# Classical Physics: Journey Through The Laws Of The Universe

Category: Tutorials

- views: 89
- date: 13 July 2024
- posted by: AD-TEAM

**Classical Physics: Journey Through The Laws Of The Universe**

Last updated 3/2022

MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz

Language: English | Size: 24.73 GB | Duration: 20h 18m

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

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

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)

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 equationsLearn how to develop computational algorithms (the Finite Difference Approximation, Runge-Kutta Method, ODE45, etc) for solving complex equations of motion commonly found in physicsEstablish 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 moviesExperience 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 setsMASTER 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 OVERVIEWThrough 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

Overview

Section 1: The Language of Classical Physics/Lagrangian Mechanics: Calculus of Variations

Lecture 1 Introductory Concepts

Lecture 2 The Problem With Functionals

Lecture 3 A New Framework For Minimizing Functionals

Lecture 4 Deriving The Euler-Lagrange Equation

Lecture 5 Math Example #1: Determining The Shortest Distance Between Two Points

Lecture 6 Math Example #2: Calculus of Variations Recap & Geodesics Introduction

Lecture 7 Math Example #2 - Part 2: Geodesics & Deriving the Functional

Lecture 8 Math Example #2 - Part 3: Geodesics & Applying the Euler-Lagrange Equation

Lecture 9 Physics Example #3 - The Brachistochrone Problem

Lecture 10 Physics Example #3 - Part 2: Applying the Euler-Lagrange Equation

Lecture 11 Physics Example #3 - Part 3: Interpretting the Brachistochrone Problem's Results

Lecture 12 Quiz 1 - Solution

Section 2: Core Physics Principles: The Lagrangian and Lagrange's Equations

Lecture 13 Hamilton's Principle / The Principle of Least Action

Lecture 14 The Lagrangian and Lagrange's Equations

Lecture 15 Formal Proof - Part 1: Why The Lagrangian Satisfies Hamilton's Principle

Lecture 16 Formal Proof - Part 2: Why The Lagrangian Satisfies Hamilton's Principle

Lecture 17 Major Advantages of Lagrangian Mechanics & Coordinate Invariance

Lecture 18 A Particle in Polar Coordinates Demonstrating Coordinate Invariance

Lecture 19 Lagrangian Mechanics Vs. Newtonian Mechanics: A Quick Comparison

Lecture 20 A General Procedure for Lagrangian Mechanics Analyses

Lecture 21 Physics Example #1: Simple Harmonic Motion Revisited

Lecture 22 Example #1 - Part 2: Lagrangian Mechanics Approach with Simple Harmonic Motion

Lecture 23 Physics Example #2: The Atwood Machine Revisited

Lecture 24 Example # 2 - Part 2: Lagrangian Mechanics Approach with the Atwood Machine 1/2

Lecture 25 Example # 2 - Part 3: Lagrangian Mechanics Approach with the Atwood Machine 2/2

Lecture 26 Quiz 2 Solution Part 1 - General Methodology and Approach

Lecture 27 Quiz 2 Solution Part 2 - Calculating Potential Energy Method #1

Lecture 28 Quiz 2 Solution Part 3 - Calculating Potential Energy Method #2

Lecture 29 Quiz 2 Solution Part 4 - Determining the Lagrangian & Equation of Motion

Lecture 30 Quiz 2 Solution Part 5 - Solving the Differential Equation & Conclusion

Section 3: Application: Developing Physics Simulations of the Universe

Lecture 31 Linearizing Equations of Motion to Obtain Analytical Solutions

Lecture 32 Obtaining Analytical Solutions to Verify Physics Simulations

Lecture 33 The Finite Difference Method: Forward Difference Approximations

Lecture 34 The Finite Difference Method Part 2: Backward & Central Difference Approximation

Lecture 35 Applying the Finite Difference Method to Simple Harmonic Motion

Lecture 36 Simulating Simple Harmonic Motion: Introduction & Setting Up the Simulation

Lecture 37 Simulating SHM - Part 2: Programming the Algorithm & Verification

Lecture 38 Exploring Our First Physics Simulation: Balancing Accuracy & Computation Time

Section 4: Physics Example: The Double Pendulum and Chaotic Motion

Lecture 39 Problem Introduction and Deriving the System's Kinetic Energy

Lecture 40 Determining the Double Pendulum Lagrangian

Lecture 41 Deriving the Double Pendulum's First Lagrange Equation

Lecture 42 Deriving the Double Pendulum's Second Lagrange Equation

Lecture 43 Eliminating the System's Non-linearities to Find Analytical Solutions

Lecture 44 The Art of Solving Differential Equations: Setting up an Analytical Framework

Lecture 45 Determining the Natural Frequencies for Each Normal Mode

Lecture 46 Figuring Out How Each Normal Mode's Amplitudes Are Related

Lecture 47 Constructing the Full, Generalized Solution

Lecture 48 Using ODE45 to Computationally Solve Coupled, 2nd Order Differential Equations

Lecture 49 Programming the Simulation in MATLAB

Lecture 50 Assessing the Physics Simulation's Level of Accuracy

Lecture 51 Exploring the Simulation: Chaotic Motion, Phase Portraits, & Real-life Examples

Section 5: Physics Example: The Bead in a Hoop Problem

Lecture 52 Problem Introduction and Geometry Analysis

Lecture 53 Deriving the System's Total Kinetic Energy

Lecture 54 Determining The System's Lagrangian

Lecture 55 Utilizing the Lagrangian to Derive Each Case's Equation of Motion

Lecture 56 Case 1: Eliminating Non-linearities to Obtain an Analytical Solution

Lecture 57 Case 2: Why the Analytical Solution Consists of Transient & Stationary Solutions

Lecture 58 The Method of Undetermined Coefficients & Why it Needs to be Modified

Lecture 59 Case 2: Determining the Steady-State Solution

Lecture 60 Case 2: Using Superposition to Yield the Full Analytical Solution

Lecture 61 Physics Simulation: Developing Algorithms for Rendering the Geoemtry

Lecture 62 Case 1: Programming the Finite Difference Algorithm & Verifying the Solution

Lecture 63 Case 2: Modifying the Physics Simulation & Verifying Results

Lecture 64 Case 2: Exploring the Effects of Centripetal Acceleration & Gravity

Section 6: The Two-Rod Arm Physics Problem: Analyzing Rigid-Body Systems

Lecture 65 Problem Introduction & Geometry Analysis

Lecture 66 Determining Each Rigid Body's Center of Mass

Lecture 67 Determing Each Rigid Body's Center of Mass - Part 2

Lecture 68 The Difference Between Translational & Rotational Kinetic Energy

Lecture 69 The Center of Mass Trick

Lecture 70 Determining the System's Total Kinetic Energy: Fail-Safe Method

Lecture 71 Simplifying Our Kinetic Energy Derivation: A More Efficient Approach

Lecture 72 Deriving the Full System's Lagrangian

Lecture 73 Determining the Equation of Motion - Part 1: Finding the Puzzle Pieces

Lecture 74 Determining the Equation of Motion - Part 2: Putting the Pieces Together

Lecture 75 Eliminating Non-Linearities to Obtain an Analytical Solution

Lecture 76 Finding the Transient & Steady-State Solutions

Lecture 77 Constructing the Full Analytical Solution

Lecture 78 Exploring the

Lecture 79 Developing the ODE45 Computational Algorithm for Our Simulation

Lecture 80 Developing an Algorithm for Rendering the Geometry and Motion

Lecture 81 Implementing Collision Detection Algorithms into the Simulation

Lecture 82 Verifying the Computational Solution & Assessing its Accuracy

Lecture 83 Visually Experimenting with Each Case & Exploring Static Equilibrium Conditions

Lecture 84 Demonstrating How Non-Linearities Grow as the Simulation Runs & More Exploration

Lecture 85 Concluding Remarks

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**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

**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 equationsLearn how to develop computational algorithms (the Finite Difference Approximation, Runge-Kutta Method, ODE45, etc) for solving complex equations of motion commonly found in physicsEstablish 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 moviesExperience 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 setsMASTER 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 OVERVIEWThrough 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

Overview

Section 1: The Language of Classical Physics/Lagrangian Mechanics: Calculus of Variations

Lecture 1 Introductory Concepts

Lecture 2 The Problem With Functionals

Lecture 3 A New Framework For Minimizing Functionals

Lecture 4 Deriving The Euler-Lagrange Equation

Lecture 5 Math Example #1: Determining The Shortest Distance Between Two Points

Lecture 6 Math Example #2: Calculus of Variations Recap & Geodesics Introduction

Lecture 7 Math Example #2 - Part 2: Geodesics & Deriving the Functional

Lecture 8 Math Example #2 - Part 3: Geodesics & Applying the Euler-Lagrange Equation

Lecture 9 Physics Example #3 - The Brachistochrone Problem

Lecture 10 Physics Example #3 - Part 2: Applying the Euler-Lagrange Equation

Lecture 11 Physics Example #3 - Part 3: Interpretting the Brachistochrone Problem's Results

Lecture 12 Quiz 1 - Solution

Section 2: Core Physics Principles: The Lagrangian and Lagrange's Equations

Lecture 13 Hamilton's Principle / The Principle of Least Action

Lecture 14 The Lagrangian and Lagrange's Equations

Lecture 15 Formal Proof - Part 1: Why The Lagrangian Satisfies Hamilton's Principle

Lecture 16 Formal Proof - Part 2: Why The Lagrangian Satisfies Hamilton's Principle

Lecture 17 Major Advantages of Lagrangian Mechanics & Coordinate Invariance

Lecture 18 A Particle in Polar Coordinates Demonstrating Coordinate Invariance

Lecture 19 Lagrangian Mechanics Vs. Newtonian Mechanics: A Quick Comparison

Lecture 20 A General Procedure for Lagrangian Mechanics Analyses

Lecture 21 Physics Example #1: Simple Harmonic Motion Revisited

Lecture 22 Example #1 - Part 2: Lagrangian Mechanics Approach with Simple Harmonic Motion

Lecture 23 Physics Example #2: The Atwood Machine Revisited

Lecture 24 Example # 2 - Part 2: Lagrangian Mechanics Approach with the Atwood Machine 1/2

Lecture 25 Example # 2 - Part 3: Lagrangian Mechanics Approach with the Atwood Machine 2/2

Lecture 26 Quiz 2 Solution Part 1 - General Methodology and Approach

Lecture 27 Quiz 2 Solution Part 2 - Calculating Potential Energy Method #1

Lecture 28 Quiz 2 Solution Part 3 - Calculating Potential Energy Method #2

Lecture 29 Quiz 2 Solution Part 4 - Determining the Lagrangian & Equation of Motion

Lecture 30 Quiz 2 Solution Part 5 - Solving the Differential Equation & Conclusion

Section 3: Application: Developing Physics Simulations of the Universe

Lecture 31 Linearizing Equations of Motion to Obtain Analytical Solutions

Lecture 32 Obtaining Analytical Solutions to Verify Physics Simulations

Lecture 33 The Finite Difference Method: Forward Difference Approximations

Lecture 34 The Finite Difference Method Part 2: Backward & Central Difference Approximation

Lecture 35 Applying the Finite Difference Method to Simple Harmonic Motion

Lecture 36 Simulating Simple Harmonic Motion: Introduction & Setting Up the Simulation

Lecture 37 Simulating SHM - Part 2: Programming the Algorithm & Verification

Lecture 38 Exploring Our First Physics Simulation: Balancing Accuracy & Computation Time

Section 4: Physics Example: The Double Pendulum and Chaotic Motion

Lecture 39 Problem Introduction and Deriving the System's Kinetic Energy

Lecture 40 Determining the Double Pendulum Lagrangian

Lecture 41 Deriving the Double Pendulum's First Lagrange Equation

Lecture 42 Deriving the Double Pendulum's Second Lagrange Equation

Lecture 43 Eliminating the System's Non-linearities to Find Analytical Solutions

Lecture 44 The Art of Solving Differential Equations: Setting up an Analytical Framework

Lecture 45 Determining the Natural Frequencies for Each Normal Mode

Lecture 46 Figuring Out How Each Normal Mode's Amplitudes Are Related

Lecture 47 Constructing the Full, Generalized Solution

Lecture 48 Using ODE45 to Computationally Solve Coupled, 2nd Order Differential Equations

Lecture 49 Programming the Simulation in MATLAB

Lecture 50 Assessing the Physics Simulation's Level of Accuracy

Lecture 51 Exploring the Simulation: Chaotic Motion, Phase Portraits, & Real-life Examples

Section 5: Physics Example: The Bead in a Hoop Problem

Lecture 52 Problem Introduction and Geometry Analysis

Lecture 53 Deriving the System's Total Kinetic Energy

Lecture 54 Determining The System's Lagrangian

Lecture 55 Utilizing the Lagrangian to Derive Each Case's Equation of Motion

Lecture 56 Case 1: Eliminating Non-linearities to Obtain an Analytical Solution

Lecture 57 Case 2: Why the Analytical Solution Consists of Transient & Stationary Solutions

Lecture 58 The Method of Undetermined Coefficients & Why it Needs to be Modified

Lecture 59 Case 2: Determining the Steady-State Solution

Lecture 60 Case 2: Using Superposition to Yield the Full Analytical Solution

Lecture 61 Physics Simulation: Developing Algorithms for Rendering the Geoemtry

Lecture 62 Case 1: Programming the Finite Difference Algorithm & Verifying the Solution

Lecture 63 Case 2: Modifying the Physics Simulation & Verifying Results

Lecture 64 Case 2: Exploring the Effects of Centripetal Acceleration & Gravity

Section 6: The Two-Rod Arm Physics Problem: Analyzing Rigid-Body Systems

Lecture 65 Problem Introduction & Geometry Analysis

Lecture 66 Determining Each Rigid Body's Center of Mass

Lecture 67 Determing Each Rigid Body's Center of Mass - Part 2

Lecture 68 The Difference Between Translational & Rotational Kinetic Energy

Lecture 69 The Center of Mass Trick

Lecture 70 Determining the System's Total Kinetic Energy: Fail-Safe Method

Lecture 71 Simplifying Our Kinetic Energy Derivation: A More Efficient Approach

Lecture 72 Deriving the Full System's Lagrangian

Lecture 73 Determining the Equation of Motion - Part 1: Finding the Puzzle Pieces

Lecture 74 Determining the Equation of Motion - Part 2: Putting the Pieces Together

Lecture 75 Eliminating Non-Linearities to Obtain an Analytical Solution

Lecture 76 Finding the Transient & Steady-State Solutions

Lecture 77 Constructing the Full Analytical Solution

Lecture 78 Exploring the

**Requirements**for Static EquilibriumLecture 79 Developing the ODE45 Computational Algorithm for Our Simulation

Lecture 80 Developing an Algorithm for Rendering the Geometry and Motion

Lecture 81 Implementing Collision Detection Algorithms into the Simulation

Lecture 82 Verifying the Computational Solution & Assessing its Accuracy

Lecture 83 Visually Experimenting with Each Case & Exploring Static Equilibrium Conditions

Lecture 84 Demonstrating How Non-Linearities Grow as the Simulation Runs & More Exploration

Lecture 85 Concluding Remarks

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