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Theoretical Classical Mechanics: From Beginner to Expert

Name: Theoretical Classical Mechanics: From Beginner to Expert
Rating: 4.6 (368 reviews)

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

Created byDr. Börge Göbel

Last updated 1/2024

English

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

Course content

12 sections • 203 lectures • 23h 11m total length

Structure of this course4:35
About the following videos0:06
[Mathematical Basics] Derivatives12:59
[Mathematical Basics] Integrals6:51
[Mathematical Basics] Vectors19:51

Section intro0:37
Kinematics overview5:44
Uniform motion in one dimension3:33
Definition of velocity and acceleration11:47
Derivatives and integrals in kinematics10:00
Example: Derivatives and integrals in kinematics8:12
Motion with constant acceleration3:12
Superposition principle: Throw in multiple dimensions11:15
About exercises and quizzes0:49
Test your knowledge!
[Exercise] Acceleration in a roller coaster2:59
[Solution] Acceleration in a roller coaster5:34
Circular motion7:18
Uniform circular motion: Constant angular velocity5:28
Constant angular acceleration4:23
Difficult example: Pendulum, swing & carousel9:38
Simplified example: Harmonic oscillator8:29
Test your knowledge!
[Exercise] Roundabout, carousel, merry-go-round2:00
[Solution] Roundabout, carousel, merry-go-round5:22
Section outro0:43
Slides of this section0:05

Section intro1:19
Explanation: Please read before starting with this section!0:30
Differentiation: From derivatives in 1D to partial and directional derivatives12:45
Multidimensional derivatives: Nabla operator, gradient, curl and divergence26:54
Test your knowledge about the Nabla operator.
[Exercise] 3-dimensional derivatives0:07
[Solutions] 3-dimensional derivatives15:33
Integration: From 1D to multidimensional integrals17:49
Line integrals11:00
Alternative coordinate systems13:50
Integration in spherical coordinates14:48
Taylor expansion10:42
Section outro0:21
Slides of this section0:05

Section intro1:07
Mass, Inertia & Forces8:39
Newton's axioms4:26
Weight & Gravity14:41
Pulley7:49
Forces of an inclined plane4:54
Pendulum & Harmonic oscillator10:00
Friction forces11:15
Test your knowledge!
[Exercise] Forces: Inclined plane and friction2:50
[Solution] Forces: Inclined plane and friction7:47
Conservative forces & Potentials10:36
Work & Relation to potentials14:21
Work of pulleys5:14
Energy & Energy conservation14:46
Power5:47
Test your knowledge!
[Exercise] Energy: Spaceship4:46
[Solution] Energy: Spaceship20:24
Momentum & Momentum conservation5:15
Inelastic collisions9:43
Use momentum conservation to analyze inelastic collisions where deforming objects form a single body with a common velocity, illustrating energy conversion to heat and deformation during crashes.
Elastic collisions12:07
Test your knowledge!
[Exercise] Collision analysis3:25
[Solution] Collision analysis10:55
Determine the impact velocity from kinetic friction, then apply momentum conservation for a 45-degree inelastic collision to compute post-collision speeds and assess which driver exceeded the speed limit.
Section outro0:42
Slides of this section0:05

Section intro1:19
Centripetal force15:10
Centripetal versus centrifugal force2:38
Work of centripetal force1:54
Test your knowledge!
[Exercise] Roller coaster3:45
[Solution] Roller coaster19:56
Rotational energy5:32
Moment of inertia9:59
Moment of inertia: Stick6:30
Moment of inertia: Sphere11:50
Test your knowledge!
[Exercise] Rolling objects2:34
[Solution] Rolling objects10:46
Torque5:35
Levers4:17
Angular momentum & Angular momentum conservation10:27
Test your knowledge!
[Exercise] Torque & Angular momentum5:17
[Solution] Torque & Angular momentum11:36
Comparison: Translation versus rotation5:40
Spinning top: Rotation, precession & nutation15:16
Inertial versus accelerated frame of reference: Velocity11:16
Inertial versus accelerated frame of reference: Forces14:46
Coriolis force7:19
Motion of planets: Kepler’s 1st law17:07
Motion of planets: Kepler’s 2nd law6:00
Motion of planets: Kepler’s 3rd law5:58
Section outro1:00
Slides of this section0:05

Section intro1:46
Constraints8:09
D'Alembert's principle3:43
D’Alembert’s principle: Generalized coordinates & Example: Pendulum14:38
[Exercise] D'Alembert's principle: Inclined plane3:44
[Solution] D'Alembert's principle: Inclined plane5:31
Generalized forces3:39
Lagrange equation9:16
Euler-Lagrange equation (2nd kind)4:17
Euler-Lagrange equation: Harmonic oscillator3:44
[Exercise] Lagrangian mechanics: Pendulum & Kepler problem5:16
[Solution] Lagrangian mechanics: Pendulum4:53
[Solution] Lagrangian mechanics: Kepler problem9:24
Test your knowledge about Lagrangian mechanics
Lagrangian & Action3:49
Hamilton’s principle of stationary action9:28
Euler-Lagrange equation derived from Hamilton's principle6:42
Why is Hamilton’s principle true? - Example: Vertical throw7:17
Mathematical detour on action: Calculus of variations7:27
Euler-Lagrange equation (1st kind)2:04
Euler-Lagrange equation: Atwood's machine11:31
Noether theorem6:33
Noether theorem: Rotation invariance & Angular momentum4:33
Noether theorem: Time invariance & Hamiltonian6:32
Test your knowledge about Lagrangian mechanics
Section outro0:34
Slides of this section0:05

Section intro0:55
Hamiltonian3:18
Mathematical detail: Legendre transformation10:20
Hamilton’s equations of motion5:24
Phase space & Example: Harmonic oscillator6:53
[Exercise] Hamiltonian mechanics: Pendulum & Kepler problem4:12
[Solution] Hamiltonian mechanics: Pendulum5:32
[Exercise] Hamiltonian mechanics: Kepler problem9:22
Time evolution & Poisson bracket4:13
Hamilton-Jacobi equation & Alternative formulations of classical mechanics7:16
Test your knowledge about Hamiltonian mechanics!
Section outro0:34
Slides of this section0:05

Section intro1:28
Explanation: Please read before starting with this section!0:25
Complex numbers 1 - What are complex numbers?9:25
Complex numbers 2 - Addition, subtraction & Complex plane13:08
Complex numbers 3 - Multiplication & division9:37
Complex numbers 4 - Exponentials & Polar representation19:43
[Exercise] Complex numbers0:07
[Solution] Complex numbers14:10
Test your knowledge about complex numbers.
Matrices 1 - What is a matrix?10:16
Matrices 2 - Matrix addition & subtraction2:11
Matrices 3 - Matrix multiplication13:54
Matrices 4 - Calculating the determinant of a matrix14:44
Matrices 5 - Eigensystems: Eigenvalues & Eigenvectors of a matrix18:44
[Exercise] Matrices
[Solution] Matrices14:49

Section intro1:47
What are differential equations? Motivation & Example12:59
Classification of differential equations4:58
Classification of ordinary differential equations (ODE)6:26
Trivial case: Direct integration3:48
Example: Free fall2:37
Homogeneous linear differential equations & Exponential ansatz12:25
Example of exponential ansatz: Harmonic oscillator9:15
[Exercise] Homogeneous differential equations
[Solution] Homogeneous differential equations11:05
[Exercise] Damped harmonic oscillator
[Solution] Damped harmonic oscillator17:26
Inhomogeneous linear differential equations3:32
Example: Driven harmonic oscillator4:07
[Exercise] Inomogeneous differential equation
[Solution] Inomogeneous differential equation7:42
How to continue2:50
Section outro0:32
Slides of this section0:05

Section intro2:00
Learn to solve differential equations numerically with Python 3, starting with simple code and using SciPy Runge-Kutta to model earth–moon orbit, spaceship trajectories, and eigenvalues from coupled oscillators.
[How to] Download and install Python3 & Jupyter Notebook8:43
Learn to download and install Python via the Anaconda individual edition with a graphical installer, select Python 3.9, and launch Jupyter Notebook through the Anaconda Navigator.
Download the template file0:05
Background: Euler method15:13
Example 1: Radioactive decay solved with a function7:26
Example 2: Free fall - Higher-order differential equations12:43
Example 3: Pendulum as a harmonic oscillator7:10
Accurate solution of the pendulum5:05
Adding damping and driving forces6:55
Improvement: Use the SciPy function solve_ivp8:05
Example 4: Simulating a rolling ball - Two decoupled oscillators14:21
Download the final notebook0:01
Rolling ball in Wolfram Mathematica7:33
Download the template file0:05
3-body problem 1/5: Coupled differential equations for sun, earth & moon6:22
3-body problem 2/5: Coding the differential equations for sun, earth & moon12:23
3-body problem 3/5: Solving the differential equations for sun, earth & moon6:27
3-body problem 4/5: Analyzing the orbital motion of earth & moon13:13
3-body problem 5/5: Comment on inclination of the moon1:52
Spaceship 1/5: Coding & Solving the differential equations8:57
Spaceship 2/5: Changing starting velocity: Elliptical orbit around earth6:59
Spaceship 3/5: Simulating earth escape8:54
Spaceship 4/5: Simulating a moon encounter5:17
Spaceship 5/5: Brake maneuver to reach moon orbit15:47
Download the final notebook0:03

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 solution
Numerical solution with Python3

Why 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!

Who this course is for:

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)

Theoretical Classical Mechanics: From Beginner to Expert

What you'll learn

Explore related topics

Course content

[Level 1] Overview & Mathematical basics5 lectures • 44min

Kinematics20 lectures • 1hr 47min

[Level 2] More mathematical basics13 lectures • 2hr 6min

Dynamics: Newton's approach24 lectures • 3hr 12min

Dynamics of the circular motion27 lectures • 3hr 34min

[Level 3] Theoretical mechanics: Lagrange's approach25 lectures • 2hr 25min

Theoretical mechanics: Hamilton's approach & beyond12 lectures • 58min

[Level 4] Advanced mathematical basics15 lectures • 2hr 23min

Differential equations: Analytical methods and simple examples from physics19 lectures • 1hr 42min

Differential equations: Solving advanced physics problems numerically [Python]25 lectures • 3hr 2min

Requirements

Description

Who this course is for: