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

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

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

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

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

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

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

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

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

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

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

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