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Classical Physics: Journey Through the Laws of the Universe

Peer beneath the surface of reality, understand how the universe works, and develop physics apps that make a difference!

Created byEdward Reilly

Last updated 3/2022

English

What you'll learn

Fluently speak and utilize the language of Lagrangian Mechanics: Calculus of Variations
Confidently apply Classical Physics / Lagrangian Mechanics concepts to analyze any system or phenomenon throughout the universe
Program algorithms for computationally solving any equation of motion
Develop physics simulations similar to those powering your favorite video games and animated movies
Establish foundational MATLAB programming skills essential in any STEM industry
Develop easy-to-understand techniques for linearizing and solving differential equations

Course content

6 sections • 85 lectures • 20h 18m total length

Introductory Concepts12:52
"Physics is the philosophy of life translated into the language of mathematics, and, in this case, that language is calculus of variations"
Welcome to the course and thanks so much for both your support and curiosity; it means the world to me, and you're about to embark on a transformative journey through the depths of the universe. In this lecture, we'll explore the following topics:
Lagrange's motives and original inspiration
The underlying philosophy of Lagrangian Mechanics
The Principle of Least Action
The Problem With Functionals11:24
In Calculus 101, you learned a precise method for minimizing any function y(x). This method involved finding the point at which there is no change in the function y(x) for infinitesimal changes in the INDEPENDENT variable x (ie the point at which the derivative y'(x) = 0). In this lesson, we're going to divulge the major issue associated with functionals, and you'll find that the same framework from Calculus 101 no longer applies. This is why we're going to need a new framework and language for minimizing functionals: Calculus of Variations

Major topics in this lesson:
A brief review of how to minimize any function y(x)
The definition of a functional and how it differs from functions like y(x)
Why principles from Calculus 101 can not be applied to functionals
Establishing the framework for an example: deriving the shortest distance between any 2 points
A New Framework For Minimizing Functionals8:42
Now that we understand the issues surrounding functionals, it's time to find an appropriate solution. In this lesson, we're going to do just that; we're going to develop a new framework just for minimizing functionals. To do so, the lesson will use principles from Calculus 101 as a guide and slightly alter them to work with functionals too. By the end of this lesson, you should understand the approach we'll take in minimizing functionals; this will establish the basic framework used to derive the Euler-Lagrange Equation (See Lecture 4).
Major topics in this lesson:
Why functionals need a new "independent" variable that directly varies the functional
Redefining functions to implement this "independent" variable
Laying out the framework that will be used to minimize functionals and derive the Euler-Lagrange Equation
Deriving The Euler-Lagrange Equation18:18
Using the framework established in lecture 3, familiar Calculus 101 principles will now be utilized to minimize a functional. Such a framework essentially allows us to differentiate the functional and set it equal to zero (in other words, it allows us to find extrema as we typically would). After doing so, we'll use integration by parts and conditions asserted in lecture 3 to simplify the resulting expression. In the end, this process will yield a second-order partial differential equation - known as the Euler-Lagrange Equation - which must be satisfied to obtain stationary paths.

Major topics in this lesson:
How to use Calculus 101 principles along with our new framework to minimize functionals
The importance of integration by parts for deriving critical results
Why the Euler-Lagrange Equation forms the basis of both Calculus of Variations and Lagrangian Mechanics
Math Example #1: Determining The Shortest Distance Between Two Points12:46
Now that we know how to minimize functionals, it's time to unleash the power of the Euler-Lagrange Equation and discover what it's capable of. In this lesson, we're going to utilize our new tools to prove that the shortest distance between two points is a straight line. In doing so, you'll learn both how to use the Euler-Lagrange Equation and obtain useful results. The main goal here is to learn how to speak the language of Lagrangian Mechancs: Calculus of Variations.

Major topics in this lesson:
How to use the Euler-Lagrange Equation to obtain meaningful results
How to work with and interpret results to better understand geometric problems
Why Calculus of Variations is the language of Lagrangian Mechanics
Math Example #2: Calculus of Variations Recap & Geodesics Introduction5:56
As mentioned in the previous lesson, the main goal for this section's remainder is to learn how to speak the language of Lagrangian Mechanics: Calculus of Variations. To do so, first we'll revisit the major topics in this course so far, such as the following: The Principle of Least Action, the dilemmas associated with functionals, and the Euler-Lagrange Equation. Then, I'll introduce you to our second Calculus of Variations example: determining geodesics for any curved surface. As we continue this example in the following video, the goal is to generate familiarity with these concepts and help you speak this new language fluently.
Major topics in this lesson:
A Recap of major principles discussed in this course: Principle of Least Actions, functionals, and the Euler-Lagrange Equation
What is a geodesic and how does it involve Calculus of Variations?
A plan for running through this example in the next 2 video lectures
Math Example #2 - Part 2: Geodesics & Deriving the Functional19:28
In the previous lesson, the main topic of this example was briefly introduced: how to determine geodesics for any curved surface. Now, it's time to delve into that example and divulge the process for determining geodesics. In this video, I'll show you how to derive the functional required to used the Euler-Lagrange Equation. This will be accomplished using two generalized methods - which are applicable for any curved surface. Then, the surface of a cylinder will be utilized to demonstrate these methods directly.
Major topics in this lesson:
Two generalized methods for determining functionals used to find geodesics
Approximating functions using Taylor series expansions
Deriving the distance functional for a cylindrical surface
Math Example #2 - Part 3: Geodesics & Applying the Euler-Lagrange Equation19:36
In this video, we're going to continue and wrap up the same example discussed in the previous two videos. Again, the main intent is to help you further understand the language of Lagrangian Mechanics: Calculus of Variations. With the underlying framework now derived for our example, it's time to employ the Euler-Lagrange equation and minimize the functional. In doing so, we're going to determine both the shortest path between points on a cylinder and a generalized approach for determining geodesics. While running through the video, make sure you pay close attention to the role of the problem's path and independent variables - as they dictate how the Euler-Lagrange equation will be applied.
Major topics in this lesson:
How to properly apply the Euler-Lagrange Equation
The role of both path and independent variables when minimizing functionals
Determining the geodesic for a cylindrical surface
Physics Example #3 - The Brachistochrone Problem21:29
Having run through two examples so far, you should be starting to understand and feel comfortable with this section's core, fundamental principles. Now, in this final example, we're going to further enhance that understanding and ensure that you can speak this new language - Calculus of Variations - fluently. The problem which will help us achieve this goal, which is formerly known as the brachistochrone problem, is one of historic significance. Solved by some of the greatest mathematicians/physicists of all time, this problem originally gave birth to Calculus of variations; it essentially seeks to find the path between two points which minimizes travel time. Running through this example will also give you more practice in determining the appropriate path and independent variables.

Major topics in this lesson:

The brachistochrone Problem - finding the path between two points resulting in the shortest travel time
Using basic physics principles to derive the problem's functional
The historic significance of the brachistochrone problem and how it gave birth to Calculus of Variations
Physics Example #3 - Part 2: Applying the Euler-Lagrange Equation18:41
In the previous lecture, we explored this section's final example (the brachistochrone problem) and utilized our physical intuition to derive a functional. Now, it's time to speak the language of Calculus of Variations and further enhance our understanding of it. In this video, we're going to apply the Euler-Lagrange Equation to the functional to find the path of shortest travel time. While doing so, we'll also take a look at how both the path and independent variables are chosen, and you'll find that, sometimes, this process is not as intuitive as you'd think. However, it's ABSOLUTELY important that you understand how to appropriately choose these variables before proceeding; so, pay attention and review older lectures if necessary :)! Please reach out to me if you still have trouble identifying these variables after watching the lecture.

Major topics in this lesson:

How to identify the appropriate path and independent variables
Applying the Euler-Lagrange Equation to a functional to find its stationary points
Trigonometric substitution in deriving the brachistochrone curve
Physics Example #3 - Part 3: Interpretting the Brachistochrone Problem's Results12:27
Now that we have a general parametric expression for the brachistochrone curve, we're ready to further define that expression and interpret it. That's exactly what we're going to do in this section's final video; we're going to wrap up the example and tie a bow on top. In the end, we'll find that this expression defines what is known as a cycloid, and its radius is governed by the parameter "a". Furthermore, we'll find that this curve also solves the isochrone/tautochrone problem - which is pretty intriguing. Thanks for making it to the end of this section, and I hope you had a ton of fun while learning some really beautiful stuff :)! In the next section, we're going to finally make use of your new language skills as the main principles of Lagrangian Mechanics (and also the basic laws of the universe!) are explored.
Major topics in this lesson:
Why the cycloid solves both the brachistochrone and Isochrone/Tautochrone problems
The importance of correctly identifying variables as either path or independent variables
A brief review of this section's most important ideas
Minimizing the Distance Between Two Points on a 3D Surface
Quiz 1 - Solution32:07

Hamilton's Principle / The Principle of Least Action8:50
In the previous section, we learned how to fluently speak a new, intriguing language: Calculus of Variations. This allowed us to minimize functionals and solve some tricky problems, but the unlimited power inherent in this new language was ultimately left untapped. In this new section, we're going to finally unleash that power to completely reformulate classical physics. It's time to talk about Lagrangian Mechanics using our new, trusty language. This will be accomplished by first exploring the disadvantages associated with Newtonian Mechanics - such as its reliance on free-body diagrams, vector-based system, and fictitious forces. Then, we'll see how Hamilton's Principle can be used to reformulate Newtonian Mechanics and resolve its main weaknesses.
Major topics in this lesson:
How to utilize Calculus of Variations to reformulate classical physics
The disadvantages associated with the Newtonian method
Hamilton's Principle and how it can be used to reformulate Newtonian Mechanics
The Lagrangian and Lagrange's Equations16:57
In this video, we're going to take our first, exciting step into the world of Lagrangian Mechanics - as we explore the concept of a Lagrangian. Defined as an expression of the system's kinetic and potential energies, the Lagrangian holds all the information one would need to fully describe any physical system. This is due to the fact that stationary curves of the Lagrangian satisfy Hamilton's principle; they guarantee the laws of the universe. In other words, they yield the same exact equations of motion one would obtain from Newtonian Mechanics. As a result, this allows us to reformulate classical physics using a more efficient and powerful framework - which was a MONUMENTAL accomplishment. To conclude the video, we're also going to run through a quick example (analyzing an unconstrained particle in 3 dimensions) to demonstrate these claims.

Major topics in this lesson:

The definition of the Lagrangian and how it fulfills Hamilton's principle
Generalized coordinates and their relation to Lagrange's Equations
The equivalence of Lagrange's Equations to the equation of motion and generalized forces / momenta
Formal Proof - Part 1: Why The Lagrangian Satisfies Hamilton's Principle15:58
Formal Proof - Part 2: Why The Lagrangian Satisfies Hamilton's Principle13:00
Major Advantages of Lagrangian Mechanics & Coordinate Invariance7:01
A Particle in Polar Coordinates Demonstrating Coordinate Invariance16:46
Lagrangian Mechanics Vs. Newtonian Mechanics: A Quick Comparison8:37
A General Procedure for Lagrangian Mechanics Analyses8:18
Physics Example #1: Simple Harmonic Motion Revisited13:43
Example #1 - Part 2: Lagrangian Mechanics Approach with Simple Harmonic Motion19:52
Physics Example #2: The Atwood Machine Revisited18:42
Example # 2 - Part 2: Lagrangian Mechanics Approach with the Atwood Machine 1/214:34
Example # 2 - Part 3: Lagrangian Mechanics Approach with the Atwood Machine 2/29:37
Rope Sliding Off of a Table
Quiz 2 Solution Part 1 - General Methodology and Approach10:17
Quiz 2 Solution Part 2 - Calculating Potential Energy Method #117:23
Quiz 2 Solution Part 3 - Calculating Potential Energy Method #215:26
Quiz 2 Solution Part 4 - Determining the Lagrangian & Equation of Motion10:37
Quiz 2 Solution Part 5 - Solving the Differential Equation & Conclusion15:20

Linearizing Equations of Motion to Obtain Analytical Solutions14:36
Obtaining Analytical Solutions to Verify Physics Simulations15:42
The Finite Difference Method: Forward Difference Approximations15:59
The Finite Difference Method Part 2: Backward & Central Difference Approximation15:55
Applying the Finite Difference Method to Simple Harmonic Motion14:21
Simulating Simple Harmonic Motion: Introduction & Setting Up the Simulation16:55
Simulating SHM - Part 2: Programming the Algorithm & Verification11:05
Exploring Our First Physics Simulation: Balancing Accuracy & Computation Time17:49
Developing Your First Physics Simulation

Problem Introduction and Deriving the System's Kinetic Energy20:12
Determining the Double Pendulum Lagrangian15:05
Deriving the Double Pendulum's First Lagrange Equation17:36
Deriving the Double Pendulum's Second Lagrange Equation15:42
Eliminating the System's Non-linearities to Find Analytical Solutions20:55
The Art of Solving Differential Equations: Setting up an Analytical Framework14:48
Determining the Natural Frequencies for Each Normal Mode18:24
Figuring Out How Each Normal Mode's Amplitudes Are Related17:05
Constructing the Full, Generalized Solution20:15
Using ODE45 to Computationally Solve Coupled, 2nd Order Differential Equations18:28
Programming the Simulation in MATLAB4:45
Assessing the Physics Simulation's Level of Accuracy12:02
Exploring the Simulation: Chaotic Motion, Phase Portraits, & Real-life Examples17:16
The Coupled Pendulum Physics Problem

Problem Introduction and Geometry Analysis16:17
Deriving the System's Total Kinetic Energy14:12
Determining The System's Lagrangian14:54
Utilizing the Lagrangian to Derive Each Case's Equation of Motion17:31
Case 1: Eliminating Non-linearities to Obtain an Analytical Solution16:57
Case 2: Why the Analytical Solution Consists of Transient & Stationary Solutions10:35
The Method of Undetermined Coefficients & Why it Needs to be Modified14:40
Case 2: Determining the Steady-State Solution13:06
Case 2: Using Superposition to Yield the Full Analytical Solution12:23
Physics Simulation: Developing Algorithms for Rendering the Geoemtry18:08
Case 1: Programming the Finite Difference Algorithm & Verifying the Solution19:55
Case 2: Modifying the Physics Simulation & Verifying Results15:13
Case 2: Exploring the Effects of Centripetal Acceleration & Gravity15:12
The Rotating Pendulum Physics Problem

Problem Introduction & Geometry Analysis19:16
Determining Each Rigid Body's Center of Mass20:12
Determing Each Rigid Body's Center of Mass - Part 27:10
The Difference Between Translational & Rotational Kinetic Energy16:31
The Center of Mass Trick5:49
Determining the System's Total Kinetic Energy: Fail-Safe Method12:51
Simplifying Our Kinetic Energy Derivation: A More Efficient Approach11:30
Deriving the Full System's Lagrangian20:12
Determining the Equation of Motion - Part 1: Finding the Puzzle Pieces14:01
Determining the Equation of Motion - Part 2: Putting the Pieces Together9:36
Eliminating Non-Linearities to Obtain an Analytical Solution11:01
Finding the Transient & Steady-State Solutions10:02
Constructing the Full Analytical Solution11:15
Exploring the Requirements for Static Equilibrium16:04
Developing the ODE45 Computational Algorithm for Our Simulation10:26
Developing an Algorithm for Rendering the Geometry and Motion6:34
Implementing Collision Detection Algorithms into the Simulation16:51
Verifying the Computational Solution & Assessing its Accuracy9:24
Visually Experimenting with Each Case & Exploring Static Equilibrium Conditions10:35
Demonstrating How Non-Linearities Grow as the Simulation Runs & More Exploration5:57
Concluding Remarks4:15
Rigid Body Physics Problem

Requirements

Any introduction to elementary math and physics concepts (such as Calc 101, Linear Algebra, and Newtonian Mechanics) would GREATLY supplement your journey throughout this course but is not essential.
If you wish to follow along with our MATLAB simulations, you should have a copy of the program. However, note that MATLAB is available as a free trial download through Mathworks (Contact me for extra details if needed)

Description

In this course, you're going to journey through the deepest depths of the universe, understand the underlying physics governing it, and develop practical skills and applications that set you apart from the crowd. Most of these skills are essential for success in any STEM industry (Physics, Engineering, Economics, Computer Science, etc), but we'll mostly be using them to develop the same physics simulations that power your favorite video games and animated movies!

EXPLORE THE WONDERS OF THE UNIVERSE WHILE DEVELOPING SKILLS ESSENTIAL FOR SUCCESS!

Learn to fluently speak the language of Lagrangian Mechanics / Classical Physics: Calculus of Variations (Apply the Euler-Lagrange equation, find stationary points to functionals, and extend your Calculus knowledge)
Master the basic tenets of Lagrangian Mechanics / Classical Physics (The Principle of Least Action, the Lagrangian, and Lagrange's Equations of Motion)
Apply physics principles to solve an assortment of classic examples (The Brachistochrone Problem, Simple Harmonic Motion, Chaotic Motion with Double Pendulums, etc)
Develop easy-to-understand techniques for linearizing and solving differential equations
Learn how to develop computational algorithms (the Finite Difference Approximation, Runge-Kutta Method, ODE45, etc) for solving complex equations of motion commonly found in physics
Establish foundational MATLAB programming skills which are essential in most STEM industries (Physics, Engineering, Research and Development, Computer Science, etc)
Develop real-world physics simulations similar to those powering your favorite video games and animated movies
Experience the beauty of physics in a fun, exciting atmosphere you'll never find in a classroom; I'll make you love and enjoy physics, while enhancing your skill sets

MASTER THE PHYSICS PRINCIPLES THAT MAKE MODERN SOCIETY A POSSIBILITY!

Developed back in the 18th century, the basic principles of Classical Physics - which strive to explain the universe's underlying behavior - are some of the oldest academic achievements to-date. However, their innate power and practicality have barely been tarnished over the years. In fact, they've never been more prominent and influential as they are today. For example, most of the essential products you use on a daily basis (such as your house, car, phone, computer, etc) were designed using these same basic principles. The world you now know - along with all of its beauty and thrills - is literally a direct product of classical physics; it would not be possible without these marvelous ideas.

As a result, it is no wonder that Classical Physics concepts are invaluable assets with extremely high demand. Whether you want to be an engineer, scientist, or financial analyst, these skills are not only essential for success in any STEM industry, but their intrinsic value will get you some of the highest salaries throughout the world. For example, in the Forbes article "15 Most Valuable College Majors", not only does physics appear in the list, but 12 out of the 15 listed majors require physics / programming as an essential skill!

In this course, you're going to learn these highly coveted physics concepts that make modern society a possibility. Furthermore, you'll learn how to employ this knowledge to develop practical applications that make a difference. The main application you'll focus on is developing the same simulations that power your favorite video games and animated movies! So, in the end, not only will you be a master of the universe's underlying principles, but you'll also learn practical skills - such as MATLAB programming - that set you apart from the competition.

COURSE CONTENT AND OVERVIEW

Through 79 lectures and 18.5 hours worth of HD-quality content, you're going to journey through some of the deepest depths of the universe - experiencing all the beauty it has to offer. By the end of this journey, you'll be able to confidently apply classical physics concepts to analyze ANY system or phenomenon throughout the universe - whether that be the orbital motion of our planets or the dynamics of automotive components. Each step of this immersive journey also concludes with a comprehensive quiz; so, you'll have a chance to practice these concepts first-hand, receive immediate feedback, and quickly become a master of the universe! Here's a brief overview of each component you'll explore as you journey through the course:

Component I: As you embark on your journey, you'll first learn to fluently speak the language of Classical Physics / Lagrangian Mechanics: Calculus of Variations. This mathematical framework involves finding stationary points to "functionals", using your new favorite equation: the Euler-Lagrange Equation. Not only will this mathematical language allow us to explore and understand Classical Physics principles, but it can also be used to derive catenary curves - which are extensively utilized throughout architecture and civil engineering.

Component II: Once you've learned to fluently speak this intriguing language, you'll then put it to use to peer beneath the surface of reality. In this portion of your journey, we're going to explore the basic tenets of Lagrangian Mechanics / Classical Physics, which govern all behavior throughout the universe. Some essential ideas you'll discover include the Principle of Least Action, Lagrangian, and Lagrange's Equations of Motion.

Component III: ACTIONABLE knowledge is the true source of power and influence though, right? That's why this journey was also designed to instantly apply your new skills and use them practically. Once you've mastered the basic principles of the universe, you'll first apply these beautiful theories to tackle an assortment of classic problems. Some examples we'll work through together include the Brachistochrone Problem, straightforward problems involving Simple Harmonic Motion (SHM), and more challenging problems such as chaotic motion with double pendulums. You'll also have a chance to work through your own examples/problems to test your comprehension and address any gaps in knowledge.

Component IV: At this point, you'll be a master of the universe; not only will you understand how it works and operates, but you'll also have experience applying that understanding to problems. So, in the final component of your journey, you'll get an opportunity to fully unleash your new skills' potential. You're going to develop your own practical applications: simulations similar to those powering your favorite video games and animated movies! First, we'll focus on computational algorithms (such as the FInite Difference Approximation, Runge-Kutta Method, and ODE45) for solving complex equations of motion. Then, you'll learn how to program them into MATLAB and bring your simulations to life. As you journey through this component of the course, you'll run through 5 full-fledged walkthroughs on developing real-world simulations - where we walk you through the entire process step-by-step and ensure full comprehension.

At the end of your journey, you will have peered beneath the surface of reality, understood how the universe works and operates, and applied your newfound physics knowledge to develop real-world applications. You'll be a master of the universe who knows how to utilize his/her understanding, and that alone will make you shine in the STEM industry or effortlessly excel in-class.

So, feel free to explore our comprehensive curriculum or preview videos, and I look forward to seeing and working with you inside the course. I can't wait to guide you along the same journey which personally made me love physics, engineering, and mathematics :)!

Here's what some of my YouTube subscribers had to say about the course's content:

"Well, you just blew my mind. I seriously can´t thank you enough. I'm learning this in Classical Mechanics and I was having some difficulty understanding it. I am totally recommending this to my classmates! Thanks once more, I will keep watching your next videos!" - lp

"This is what i can't help but say after finishing watching your explanation: 'Dude,you are awesome! You actually got me to do some calculus solving! It's hard to make someone who despises a subject into getting interested and curious about it. But you managed to pull it off on me!' " -Yassa Moin

"This is what I need - A HIGH ENERGY LECTURER who claps his hands to get my attention! :-) Too many old men droning on in the rest of youtube lectures. Something like this is engaging and keeps you awake :-)!" -Hugh Jones

"As a retiree,finally finding time in life to start learning beautiful maths from a young man like you is very gratifying! You are tearing it up kid! Bravo!" -Blue StarFractal

"Thank you very much for this lesson. The enthusiasm and feeling for a clear explanation is very impressive." -FA Videos

"What I liked best about the video is that you first spent time explaining the intuition behind converting the problem (from, say, cartesian space) to a more abstract space and finding the solution there. Explaining such philosophy greatly helps put things in context as it answers what question really we are trying to answer. Following the content becomes much easier then. If you are not doing that in other videos, please do so :)!

TBH I was a bit skeptical about following through the whole video because I'd like to think I understand problems best when there is a geometrical intuition behind it, and have run away from any kind of analytical math all my life (even though I understood geometrical intuition behind complex calculus operations I'd still fail terribly in exams because I have a problem stating the questions analytically). This video helped me change my perspective. I wish I had teachers like you in college!

Thank you for the wonderful lecture, man!" -nilspin

Who this course is for:

University students looking to effortlessly excel in physics & math classes and get that top mark
Die-hard scientists, engineers, or entrepreneurs looking to design the next breakthrough product that will change how we all live
Any curious individual looking to master the principles of physics, develop a rich foundation in mathematics, and use that knowledge practically - whether you want to design simulations or develop engineering masterpieces

Classical Physics: Journey Through the Laws of the Universe

What you'll learn

Explore related topics

Course content

The Language of Classical Physics/Lagrangian Mechanics: Calculus of Variations12 lectures • 3hr 14min

Core Physics Principles: The Lagrangian and Lagrange's Equations18 lectures • 4hr 1min

Application: Developing Physics Simulations of the Universe8 lectures • 2hr 2min

Physics Example: The Double Pendulum and Chaotic Motion13 lectures • 3hr 33min

Physics Example: The Bead in a Hoop Problem13 lectures • 3hr 19min

The Two-Rod Arm Physics Problem: Analyzing Rigid-Body Systems21 lectures • 4hr 10min

Requirements

Description

Who this course is for: