​​​​Juan David Vasquez Jaramillo, Ph.D​

1. Newtonian Mechanics

    1.1 Introductory Problem: The Inverted Pendulum

    1.2 Computational Solution to The Inverted Pendulum

    1.3 The Problem of Dissipation and Friction.

    1.4 The Problem of the Wave on a String.

    1.5 The Problem of a Wave on a Two Dimensional Membrane.

    1.6 Phase Space and the N-Body Problem

    1.7 Energy, Work and Conservation of Mechanical Energy.

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2. Lagrangian Mechanics

    2.1 Coordinates and Constraints

    2.2 Elements of Functional Analysis.

    2.3 Stationary Action Principle.

    2.4 Euler-Lagrange Equations and the Ostrogradsky Equation.

    2.5 Application: Double Pendulum

    2.6 Computational Solution to the Double Pendulum.

    2.7 Point Transformations in Lagrangian Mechanics.

    2.8 Symmetries and Noether Theorem.

    2.9 Virtual Work and D'alambert Principle.

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3. Central Force Problems and Classical Scattering

    3.1 Reduction of the Two Body Problem to a Single Body

          Problem.

    3.2 Central Field Force and Lagrangian Formulation.

    3.3 Kepler's Problem and Celestial Mechanics.

    3.4 Inverse Square Central Force Field.

    3.5 Classical Theory of Scattering: Impact Parameter and

          Differential Cross Section.

    3.6 Rutherford and Yukawa Scattering.

    3.7 Kinematics in Lab and Center of Mass Coordinates.

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4. Introduction to Hamiltonian Dynamics

    4.1 Legendre Transformation.

    4.2 Hamilton Canonical Equations.

    4.3 Liouville's Theorem and Classical Dynamics.

    4.4 Phase Space Density Function.

    4.5 Classical Statistical Mechanics.

    4.6 Boltzmann Equation and the H-Theorem

    4.7 Maxwell-Boltzmann Distribution

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5. Dynamical Systems and State Space Models

    5.1 Stationary Dynamics From the Green's Function.

    5.2 Transition Matrix and State Space Models.

    5.3 Canonical Realizations: Phase Space Variables.

    5.4 Canonical Realizations: Jordan Variables.

    5.5 Non-Linear Dynamical Systems.

    5.6 Analogic Computation for Linear and Non-Linear Dynamical

          Systems.

    5.7 Spatial Dynamics and the Poisson Equation.

    5.8 Temporal Dynamics, Causality and The Lorentz Oscillator.

    5.9 Spatio-Temporal Dynamics and Retarded Green's Functions.

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6. Special Theory of Relativity

    6.1 The Principle of General Covariance.

    6.2 Lorentz Transformations and Invariance.

    6.3 Minkowski Space and Euclidean Norms.

    6.4 Length Contraction, Time Dilations and Addition of

          Relativistic Velocities.

    6.5 Four Vector Formalism For Momentum.

    6.6 Relativistic Electrodynamics.

    6.7 Covariant Formulation.

    6.8 Relativistic Kinematics.

    6.9 Relativistic Dynamics, The Photon and the Compton Effect.

  

7. Classical Theory of Fields

    7.1 The Continuum Limit of a Chain on Non-interacting

          Oscillators.

    7.2 Stationary Principle and Euler-Lagrange Field Equation.

    7.3 Klein Gordon Equation and Sources of a Classical Field.

    7.4 Complex and Non-Linear Field Theories.

    7.5 Stationary vs Time Dependent Field Theories.

    7.6 Electromagnetic Field Theory and the Maxwell Stress

          Tensor.

    7.7 Electromagnetic Waves on a Disperse Medium.

    7.8 Quantization of a Classical Field and Mass of a Field.

    7.9 Putting it All Together: The Higgs Boson