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Variational Principles in Classical Mechanics

This text emphasizes the important philosophical advantages of using variational principles, rather than the vectorial approach adopted by Newton, and attempts to bridge the chasm that exists between the approaches used in classical and quantam physics.

Classical Mechanics Physics

Author: Douglas Cline
Publisher: University of Rochester
Source: LibreTexts
License: CC BY-NC-SA
Language: EN
Category: Basic Sciences

Read online Download PDF (open licence)

Contents· 245 sections

01: A brief History of Classical Mechanics
1.01: Introduction
1.02: Greek Antiquity
1.03: Middle Ages
1.04: Age of Enlightenment
1.05: Variational methods in physics
1.06: The 20th Century Revolution in Physics
02: Review of Newtonian Mechanics
2.01: Introduction to Newtonian Mechanics
2.02: Newton's Laws of motion
2.03: Inertial Frames of reference
2.04: First-order Integrals in Newtonian mechanics
2.05: Conservation Laws in Classical Mechanics
2.06: Motion of finite-sized and many-body systems
2.07: Center of Mass of a Many-Body System
2.08: Total Linear Momentum of a Many-body System
2.09: Angular Momentum of a Many-Body System
2.10: Work and Kinetic Energy for a Many-Body System
2.11: Virial Theorem
2.12: Applications of Newton's Equations of Motion
2.13: Solution of many-body equations of motion
2.14: Newton's Law of Gravitation
2.E: Review of Newtonian Mechanics (Exercises)
2.S: Newtonian Mechanics (Summary)
03: Linear Oscillators
3.01: Introduction to Linear Oscillators
3.02: Linear Restoring Forces
3.03: Linearity and Superposition
3.04: Geometrical Representations of Dynamical Motion
3.05: Linearly-damped Free Linear Oscillator
3.06: Sinusoidally-driven linearly-damped linear oscillator
3.07: Wave equation
3.08: Travelling and standing wave solutions of the wave equation
3.09: Waveform Analysis
3.10: Signal Processing
3.11: Wave Propagation
3.E: Linear Oscillators (Exercises)
3.S: Linear Oscillators (Summary)
04: Nonlinear Systems and Chaos
4.01: Introduction to Nonlinear Systems and Chaos
4.02: Weak Nonlinearity
4.03: Bifurcation and Point Attractors
4.04: Limit Cycles
4.05: Harmonically-driven linearly-damped plane pendulum
4.06: Differentiation Between Ordered and Chaotic Motion
4.07: Wave Propagation for Non-linear Systems
4.E: Nonlinear Systems and Chaos (Exercises)
4.S: Nonlinear Systems and Chaos (Summary)
05: Calculus of Variations
5.01: Introduction to the Calculus of Variations
5.02: Eulers Differential Equation
5.03: Applications of Eulers Equation
5.04: Selection of the Independent Variable
5.05: Functions with Several Independent Variables
5.06: Eulers Integral Equation
5.07: Constrained Variational Systems
5.08: Generalized coordinates in Variational Calculus
5.09: Lagrange multipliers for Holonomic Constraints
5.10: Geodesic
5.11: Variational Approach to Classical Mechanics
5.E: Calculus of Variations (Exercises)
5.S: Calculus of Variations (Summary)
06: Lagrangian Dynamics
6.01: Introduction to Lagrangian Dynamics
6.02: Newtonian plausibility argument for Lagrangian mechanics
6.03: Lagrange Equations from dAlemberts Principle
6.04: Lagrange equations from Hamiltons Principle
6.05: Constrained Systems
6.06: Applying the Euler-Lagrange equations to classical mechanics
6.07: Applications to unconstrained systems
6.08: Applications to systems involving holonomic constraints
6.09: Applications involving Non-holonomic Constraints
6.10: Velocity-dependent Lorentz force
6.11: Time-dependent forces
6.12: Impulsive Forces
6.13: The Lagrangian versus the Newtonian approach to classical mechanics
6.E: Lagrangian Dynamics (Exercises)
6.S: Lagrangian Dynamics (Summary)
07: Symmetries Invariance and the Hamiltonian
7.01: Introduction to Symmetries Invariance and the Hamiltonian
7.02: Generalized Momentum
7.03: Invariant Transformations and Noethers Theorem
7.04: Rotational invariance and conservation of angular momentum
7.05: Cyclic Coordinates
7.06: Kinetic Energy in Generalized Coordinates
7.07: Generalized Energy and the Hamiltonian Function
7.08: Generalized energy theorem
7.09: Generalized energy and total energy
7.10: Hamiltonian Invariance
7.11: Hamiltonian for Cyclic Coordinates
7.12: Symmetries and Invariance
7.13: Hamiltonian in Classical Mechanics
7.E: Symmetries Invariance and the Hamiltonian (Exercises)
7.S: Symmetries Invariance and the Hamiltonian (Summary)
08: Hamiltonian Mechanics
8.01: Introduction
8.02: Legendre Transformation between Lagrangian and Hamiltonian mechanics
8.03: Hamiltons Equations of Motion
8.04: Hamiltonian in Different Coordinate Systems
8.05: Applications of Hamiltonian Dynamics
8.06: Routhian Reduction
8.07: Variable-mass systems
8.E: Hamiltonian Mechanics (Exercises)
8.S: Hamiltonian Mechanics (Summary)
09: Hamilton's Action Principle
9.01: Introduction to Hamilton's Action Principle
9.02: Hamilton's Principle of Stationary Action
9.03: Lagrangian
9.04: Application of Hamilton's Action Principle to Mechanics
9.S: Hamilton's Action Principle (Summary)
10: Nonconservative Systems
10.01: Introduction to Nonconservative Systems
10.02: Origins of Nonconservative Motion
10.03: Algebraic Mechanics for Nonconservative Systems
10.04: Rayleighs Dissipation Function
10.05: Dissipative Lagrangians
10.S: Nonconservative systems (Summary)
11: Conservative two-body Central Forces
11.01: Introduction to Conservative two-body Central Forces
11.02: Equivalent one-body Representation for two-body motion
11.03: Angular Momentum
11.04: Equations of Motion
11.05: Differential Orbit Equation
11.06: Hamiltonian
11.07: General Features of the Orbit Solutions
11.08: Inverse-square two-body central force
11.09: Isotropic linear two-body central force
11.10: Closed-orbit Stability
11.11: The Three-Body Problem
11.12: Two-body Scattering
11.13: Two-body kinematics
11.E: Conservative two-body Central Forces (Exercises)
11.S: Conservative two-body Central Forces (Summary)
12: Non-inertial Reference Frames
12.01: Introduction to Non-inertial Reference Frames
12.02: Translational acceleration of a reference frame
12.03: Rotating Reference Frame
12.04: Reference Frame Undergoing Rotation Plus Translation
12.05: Newtons Law of Motion in a Non-Inertial Frame
12.06: Lagrangian Mechanics in a Non-Inertial Frame
12.07: Centrifugal Force
12.08: Coriolis Force
12.09: Routhian Reduction for Rotating Systems
12.10: Effective gravitational force near the surface of the Earth
12.11: Free Motion on the Earth
12.12: Weather systems
12.13: Foucault pendulum
12.E: Non-inertial reference frames (Exercises)
12.S: Non-inertial reference frames (Summary)
13: Rigid-body Rotation
13.01: Introduction to Rigid-body Rotation
13.02: Rigid-body Coordinates
13.03: Rigid-body Rotation about a Body-Fixed Point
13.04: Inertia Tensor
13.05: Matrix and Tensor Formulations of Rigid-Body Rotation
13.06: Principal Axis System
13.07: Diagonalize the Inertia Tensor
13.08: Parallel-Axis Theorem
13.09: Perpendicular-axis Theorem for Plane Laminae
13.10: General Properties of the Inertia Tensor
13.11: Angular Momentum and Angular Velocity Vectors
13.12: Kinetic Energy of Rotating Rigid Body
13.13: Euler Angles
13.14: Angular Velocity
13.15: Kinetic energy in terms of Euler angular velocities
13.16: Rotational Invariants
13.17: Eulers equations of motion for rigid-body rotation
13.18: Lagrange equations of motion for rigid-body rotation
13.19: Hamiltonian equations of motion for rigid-body rotation
13.20: Torque-free rotation of an inertially-symmetric rigid rotor
13.21: Torque-free rotation of an asymmetric rigid rotor
13.22: Stability of torque-free rotation of an asymmetric body
13.23: Symmetric rigid rotor subject to torque about a fixed point
13.24: The Rolling Wheel
13.25: Dynamic balancing of wheels
13.26: Rotation of Deformable Bodies
13.E: Rigid-body Rotation (Exercises)
13.S: Rigid-body Rotation (Summary)
14: Coupled Linear Oscillators
14.01: Introduction to Coupled Linear Oscillators
14.02: Two Coupled Linear Oscillators
14.03: Normal Modes
14.04: Center of Mass Oscillations
14.05: Weak Coupling
14.06: General Analytic Theory for Coupled Linear Oscillators
14.07: Two-body coupled oscillator systems
14.08: Three-body coupled linear oscillator systems
14.09: Molecular coupled oscillator systems
14.10: Discrete Lattice Chain
14.11: Damped Coupled Linear Oscillators
14.12: Collective Synchronization of Coupled Oscillators
14.E: Coupled linear oscillators (Exercises)
14.S: Coupled linear oscillators (Summary)
15: Advanced Hamiltonian Mechanics
15.01: Introduction to Advanced Hamiltonian Mechanics
15.02: Poisson bracket Representation of Hamiltonian Mechanics
15.03: Canonical Transformations in Hamiltonian Mechanics
15.04: Hamilton-Jacobi Theory
15.05: Action-angle Variables
15.06: Canonical Perturbation Theory
15.07: Symplectic Representation
15.08: Comparison of the Lagrangian and Hamiltonian Formulations
15.E: Advanced Hamiltonian Mechanics (Exercises)
15.S: Advanced Hamiltonian mechanics (Summary)
16: Analytical Formulations for Continuous Systems
16.01: Introduction
16.02: The Continuous Uniform Linear Chain
16.03: The Lagrangian density formulation for continuous systems
16.04: The Hamiltonian density formulation for continuous systems
16.05: Linear Elastic Solids
16.06: Electromagnetic Field Theory
16.07: Ideal Fluid Dynamics
16.08: Viscous Fluid Dynamics
16.09: Summary and implications
17: Relativistic Mechanics
17.01: Introduction to Relativistic Mechanics
17.02: Galilean Invariance
17.03: Special Theory of Relativity
17.04: Relativistic Kinematics
17.05: Geometry of Space-time
17.06: Lorentz-Invariant Formulation of Lagrangian Mechanics
17.07: Lorentz-invariant formulations of Hamiltonian Mechanics
17.08: The General Theory of Relativity
17.09: Implications of Relativistic Theory to Classical Mechanics
17.E: Relativistic Mechanics (Exercises)
17.S: Relativistic Mechanics (Summary)
18: The Transition to Quantum Physics
18.01: Introduction to Quantum Physics
18.02: Brief summary of the origins of quantum theory
18.03: Hamiltonian in Quantum Theory
18.04: Lagrangian Representation in Quantum Theory
18.05: Correspondence Principle
18.S: The Transition to Quantum Physics (Summary)
19: Mathematical Methods for Classical Mechanics
19.01: Introduction
19.02: Appendix - Matrix Algebra
19.03: Appendix - Vector algebra
19.04: Appendix - Orthogonal Coordinate Systems
19.05: Appendix - Coordinate transformations
19.06: Appendix - Tensor Algebra
19.07: Appendix - Aspects of Multivariate Calculus
19.08: Appendix - Vector Differential Calculus
19.09: Appendix - Vector Integral Calculus
19.10: Appendix - Waveform analysis
19.11: Bibliography