
1.3.1 Vector Fields on a Manifold that Arise from Differential Equations
28(1)
1.3.2 Vector Fields on a Manifold that Arise from Differential-Algebraic Equations
29(3)
1.3.3 Linearized Vector Fields
32(1)
1.3.4 Stability of an Equilibrium
33(1)
1.3.5 Examples of Vector Fields
34(8)
1.3.6 Geometric Integrators
42(1)
1.4 Covector Fields on a Manifold
43(1)
1.5 Problems
43(10)
2 Kinematics
53(36)
2.1 Multi-Body Systems
53(1)
2.2 Euclidean Frames
54(1)
2.3 Kinematics of Ideal Mass Particles
55(1)
2.4 Rigid Body Kinematics
56(1)
2.5 Kinematics of Deformable Bodies
57(1)
2.6 Kinematics on a Manifold
58(1)
2.7 Kinematics as Descriptions of Velocity Relationships
59(23)
2.7.1 Translational Kinematics of a Particle on an Inclined Plane
59(1)
2.7.2 Translational Kinematics of a Particle on a Hyperbolic Paraboloid
60(2)
2.7.3 Rotational Kinematics of a Planar Pendulum
62(2)
2.7.4 Rotational Kinematics of a Spherical Pendulum
64(1)
2.7.5 Rotational Kinematics of a Double Planar Pendulum
65(2)
2.7.6 Rotational Kinematics of a Double Spherical Pendulum
67(1)
2.7.7 Rotational Kinematics of a Planar Pendulum Connected to a Spherical Pendulum
68(2)
2.7.8 Kinematics of a Particle on a Torus
70(3)
2.7.9 Rotational Kinematics of a Free Rigid Body
73(2)
2.7.10 Rotational and Translational Kinematics of a Rigid Body Constrained to a Fixed Plane
75(1)
2.7.11 Rotational and Translational Kinematics of a Free Rigid Body
76(2)
2.7.12 Translational Kinematics of a Rigid Link with Ends Constrained to Slide Along a Straight Line and a Circle in a Fixed Plane
78(2)
2.7.13 Rotational and Translational Kinematics of a Constrained Rigid Rod
80(2)
2.8 Problems
82(7)
3 Classical Lagrangian and Hamiltonian Dynamics
89(42)
3.1 Configurations as Elements in Rn
89(1)
3.2 Lagrangian Dynamics on Rn
90(4)
3.2.1 Lagrangian Function
90(1)
3.2.2 Variations on Rn
90(1)
3.2.3 Hamilton's Variational Principle
91(1)
3.2.4 Euler-Lagrange Equations
92(2)
3.3 Hamiltonian Dynamics on Rn
94(5)
3.3.1 Legendre Transformation and the Hamiltonian
95(1)
3.3.2 Hamilton's Equations and Euler-Lagrange Equations
95(2)
3.3.3 Hamilton's Phase Space Variational Principle
97(1)
3.3.4 Hamilton's Equations
98(1)
3.4 Flow Properties of Lagrangian and Hamiltonian Dynamics
99(6)
3.4.1 Energy Properties
100(1)
3.4.2 Cyclic Coordinates, Conserved Quantities, and Classical Reduction
101(3)
3.4.3 Symplectic Property
104(1)
3.5 Lagrangian and Hamiltonian Dynamics with Holonomic Constraints
105(3)
3.6 Lagrange-d'Alembert Principle
108(2)
3.7 Classical Particle Dynamics
110(13)
3.7.1 Dynamics of a Particle in Uniform, Constant Gravity
110(2)
3.7.2 Dynamics of a Particle, Constrained to an Inclined Plane, in Uniform, Constant Gravity
112(3)
3.7.3 Dynamics of a Particle, Constrained to a Hyperbolic Paraboloid, in Uniform, Constant Gravity
115(2)
3.7.4 Keplerian Dynamics of a Particle in Orbit
117(3)
3.7.5 Dynamics of a Particle Expressed in a Rotating Euclidean Frame
120(3)
3.8 Problems
123(8)
4 Lagrangian and Hamiltonian Dynamics on (S1)n
131(76)
4.1 Configurations as Elements in (S1)n
131(1)
4.2 Kinematics on (S1)n
132(1)
4.3 Lagrangian Dynamics on (S1)n
133(9)
4.3.1 Hamilton's Variational Principle in Terms of (q, q)
133(2)
4.3.2 Euler-Lagrange Equations in Terms of (q, μ)
135(3)
4.3.3 Hamilton's Variational Principle in Terms of (q, π)
138(1)
4.3.4 Euler-Lagrange Equations in Terms of (q, π)
139(3)
4.4 Hamiltonian Dynamics on (S1)n
142(9)
4.4.1 Hamilton's Phase Space Variational Principle in Terms of (q, μ)
142(2)
4.4.2 Hamilton's Equations in Terms of (q, μ)
144(4)
4.4.3 Hamilton's Phase Space Variational Principle in Terms of (q, μ)
148(1)
4.4.4 Hamilton's Equations in Terms of (q, π)
149(2)
4.5 Linear Approximations of Dynamics on (S1)n
151(1)
4.6 Dynamics of Systems on (S1)n
152(43)
4.6.1 Dynamics of a Planar Pendulum
152(5)
4.6.2 Dynamics of a Particle Constrained to a Circular Hoop That Rotates with Constant Angular Velocity
157(6)
4.6.3 Dynamics of Two Elastically Connected Planar Pendulums
163(6)
4.6.4 Dynamics of a Double Planar Pendulum
169(6)
4.6.5 Dynamics of a Particle on a Torus
175(6)
4.6.6 Dynamics of a Furuta Pendulum
181(6)
4.6.7 Dynamics of a Three-Dimensional Revolute Joint Robot
187(8)
4.7 Problems
195(12)
5 Lagrangian and Hamiltonian Dynamics on (S2)n
207(66)
5.1 Configurations as Elements in (S2)n
207(1)
5.2 Kinematics on (S2)n
208(1)
5.3 Lagrangian Dynamics on (S2)n
209(13)
5.3.1 Hamilton's Variational Principle in Terms of (q, q)
209(3)
5.3.2 Euler-Lagrange Equations Expressed in Terms of (q, q)
212(2)
5.3.3 Hamilton's Variational Principle in Terms of (q, ω))
214(2)
5.3.4 Euler-Lagrange Equations in Terms of (q, ω)
216(6)
5.4 Hamiltonian Dynamics on (S2)n
222(10)
5.4.1 Hamilton's Phase Space Variational Principle in Terms of (q, ω)
222(2)
5.4.2 Hamilton's Equations in Terms of (q, μ)
224(4)
5.4.3 Hamilton's Phase Space Variational Principle in Terms of (q, π)
228(1)
5.4.4 Hamilton's Equations in Terms of (q, π)
229(3)
5.5 Linear Approximations of Dynamics on (S2)n
232(1)
5.6 Dynamics on (S2)n
233(28)
5.6.1 Dynamics of a Spherical Pendulum
233(5)
5.6.2 Dynamics of a Particle Constrained to a Sphere That Rotates with Constant Angular Velocity
238(5)
5.6.3 Dynamics of a Spherical Pendulum Connected to Three Elastic Strings
243(7)
5.6.4 Dynamics of Two Elastically Connected Spherical Pendulums
250(5)
5.6.5 Dynamics of a Double Spherical Pendulum
255(6)
5.7 Problems
261(12)
6 Lagrangian and Hamiltonian Dynamics on SO(3)
273(40)
6.1 Configurations as Elements in the Lie Group SO(3)
274(1)
6.2 Kinematics on SO(3)
275(1)
6.3 Lagrangian Dynamics on SO(3)
276(5)
6.3.1 Hamilton's Variational Principle
276(2)
6.3.2 Euler-Lagrange Equations: General Form
278(2)
6.3.3 Euler-Lagrange Equations: Quadratic Kinetic Energy
280(1)
6.4 Hamiltonian Dynamics on SO(3)
281(3)
6.4.1 Hamilton's Phase Space Variational Principle
281(1)
6.4.2 Hamilton's Equations: General Form
282(2)
6.4.3 Hamilton's Equations: Quadratic Kinetic Energy
284(1)
6.5 Linear Approximations of Dynamics on SO(3)
284(1)
6.6 Dynamics on SO(3)
285(19)
6.6.1 Dynamics of a Freely Rotating Rigid Body
285(3)
6.6.2 Dynamics of a Three-Dimensional Pendulum
288(5)
6.6.3 Dynamics of a Rotating Rigid Body in Orbit
293(5)
6.6.4 Dynamics of a Rigid Body Planar Pendulum
298(6)
6.7 Problems
304(9)
7 Lagrangian and Hamiltonian Dynamics on SE(3)
313(34)
7.1 Configurations as Elements in the Lie Group SE(3)
313(1)
7.2 Kinematics on SE(3)
314(2)
7.3 Lagrangian Dynamics on SE(3)
316(6)
7.3.1 Hamilton's Variational Principle
316(2)
7.3.2 Euler-Lagrange Equations: General Form
318(2)
7.3.3 Euler-Lagrange Equations: Quadratic Kinetic Energy
320(2)
7.4 Hamiltonian Dynamics on SE(3)
322(5)
7.4.1 Hamilton's Phase Space Variational Principle
322(1)
7.4.2 Hamilton's Equations: General Form
323(2)
7.4.3 Hamilton's Equations: Quadratic Kinetic Energy
325(2)
7.5 Linear Approximations of Dynamics on SE(3)
327(1)
7.6 Dynamics on SE(3)
327(12)
7.6.1 Dynamics of a Rotating and Translating Rigid Body
327(4)
7.6.2 Dynamics of an Elastically Supported Rigid Body
331(5)
7.6.3 Dynamics of a Rotating and Translating Rigid Dumbbell Satellite in Orbit
336(3)
7.7 Problems
339(8)
8 Lagrangian and Hamiltonian Dynamics on Manifolds
347(52)
8.1 Lagrangian Dynamics on a Manifold
348(5)
8.1.1 Variations on the Tangent Bundle T M
348(1)
8.1.2 Lagrangian Variational Conditions
349(2)
8.1.3 Euler-Lagrange Equations on T M
351(1)
8.1.4 Extension of the Lagrangian Vector Field from TM to T M
352(1)
8.2 Hamiltonian Dynamics on a Manifold
353(6)
8.2.1 Legendre Transformation and the Hamiltonian
353(1)
8.2.2 Variations on the Cotangent Bundle T*M
353(1)
8.2.3 Hamilton's Phase Space Variational Principle
354(1)
8.2.4 Hamilton's Equations on T*M
354(3)
8.2.5 Invariance of the Hamiltonian
357(1)
8.2.6 Extension of the Hamiltonian Vector Field from T*M to T*Rn
358(1)
8.3 Lagrangian and Hamiltonian Dynamics on Products of Manifolds
359(8)
8.3.1 Lagrangian and Hamiltonian Dynamics on a Product of Linear Manifolds
360(2)
8.3.2 Lagrangian and Hamiltonian Dynamics on (S1)n
362(2)
8.3.3 Lagrangian and Hamiltonian Dynamics on (S2)n
364(3)
8.4 Lagrangian and Hamiltonian Dynamics Using Lagrange Multipliers
367(2)
8.5 Lagrangian and Hamiltonian Dynamics on SO(3)
369(2)
8.6 Lagrangian and Hamiltonian Dynamics on a Lie Group
371(10)
8.6.1 Additional Material on Lie Groups and Lie Algebras
371(1)
8.6.2 Variations on a Lie Group
372(1)
8.6.3 Euler-Lagrange Equations
373(3)
8.6.4 Legendre Transformation and Hamilton's Equations
376(1)
8.6.5 Hamilton's Phase Space Variational Principle
377(2)
8.6.6 Reassessment of Results in the Prior
Chapters
379(2)
8.7 Lagrangian and Hamiltonian Dynamics on a Homogeneous Manifold
381(6)
8.7.1 Additional Material on Homogeneous Manifolds
381(1)
8.7.2 A Lifting Process
382(1)
8.7.3 Euler-Lagrange Equations
382(2)
8.7.4 Reassessment of Results in the Prior
Chapters
384(3)
8.8 Lagrange-d'Alembert Principle
387(1)
8.9 Problems
388(11)
9 Rigid and Multi-Body Systems
399(86)
9.1 Dynamics of a Planar Mechanism
400(5)
9.1.1 Euler-Lagrange Equations
402(1)
9.1.2 Hamilton's Equations
403(1)
9.1.3 Conservation Properties
404(1)
9.1.4 Equilibrium Properties
404(1)
9.2 Dynamics of a Horizontally Rotating Pendulum on a Cart
405(4)
9.2.1 Euler-Lagrange Equations
405(2)
9.2.2 Hamilton's Equations
407(1)
9.2.3 Conservation Properties
408(1)
9.2.4 Equilibrium Properties
408(1)
9.3 Dynamics of a Connection of a Planar Pendulum and Spherical Pendulum
409(6)
9.3.1 Euler-Lagrange Equations
410(2)
9.3.2 Hamilton's Equations
412(1)
9.3.3 Conservation Properties
413(1)
9.3.4 Equilibrium Properties
413(2)
9.4 Dynamics of a Spherical Pendulum on a Cart
415(6)
9.4.1 Euler-Lagrange Equations
416(2)
9.4.2 Hamilton's Equations
418(1)
9.4.3 Conservation Properties
419(1)
9.4.4 Equilibrium Properties
420(1)
9.5 Dynamics of a Rotating Rigid Body with Appendage
421(4)
9.5.1 Euler-Lagrange Equations
421(2)
9.5.2 Hamilton's Equations
423(1)
9.5.3 Conservation Properties
424(1)
9.5.4 Equilibrium Properties
424(1)
9.6 Dynamics of a Three-Dimensional Pendulum on a Cart
425(5)
9.6.1 Euler-Lagrange Equations
426(2)
9.6.2 Hamilton's Equations
428(1)
9.6.3 Conservation Properties
429(1)
9.6.4 Equilibrium Properties
430(1)
9.7 Dynamics of Two Rigid Bodies Constrained to Have a Common Material Point
430(5)
9.7.1 Euler-Lagrange Equations
431(2)
9.7.2 Hamilton's Equations
433(1)
9.7.3 Conservation Properties
434(1)
9.7.4 Equilibrium Properties
435(1)
9.8 Dynamics of a Rotating and Translating Rigid Body with an Appendage
435(6)
9.8.1 Euler-Lagrange Equations
436(3)
9.8.2 Hamilton's Equations
439(1)
9.8.3 Conservation Properties
440(1)
9.8.4 Equilibrium Properties
441(1)
9.9 Dynamics of a Full Body System
441(5)
9.9.1 Euler-Lagrange Equations
442(2)
9.9.2 Hamilton's Equations
444(1)
9.9.3 Conservation Properties
445(1)
9.9.4 Equilibrium Properties
446(1)
9.9.5 Relative Full Body Dynamics
446(1)
9.10 Dynamics of a Spacecraft with Reaction Wheel Assembly
446(10)
9.10.1 Euler-Lagrange Equations
448(5)
9.10.2 Hamilton's Equations
453(1)
9.10.3 Conservation Properties
454(1)
9.10.4 Equilibrium Properties
455(1)
9.11 Dynamics of a Rotating Spacecraft and Control Moment Gyroscope
456(8)
9.11.1 Euler-Lagrange Equations
457(5)
9.11.2 Hamilton's Equations
462(1)
9.11.3 Conservation Properties
463(1)
9.12 Dynamics of Two Quad Rotors Transporting a Cable-Suspended Payload
464(7)
9.12.1 Euler-Lagrange Equations
465(5)
9.12.2 Hamilton's Equations
470(1)
9.12.3 Conservation Properties
470(1)
9.12.4 Equilibrium Properties
470(1)
9.13 Problems
471(14)
10 Deformable Multi-Body Systems
485(44)
10.1 Infinite-Dimensional Multi-Body Systems
485(1)
10.2 Dynamics of a Chain Pendulum
486(6)
10.2.1 Euler-Lagrange Equations
487(2)
10.2.2 Hamilton's Equations
489(1)
10.2.3 Comments
490(1)
10.2.4 Conservation Properties
491(1)
10.2.5 Equilibrium Properties
491(1)
10.3 Dynamics of a Chain Pendulum on a Cart
492(9)
10.3.1 Euler-Lagrange Equations
493(3)
10.3.2 Hamilton's Equations
496(2)
10.3.3 Comments
498(1)
10.3.4 Conservation Properties
499(1)
10.3.5 Equilibrium Properties
499(2)
10.4 Dynamics of a Free-Free Chain
501(6)
10.4.1 Euler-Lagrange Equations
502(2)
10.4.2 Hamilton's Equations
504(2)
10.4.3 Conservation Properties
506(1)
10.4.4 Equilibrium Properties
507(1)
10.5 Dynamics of a Fixed-Free Elastic Rod
507(6)
10.5.1 Euler-Lagrange Equations
508(3)
10.5.2 Hamilton's Equations
511(1)
10.5.3 Conservation Properties
512(1)
10.5.4 Equilibrium Properties
513(1)
10.6 Problems
513(8)
A Fundamental Lemmas of the Calculus of Variations
521(2)
A.1 Fundamental Lemma of Variational Calculus on Kn
521(1)
A.2 Fundamental Lemma of the Calculus of Variations on an Embedded Manifold
522(1)
A.3 Fundamental Lemma of Variational Calculus on a Lie Group
522(1)
B Linearization as an Approximation to Lagrangian Dynamics on a Manifold
523(6)
B.1 Linearization on TS1
524(2)
B.2 Linearization on TS2
526(1)
B.3 Linearization on TSO(3)
527(2)
References 529(6)
Index 535
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