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Graduate Classical Mechanics

Classical Mechanics Physics

Author: Michael Fowler
Publisher: University of Virginia
Source: LibreTexts
Language: EN
Category: Basic Sciences

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Contents· 201 sections

01: Introductory Statics - the Catenary and the Arch
1.01: The Catenary
1.02: A Tight String
1.03: Not So Tight
1.04: Ideal Arches
02: The Calculus of Variations
2.01: The Catenary and the Soap Film
2.02: A Soap Film Between Two Horizontal Rings- the Euler-Lagrange Equation
2.03: General Method for the Minimization Problem
2.04: An Important First Integral of the Euler-Lagrange Equation
2.05: Fastest Curve for Given Horizontal Distance
2.06: The Perfect Pendulum
2.07: Calculus of Variations with Many Variables
2.08: Multivariable First Integral
2.09: The Soap Film and the Chain
2.10: Lagrange Multipliers
2.11: Lagrange Multiplier for the Chain
2.12: The Brachistochrone
03: Fermat's Principle of Least Time
3.01: Another Minimization Problem
3.02: Huygens' Picture of Wave Propagation
3.03: Fermats Principle
3.04: Reflection Too
3.05: The Bottom Line- Geometric Optics and Wave Optics
04: Hamilton's Principle and Noether's Theorem
4.01: Introduction- Galileo and Newton
4.02: Derivation of Hamiltons Principle from Newtons Laws in Cartesian Co-ordinates- Calculus of Variations Done Backwards
4.03: But Why
4.04: Lagranges Equations from Hamiltons Principle Using Calculus of Variations
4.05: Generalized Momenta and Forces
4.06: Non-uniqueness of the Lagrangian
4.07: First Integral- Energy Conservation and the Hamiltonian
4.08: Example 1- One Degree of Freedom- Atwoods Machine
4.09: Example 2- Lagrangian Formulation of the Central Force Problem
4.10: Conservation Laws and Noethers Theorem
4.11: Momentum Conservation
4.12: Center of Mass
4.13: Angular Momentum Conservation
05: Mechanical Similarity and the Virial Theorem
5.01: Some Examples
5.02: Lagrangian Treatment
5.03: The Virial Theorem
06: Hamiltons Equations
6.01: A Dynamical Systems Path in Configuration Space and in State Space
6.02: Phase Space
6.03: Going From State Space to Phase Space
6.04: How It's Done in Thermodynamics
6.05: Math Note - the Legendre Transform
6.06: Hamilton's Use of the Legendre Transform
6.07: Checking that We Can Eliminate the qi's
6.08: Hamiltons Equations
6.09: A Simple Example
07: Time Evolution in Phase Space- Poisson Brackets and Constants of the Motion
7.01: The Poisson Bracket
7.02: Interlude - a Bit of History of Quantum Mechanics
7.03: The Jacobi Identity
7.04: Poissons Theorem
7.05: Example- Angular Momentum Components
08: A New Way to Write the Action Integral
8.01: Function of Endpoint Position
8.02: Function of Endpoint Time
8.03: Varying Both Ends
8.04: Another Way of Writing the Action Integral
8.05: How this Classical Action Relates to Phase in Quantum Mechanics
8.06: Hamiltons Equations from Action Minimization
8.07: How Can p q Really Be Independent Variables
09: Maupertuis Principle - Minimum Action Path at Fixed Energy
9.01: Divine Guidance
9.02: It's Not About Time
9.03: The Abbreviated Action
9.04: Maupertuis Principle and the Time-Independent Schrodinger Equation
10: Canonical Transformations
10.01: Point Transformations
10.02: General and Canonical Transformations
10.03: Generating Functions for Canonical Transformations
10.04: Generating Functions in Different Variables
10.05: Poisson Brackets under Canonical Transformations
10.06: Time Development is a Canonical Transformation Generated by the Action
11: Introduction to Liouville's Theorem
11.01: Paths in Simple Phase Spaces - the SHO and Falling Bodies
11.02: Following Many Systems- a Gas in Phase Space
11.03: Liouvilles Theorem- Local Gas Density is Constant along a Phase Space Path
11.04: Landaus Proof Using the Jacobian
11.05: Jacobian for Time Evolution
11.06: Jacobians 101
11.07: Jacobian proof of Liouvilles Theorem
11.08: Simpler Proof of Liouvilles Theorem
11.09: Energy Gradient and Phase Space Velocity
12: The Hamilton-Jacobi Equation
12.01: Back to Configuration Space
12.02: The Central Role of These Constants of Integration
12.03: Separation of Variables for a Central Potential Cyclic Variables
13: Adiabatic Invariants and Action-Angle Variables
13.01: Adiabatic Invariants
13.02: Adiabatic Invariance and Quantum Mechanics
13.03: Action-Angle Variables
13.04: Kepler Orbit Action-Angle Variables
14: Mathematics for Orbits
14.01: Preliminaries- Conic Sections
14.02: The Ellipse
14.03: The Parabola
14.04: The Hyperbola
15: Keplerian Orbits
15.01: Preliminary- Polar Equations for Conic Section Curves
15.02: Summary
15.03: Keplers Statement of his Three Laws
15.04: Dynamics of Motion in a Central Potential- Deriving Keplers Laws
15.05: A Vectorial Approach- Hamiltons Equation and the Runge Lenz Vector
16: Elastic Scattering
16.01: Billiard Balls
16.02: Discovery of the Nucleus
16.03: The Differential Cross Section
16.04: Analyzing Inverse-Square Repulsive Scattering- Kepler Again
16.05: Vectorial Derivation of the Scattering Angle
17: Small Oscillations
17.01: Particle in a Well
17.02: Two Coupled Pendulums
17.03: Normal Modes
17.04: Principle of Superposition
17.05: Three Coupled Pendulums
17.06: Normal Coordinates
17.07: Three Equal Pendulums Equally Coupled
18: Driven Oscillator
18.01: More General Energy Exchange
18.02: Damped Driven Oscillator
19: One-Dimensional Crystal Dynamics
19.01: The Model
19.02: The Circulant Matrix- Nature of its Eigenstates
19.03: Comparison with Raising Operators
19.04: Finding the Eigenvectors
19.05: Eigenvectors of the Linear Chain
19.06: Allowed Wavenumbers from Boundary Conditions
19.07: Finding the Eigenvalues
19.08: The Discrete Fourier Transform
19.09: A Note on the Physics of These Waves
20: Parametric Resonance
20.01: Introduction to Parametric Resonance
20.02: Resonance near Double the Natural Frequency
20.03: Example- Pendulum Driven at near Double the Natural Frequency
21: The Ponderomotive Force
21.01: Introduction to the Ponderomotive Force
21.02: Finding the Effective Potential Generated by the Oscillating Force
21.03: Stability of a Pendulum with a Rapidly Oscillating Vertical Driving Force
21.04: Hand-Waving Explanation of the Ponderomotive Force
21.05: Pendulum with Top Point Oscillating Rapidly in a Horizontal Direction
22: Resonant Nonlinear Oscillations
22.01: Introduction to Resonant Nonlinear Oscillations
22.02: Frequency of Oscillation of a Particle is a Slightly Anharmonic Potential
22.03: Resonance in a Damped Driven Linear Oscillator- A Brief Review
22.04: Damped Driven Nonlinear Oscillator- Qualitative Discussion
22.05: Nonlinear Case - Landaus Analysis
22.06: Frequency Multiples
23: Damped Driven Pendulum- Period Doubling and Chaos
23.01: Introduction
23.02: The Road to Chaos
23.03: Lyapunov Exponents and Dimensions of Strange Attractors
24: Motion of a Rigid Body - the Inertia Tensor
24.01: Definition of Rigid
24.02: Rotation of a Body about a Fixed Axis
24.03: General Motion of a Rotating Rigid Body
24.04: The Inertia Tensor
24.05: Tensors 101
24.06: Definition of a Tensor
24.07: Diagonalizing the Inertia Tensor
24.08: Principal Axes Form of Moment of Inertia Tensor
24.09: Relating Angular Momentum to Angular Velocity
24.10: Symmetries Other Axes the Parallel Axis Theorem
25: Moments of Inertia and Rolling Motion
25.01: Examples of Moments of Inertia
25.02: Analyzing Rolling Motion
25.03: Rolling Without Slipping - Two Views
25.04: Cylinder Rolling Inside another Cylinder
26: Rigid Body Moving Freely
26.01: Angular Momentum and Angular Velocity
26.02: Precession of a Symmetrical Top
26.03: Throwing a Football
26.04: Equations of Motion for Rigid Body with External Forces
27: Euler Angles
27.01: Definition of Euler Angles
27.02: Angular Velocity and Energy in Terms of Eulers Angles
27.03: Free Motion of a Symmetrical Top
27.04: Motion of Symmetrical Top around a Fixed Base with Gravity - Nutation
27.05: Steady Precession
27.06: Stability of Top Spinning about Vertical Axis
28: Eulers Equations
28.01: Introduction to Eulers Equations
28.02: Free Rotation of a Symmetric Top Using Eulers Equations
28.03: Using Energy and Angular Momentum Conservation
28.04: The Asymmetrical Top
29: Non-Inertial Frame and Coriolis Effect
29.01: The Lagrangian in Accelerating and Rotating Frames
29.02: Uniformly Rotating Frame
29.03: Coriolis Effect - Particle Moving near Earths Surface
29.04: Gyroscopes and Gyrocompasses in Navigation
30: A Rolling Sphere on a Rotating Plane
30.01: Introduction
30.02: Holonomic Constraints and non-Holonomic Constraints
30.03: DAlemberts Principle
30.04: Ball with External Forces Rolling on Horizontal Plane
30.05: Ball Rolling on Rotating Plane
30.06: Ball Rolling on Inclined Rotating Plane