Theoretical Classical Mechanics: From Beginner To Expert

Theory & Examples: Kinematics, Dynamics, Differential Equations (Including Maths & Python3), Lagrangian & Hamiltonian

What you'll learn

Kinematics: Position, velocity & acceleration are related by derivatives and integrals

Dynamics: Forces, potentials, work, energy and momentum allow for a phenomenological description based on Newton's laws

Circular motion: Angular velocity, acceleration, centripetal and centrifugal forces, torque and angular momentum

Theoretical physics: Lagrangian and Hamiltonian approaches based on d'Alembert's principle and Hamilton's principle

Solving differential equations analytically

Programming & Numerical simulations: Solving differential equations in Python3

Mathematical methods: Derivatives, integrals, Taylor expansions, coordinate systems, complex numbers & matrices

Conservation laws based on Noether theorem and symmetries

Nice examples like: Spinning top, Kepler's laws of planetary motion, coupled, damped and driven oscillators, pulleys, levers, Coriolis force and many more

Requirements

Basic mathematics

Recommended: What are derivatives, integrals and vectors?

Description

This course is for everyone who wants to learn about classical mechanic: Beginners to experts!A bit of college mathematics (basic derivatives, integrals & vectors) is all you need to know!Classical mechanics is the foundation of all disciplines in physics. It is typically at the very beginning of the university-level physics education. But that does not mean the classical physics is always super easy or even boring. Things become extremely complicated quickly and can lead to unexpected solutions. We can describe classical mechanics on different levels. I can guarantee that you will learn a lot no matter what your current skill level is.You are kindly invited to join this carefully prepared course in which we derive the following concepts from scratch. I will present examples and have prepared quizzes and exercises for all topics.[Level 1] Beginner: Kinematics (3 hours)Overview & mathematical basics (derivatives, integrals, vectors)Kinematics: Position, velocity & acceleration[Level 2] Intermediate: Dynamics (9 hours)Mathematics (Coordinate systems, multidimensional derivatives & integrals)Dynamics: Forces & related quantities (work, potentials, energy, momentum)Dynamics of the circular motion (torque, angular momentum)[Level 3] Advanced: Theoretical mechanics (3.5 hours)Lagrange’s approach (Constraints, action, Noether's theorem)Hamilton’s approach & beyond (Legendre transformation, Hamilton's equations of motion)[Level 4] Expert: Differential equations (8 hours)Advanced mathematics (Complex numbers & matrices)Differential equations: Analytical solutionNumerical solution with Python3Why me?My name is Börge Göbel  and I am a postdoc working as a scientist in theoretical physics. Therefore, I use theoretical classical mechanics very often but I have not forgotten the time when I learned about it and still remember the problems that I and other students had. I have refined my advisor skills as a tutor of Bachelor, Master and PhD students in theoretical physics and have other successful courses here on Udemy.I hope you are excited and I kindly welcome you to our course!

Overview

Section 1: [Level 1] Overview & Mathematical basics

Lecture 1 Structure of this course

Lecture 2 About the following videos

Lecture 3 [Mathematical Basics] Derivatives

Lecture 4 [Mathematical Basics] Integrals

Lecture 5 [Mathematical Basics] Vectors

Section 2: Kinematics

Lecture 6 Section intro

Lecture 7 Kinematics overview

Lecture 8 Uniform motion in one dimension

Lecture 9 Definition of velocity and acceleration

Lecture 10 Derivatives and integrals in kinematics

Lecture 11 Example: Derivatives and integrals in kinematics

Lecture 12 Motion with constant acceleration

Lecture 13 Superposition principle: Throw in multiple dimensions

Lecture 14 About exercises and quizzes

Lecture 15 [Exercise] Acceleration in a roller coaster

Lecture 16 [Solution] Acceleration in a roller coaster

Lecture 17 Circular motion

Lecture 18 Uniform circular motion: Constant angular velocity

Lecture 19 Constant angular acceleration

Lecture 20 Difficult example: Pendulum, swing & carousel

Lecture 21 Simplified example: Harmonic oscillator

Lecture 22 [Exercise] Roundabout, carousel, merry-go-round

Lecture 23 [Solution] Roundabout, carousel, merry-go-round

Lecture 24 Section outro

Lecture 25 Slides of this section

Section 3: [Level 2] More mathematical basics

Lecture 26 Section intro

Lecture 27 Explanation: Please read before starting with this section!

Lecture 28 Differentiation: From derivatives in 1D to partial and directional derivatives

Lecture 29 Multidimensional derivatives: Nabla operator, gradient, curl and divergence

Lecture 30 [Exercise] 3-dimensional derivatives

Lecture 31 [Solutions] 3-dimensional derivatives

Lecture 32 Integration: From 1D to multidimensional integrals

Lecture 33 Line integrals

Lecture 34 Alternative coordinate systems

Lecture 35 Integration in spherical coordinates

Lecture 36 Taylor expansion

Lecture 37 Section outro

Section 4: Dynamics: Newton's approach

Lecture 38 Section intro

Lecture 39 Mass, Inertia & Forces

Lecture 40 Newton's axioms

Lecture 41 Weight & Gravity

Lecture 42 Pulley

Lecture 43 Forces of an inclined plane

Lecture 44 Pendulum & Harmonic oscillator

Lecture 45 Friction forces

Lecture 46 [Exercise] Forces: Inclined plane and friction

Lecture 47 [Solution] Forces: Inclined plane and friction

Lecture 48 Conservative forces & Potentials

Lecture 49 Work & Relation to potentials

Lecture 50 Work of pulleys

Lecture 51 Energy & Energy conservation

Lecture 52 Power

Lecture 53 [Exercise] Energy: Spaceship

Lecture 54 [Solution] Energy: Spaceship

Lecture 55 Momentum & Momentum conservation

Lecture 56 Inelastic collisions

Lecture 57 Elastic collisions

Lecture 58 [Exercise] Collision analysis

Lecture 59 [Solution] Collision analysis

Lecture 60 Section outro

Lecture 61 Slides of this section

Section 5: Dynamics of the circular motion

Lecture 62 Section intro

Lecture 63 Centripetal force

Lecture 64 Centripetal versus centrifugal force

Lecture 65 Work of centripetal force

Lecture 66 [Exercise] Roller coaster

Lecture 67 [Solution] Roller coaster

Lecture 68 Rotational energy

Lecture 69 Moment of inertia

Lecture 70 Moment of inertia: Stick

Lecture 71 Moment of inertia: Sphere

Lecture 72 [Exercise] Rolling objects

Lecture 73 [Solution] Rolling objects

Lecture 74 Torque

Lecture 75 Levers

Lecture 76 Angular momentum & Angular momentum conservation

Lecture 77 [Exercise] Torque & Angular momentum

Lecture 78 [Solution] Torque & Angular momentum

Lecture 79 Comparison: Translation versus rotation

Lecture 80 Spinning top: Rotation, precession & nutation

Lecture 81 Inertial versus accelerated frame of reference: Velocity

Lecture 82 Inertial versus accelerated frame of reference: Forces

Lecture 83 Coriolis force

Lecture 84 Motion of planets: Kepler’s 1st law

Lecture 85 Motion of planets: Kepler’s 2nd law

Lecture 86 Motion of planets: Kepler’s 3rd law

Lecture 87 Section outro

Lecture 88 Slides of this section

Section 6: [Level 3] Theoretical mechanics: Lagrange's approach

Lecture 89 Section intro

Lecture 90 Constraints

Lecture 91 D'Alembert's principle

Lecture 92 D’Alembert’s principle: Generalized coordinates & Example: Pendulum

Lecture 93 [Exercise] D'Alembert's principle: Inclined plane

Lecture 94 [Solution] D'Alembert's principle: Inclined plane

Lecture 95 Generalized forces

Lecture 96 Lagrange equation

Lecture 97 Euler-Lagrange equation (2nd kind)

Lecture 98 Euler-Lagrange equation: Harmonic oscillator

Lecture 99 [Exercise] Lagrangian mechanics: Pendulum & Kepler problem

Lecture 100 [Solution] Lagrangian mechanics: Pendulum

Lecture 101 [Solution] Lagrangian mechanics: Kepler problem

Lecture 102 Lagrangian & Action

Lecture 103 Hamilton’s principle of stationary action

Lecture 104 Euler-Lagrange equation derived from Hamilton's principle

Lecture 105 Why is Hamilton’s principle true? - Example: Vertical throw

Lecture 106 Mathematical detour on action: Calculus of variations

Lecture 107 Euler-Lagrange equation (1st kind)

Lecture 108 Euler-Lagrange equation: Atwood's machine

Lecture 109 Noether theorem

Lecture 110 Noether theorem: Rotation invariance & Angular momentum

Lecture 111 Noether theorem: Time invariance & Hamiltonian

Lecture 112 Section outro

Lecture 113 Slides of this section

Section 7: Theoretical mechanics: Hamilton's approach & beyond

Lecture 114 Section intro

Lecture 115 Hamiltonian

Lecture 116 Mathematical detail: Legendre transformation

Lecture 117 Hamilton’s equations of motion

Lecture 118 Phase space & Example: Harmonic oscillator

Lecture 119 [Exercise] Hamiltonian mechanics: Pendulum & Kepler problem

Lecture 120 [Solution] Hamiltonian mechanics: Pendulum

Lecture 121 [Exercise] Hamiltonian mechanics: Kepler problem

Lecture 122 Time evolution & Poisson bracket

Lecture 123 Hamilton-Jacobi equation & Alternative formulations of classical mechanics

Lecture 124 Section outro

Lecture 125 Slides of this section

Section 8: [Level 4] Advanced mathematical basics

Lecture 126 Section intro

Lecture 127 Explanation: Please read before starting with this section!

Lecture 128 Complex numbers 1 - What are complex numbers?

Lecture 129 Complex numbers 2 - Addition, subtraction & Complex plane

Lecture 130 Complex numbers 3 - Multiplication & division

Lecture 131 Complex numbers 4 - Exponentials & Polar representation

Lecture 132 [Exercise] Complex numbers

Lecture 133 [Solution] Complex numbers

Lecture 134 Matrices 1 - What is a matrix?

Lecture 135 Matrices 2 - Matrix addition & subtraction

Lecture 136 Matrices 3 - Matrix multiplication

Lecture 137 Matrices 4 - Calculating the determinant of a matrix

Lecture 138 Matrices 5 - Eigensystems: Eigenvalues & Eigenvectors of a matrix

Lecture 139 [Exercise] Matrices

Lecture 140 [Solution] Matrices

Section 9: Differential equations: Analytical methods and simple examples from physics

Lecture 141 Section intro

Lecture 142 What are differential equations? Motivation & Example

Lecture 143 Classification of differential equations

Lecture 144 Classification of ordinary differential equations (ODE)

Lecture 145 Trivial case: Direct integration

Lecture 146 Example: Free fall

Lecture 147 Homogeneous linear differential equations & Exponential ansatz

Lecture 148 Example of exponential ansatz: Harmonic oscillator

Lecture 149 [Exercise] Homogeneous differential equations

Lecture 150 [Solution] Homogeneous differential equations

Lecture 151 [Exercise] Damped harmonic oscillator

Lecture 152 [Solution] Damped harmonic oscillator

Lecture 153 Inhomogeneous linear differential equations

Lecture 154 Example: Driven harmonic oscillator

Lecture 155 [Exercise] Inomogeneous differential equation

Lecture 156 [Solution] Inomogeneous differential equation

Lecture 157 How to continue

Lecture 158 Section outro

Lecture 159 Slides of this section

Section 10: Differential equations: Solving advanced physics problems numerically [Python]

Lecture 160 Section intro

Lecture 161 [How to] Download and install Python3 & Jupyter Notebook

Lecture 162 Download the template file

Lecture 163 Background: Euler method

Lecture 164 Example 1: Radioactive decay solved with a function

Lecture 165 Example 2: Free fall - Higher-order differential equations

Lecture 166 Example 3: Pendulum as a harmonic oscillator

Lecture 167 Accurate solution of the pendulum

Lecture 168 Adding damping and driving forces

Lecture 169 Improvement: Use the SciPy function solve_ivp

Lecture 170 Example 4: Simulating a rolling ball - Two decoupled oscillators

Lecture 171 Download the final notebook

Lecture 172 Rolling ball in Wolfram Mathematica

Lecture 173 Download the template file

Lecture 174 3-body problem 1/5: Coupled differential equations for sun, earth & moon

Lecture 175 3-body problem 2/5: Coding the differential equations for sun, earth & moon

Lecture 176 3-body problem 3/5: Solving the differential equations for sun, earth & moon

Lecture 177 3-body problem 4/5: Analyzing the orbital motion of earth & moon

Lecture 178 3-body problem 5/5: Comment on inclination of the moon

Lecture 179 Spaceship 1/5: Coding & Solving the differential equations

Lecture 180 Spaceship 2/5: Changing starting velocity: Elliptical orbit around earth

Lecture 181 Spaceship 3/5: Simulating earth escape

Lecture 182 Spaceship 4/5: Simulating a moon encounter

Lecture 183 Spaceship 5/5: Brake maneuver to reach moon orbit

Lecture 184 Download the final notebook

Section 11: Coupled oscillators: Differential equation, Eigensystem & Fourier analysis

Lecture 185 Section intro

Lecture 186 Download the template file

Lecture 187 Three coupled oscillators: Equations of motion

Lecture 188 Numerical solution of the coupled differential equations

Lecture 189 Analytical solution: Determining the Eigensystem

Lecture 190 Recovering the eigenfrequencies by Fourier transform

Lecture 191 Fitting the numerical solution with harmonic functions

Lecture 192 Download the final notebook

Lecture 193 Section outro

Lecture 194 THANK YOU & GOODBYE!

Section 12: [Outlook] Chaos

Lecture 195 Section intro

Lecture 196 Download the template file

Lecture 197 Lorenz systems - Explanation of the differential equation

Lecture 198 Solving the Lorenz differential equation for the chaotic case

Lecture 199 Solving the Lorenz differential equation for the non-chaotic case

Lecture 200 Download the final notebook

Lecture 201 Section outro

All skill levels: From beginners to experts,[Level 1] Beginner: You know about derivatives and integrals and want to know how they are related to classical mechanics (kinematics),[Level 2] Intermediate: Your want to learn about forces and how they are related to work, potentials, energy and momenta (Dynamics),[Level 3] Advanced: You know about kinematics and dynamics and want to derive everything based on fundamental laws and principles (Theoretical physics approach),[Level 4] Expert: You want to know how to solve the equations of motion analytically and numerically (Differential equations)