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E-grāmata: Classical Mechanics: Hamiltonian and Lagrangian Formalism

3.75/5 (8 ratings by Goodreads)
  • Formāts: PDF+DRM
  • Izdošanas datums: 28-Aug-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642140372
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Classical Mechanics: Hamiltonian and Lagrangian FormalismPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 28-Aug-2010
  • Izdevniecība: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Valoda: eng
  • ISBN-13: 9783642140372
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Formalism of classical mechanics underlies a number of powerful mathematical methods that are widely used in theoretical and mathematical physics. This book considers the basics facts of Lagrangian and Hamiltonian mechanics, as well as related topics, such as canonical transformations, integral invariants, potential motion in geometric setting, symmetries, the Noether theorem and systems with constraints. While in some cases the formalism is developed beyond the traditional level adopted in the standard textbooks on classical mechanics, only elementary mathematical methods are used in the exposition of the material. The mathematical constructions involved are explicitly described and explained, so the book can be a good starting point for the undergraduate student new to this field. At the same time and where possible, intuitive motivations are replaced by explicit proofs and direct computations, preserving the level of rigor that makes the book useful for the graduate students intending to work in one of the branches of the vast field of theoretical physics. To illustrate how classical-mechanics formalism works in other branches of theoretical physics, examples related to electrodynamics, as well as to relativistic and quantum mechanics, are included.

This account of the fundamentals of Hamiltonian mechanics also covers related topics such as integral invariants and the Noether theorem. With just the elementary mathematical methods used for exposition, the book is suitable for novices as well as graduates.

Recenzijas

From the reviews:

The aim of the author of this interesting book is to present the main ideas of Hamiltonian mechanics and allied topics, by using just elementary mathematical methods. To illustrate how classical mechanics formalism works in other branches of theoretical physics, examples related to electrodynamics, as well as to relativistic and quantum mechanics, are included. This handbook is well addressed to undergraduate students in physics. (Franco Cardin, Zentralblatt MATH, Vol. 1206, 2011)

1 Sketch of Lagrangian Formalism
1(76)
1.1 Newton's Equation
1(7)
1.2 Galilean Transformations: Principle of Galilean Relativity
8(5)
1.3 Poincare and Lorentz Transformations: The Principle of Special Relativity
13(10)
1.4 Principle of Least Action
23(1)
1.5 Variational Analysis
24(5)
1.6 Generalized Coordinates, Corrdinate Transformations and Symmetries of an Action
29(7)
1.7 Examples of Continuous (Field) Systems
36(8)
1.8 Action of a Constrained System: The Recipe
44(7)
1.9 Action of a Constrained System: Justification of the Recipe
51(1)
1.10 Description of Constrained System by Singular Action
52(2)
1.11 Kinetic Versus Potential Energy: Forceless Mechanics of Hertz
54(2)
1.12 Electromagnetic Field in Lagrangian Formalism
56(21)
1.12.1 Maxwell Equations
56(3)
1.12.2 Nonsingular Lagrangian Action of Electrodynamics
59(4)
1.12.3 Manifestly Poincare-Invariant Formulation in Terms of a Singular Lagrangian Action
63(2)
1.12.4 Notion of Local (Gauge) Symmetry
65(3)
1.12.5 Lorentz Transformations of Three-Dimensional Potential: Role of Gauge Symmetry
68(1)
1.12.6 Relativistic Particle on Electromagnetic Background
69(3)
1.12.7 Poincare Transformations of Electric and Magnetic Fields
72(5)
2 Hamiltonian Formalism
77(42)
2.1 Derivation of Hamiltonian Equations
77(8)
2.1.1 Preliminaries
77(2)
2.1.2 From Lagrangian to Hamiltonian Equations
79(4)
2.1.3 Short Prescription for Hamiltonization Procedure, Physical Interpretation of Hamiltonian
83(2)
2.1.4 Inverse Problem: From Hamiltonian to Lagrangian Formulation
85(1)
2.2 Poisson Bracket and Symplectic Matrix
85(2)
2.3 General Solution to Hamiltonian Equations
87(4)
2.4 Picture of Motion in Phase Space
91(2)
2.5 Conserved Quantities and the Poisson Bracket
93(3)
2.6 Phase Space Transformations and Hamiltonian Equations
96(4)
2.7 Definition of Canonical Transformation
100(2)
2.8 Generalized Hamiltonian Equations: Example of Non-canonical Poisson Bracket
102(4)
2.9 Hamiltonian Action Functional
106(1)
2.10 Schrodinger Equation as the Hamiltonian System
107(6)
2.10.1 Lagrangian Action Associated with the Schrodinger Equation
108(3)
2.10.2 Probability as a Conserved Charge Via the Noether Theorem
111(2)
2.11 Hamiltonization Procedure in Terms of First-Order Action Functional
113(1)
2.12 Hamiltonization of a Theory with Higher-Order Derivatives
114(5)
2.12.1 First-Order Trick
114(2)
2.12.2 Ostrogradsky Method
116(3)
3 Canonical Transformations of Two-Dimensional Phase Space
119(8)
3.1 Time-Independent Canonical Transformations
119(4)
3.1.1 Time-Independent Canonical Transformations and Symplectic Matrix
119(2)
3.1.2 Generating Function
121(2)
3.2 Time-Dependent Canonical Transformations
123(4)
3.2.1 Canonical Transformations and Symplectic Matrix
123(2)
3.2.2 Generating Function
125(2)
4 Properties of Canonical Transformations
127(28)
4.1 Invariance of the Poisson Bracket (Symplectic Matrix)
128(5)
4.2 Infinitesimal Canonical Transformations: Hamiltonian as a Generator of Evolution
133(3)
4.3 Generating Function of Canonical Transformation
136(4)
4.3.1 Free Canonical Transformation and Its Function F(q1, p1, τ)
136(1)
4.3.2 Generating Function S(q, q1, τ)
137(3)
4.4 Examples of Canonical Transformations
140(5)
4.4.1 Evolution as a Canonical Transformation: Invariance of Phase-Space Volume
140(3)
4.4.2 Canomical Transformations in Perturbation Theory
143(1)
4.4.3 Coordinates Adjusted to a Surface
144(1)
4.5 Transformation Properties of the Hamiltonian Action
145(1)
4.6 Summary: Equivalent Definitions for Canonical Transformation
146(1)
4.7 Hamilton-Jacobi Equation
147(4)
4.8 Action Functional as a Generating Function of Evolution
151(4)
5 Integral Invariants
155(12)
5.1 Poincare-Cartan Integral Invariant
155(7)
5.1.1 Preliminaries
155(2)
5.1.2 Line Integral of a Vector Field, Hamiltonian Action, Poincare-Cartan and Poincare Integral Invariants
157(2)
5.1.3 Invariance of the Poincare-Cartan Integral
159(3)
5.2 Universal Integral Invariant of Poincare
162(5)
6 Potential Motion in a Geometric Setting
167(36)
6.1 Analysis of Trajectories and the Principle of Maupertuis
167(7)
6.1.1 Trajectory: Separation of Kinematics from Dynamics
168(2)
6.1.2 Equations for Trajectory in the Hamiltonian Formulation
170(1)
6.1.3 The Principle of Maupertuis for Trajectories
171(1)
6.1.4 Lagrangian Action for Trajectories
172(2)
6.2 Description of a Potential Motion in Terms of a Pair of Riemann Spaces
174(4)
6.3 Some Notions of Riemann Geometry
178(11)
6.3.1 Riemann Space
178(5)
6.3.2 Covariant Derivative and Riemann Connection
183(2)
6.3.3 Parallel Transport: Notions of Covariance and Coordinate Independence
185(4)
6.4 Definition of Covariant Derivative Through Parallel Transport: Formal Solution to the Parallel Transport Equation
189(2)
6.5 The Geodesic Line and Its Reparametrization Covariant Equation
191(2)
6.6 Example: A Surface Embedded in Euclidean Space
193(3)
6.7 Shortest Line and Geodesic Line: One More Example of a Singular Action
196(4)
6.8 Formal Geometrization of Mechanics
200(3)
7 Transformations, Symmetries and Noether Theorem
203(34)
7.1 The Notion of Invariant Action Functional
203(3)
7.2 Coordinate Transformation, Induced Transformation of Functions and Symmetries of an Action
206(5)
7.3 Examples of Invariant Actions, Galileo Group
211(3)
7.4 Poincare Group, Relativistic Particle
214(1)
7.5 Symmetries of Equations of Motion
215(3)
7.6 Noether Theorem
218(2)
7.7 Infinitesimal Symmetries
220(3)
7.8 Discussion of the Noether Theorem
223(1)
7.9 Use of Noether Charges for Reduction of the Order of Equations of Motion
224(1)
7.10 Examples
225(3)
7.11 Symmetries of Hamiltonian Action
228(9)
7.11.1 Infinitesimal Symmetries Given by Canonical Transformations
228(2)
7.11.2 Structure of Infinitesimal Symmetry of a General Form
230(4)
7.11.3 Hamiltonian Versus Lagrangian Global Symmetry
234(3)
8 Hamiltonian Formalism for Singular Theories
237(66)
8.1 Hamiltonization of a Singular Theory: The Recipe
238(9)
8.1.1 Two Basic Examples
238(4)
8.1.2 Dirac Procedure
242(5)
8.2 Justification of the Hamiltonization Recipe
247(7)
8.2.1 Configuration-Velocity Space
247(2)
8.2.2 Hamiltonization
249(3)
8.2.3 Comparison with the Dirac Recipe
252(2)
8.3 Classification of Constraints
254(1)
8.4 Comment on the Physical Interpretation of a Singular Theory
255(4)
8.5 Theory with Second-Class Constraints: Dirac Bracket
259(3)
8.6 Examples of Theories with Second-Class Constraints
262(4)
8.6.1 Mechanics with Kinematic Constraints
262(2)
8.6.2 Singular Lagrangian Action Underlying the Schrodinger Equation
264(2)
8.7 Examples of Theories with First-Class Constraints
266(8)
8.7.1 Electrodynamics
266(2)
8.7.2 Semiclassical Model for Description of Non Relativistic Spin
268(6)
8.8 Local Symmetries and Constraints
274(7)
8.9 Local Symmetry Does Not Imply a Conserved Charge
281(1)
8.10 Formalism of Extended Lagrangian
281(5)
8.11 Local Symmetries of the Extended Lagrangian: Dirac Conjecture
286(4)
8.12 Local Symmetries of the Initial Lagrangian
290(3)
8.13 Conversion of Second-Class Constraints by Deformation of Lagrangian Local Symmetries
293(10)
8.13.1 Conversion in a Theory with Hidden SO(1, 4) Global Symmetry
296(2)
8.13.2 Classical Mechanics Subject to Kinematic Constraints as a Gauge Theory
298(3)
8.13.3 Conversion in Maxwell-Proca Lagrangian for Massive Vector Field
301(2)
Bibliography 303(2)
Index 305
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