OpenCourser brand icon
Sign in
We may earn an affiliate commission when you visit our partners.
Course image
Udemy logo

Calculus of Variations

4.6 Filled star Filled star Filled star Filled star Half star
Based on 105 ratings
see reviews
Ross McGowan

This is not your average Udemy course. You will not see the calculus of variations taught as thoroughly and intuitively anywhere on the internet (I know because I looked for such a course and not having found one decided to create this course). You will not see this subject taught in this way anywhere else online. The most important point and value in this course is I want you to understand the subject , I want you to be able to say at the end of this course, '

Read more

This is not your average Udemy course. You will not see the calculus of variations taught as thoroughly and intuitively anywhere on the internet (I know because I looked for such a course and not having found one decided to create this course). You will not see this subject taught in this way anywhere else online. The most important point and value in this course is I want you to understand the subject , I want you to be able to say at the end of this course, '

If you want to understand the calculus of variations as opposed to just applying some random maths equations and wonder what is going on then this is the course for you. What matters is true comprehension. As such I have introduced this subject in a non standard way. I derive the basic building block of calculus of variations namely the Euler-Lagrange equation in the terms that Euler first derived it and leave the standard derivation to much later in the course. The course has many examples including some of the most famous but also some that you just won't see in any textbook. The derivations are thorough and explained line per line equation to equation. You just won't get this subject taught like this unless you pay to go to university (and even then you will probably get the standard formulaic version).

In this course you will get access to my original derivations , some of which are common derivations you will get in any book but some of the derivations you will not find in any books (I know I looked for them and in the end spent many hours deriving them myself). You can download a PDF copy of each lecture or a PDF of the entire lecture course.

This is It will be time well spent as you will learn lots of tips and mathematical tricks that you just won't find anywhere else. I recommend you recreate the derivations and think about them as you walk to work or have coffee at lunch. Don't get put off or dejected - remember

Good luck with the course.

Enroll now

What's inside

Learning objectives

  • A thorough grounding in the calculus of variations. covering the first variation
  • Simulations using a graphical calculator to help your understanding
  • Introduction to the subject using an intuitive geometric approach which you will not see anywhere else
  • Euler lagrange equation derived both geometrically and analytically
  • Derivation and simulation of double pendulum

Syllabus

This section covers the introduction to the calculus of variations

This is a quick introduction to the course. New course video.

Here are all the course slides. If this looks like the course for you then sign up and let's get started.

Read more

This is a motivation for the course. Please note that you can easily pick holes in the logic of this video , take it as a very rough introduction and note that much more thorough discussions will follow from it. New course video.

In this video we derive the potential energy of a hanging rope. New course video.

In this video we derive a general equation for a simple functional.New course video.

In this video we look at the bigger picture and create a map of our intended route. New course video.

In this video we derive the first part of the Euler Lagrange equation which looks at a change in position from the equilibrium point. New course video.

In this video we derive the second part of the Euler Lagrange equation which details a change in gradient. New course video.

In this video we answer the first question posed in this video series , 'what is the equation of a hanging rope' and then we prove that it is something called a catenary. New course video.

We use the calculus of variations to prove that the shortest distance between 2 points is a straight line. New course video.

In this video we derive the equation of motion of a simple pendulum. New course video.

This video shows the complete course book, in order to give you a flavour of what is to come. You can download this pdf in colour or black and white for easy printing. If you want to just get started then head onto the next video.

In this video we derive the equations of motion for a simple object falling under gravity. New course video.

In this video I ask what is the path of shortest time that a ball will take rolling down a track. New course video.

In this video we derive the equation of a cycloid. New course video.

In this video we revisit the shortest distance between 2 points and prove it in polar coordinates. New course video.

In this video we derive the geodesic on a cylinder. New course video.

In this video we derive the geodesic on a sphere and show that it is a great arc. New course video.

In this video we derive the equations of projectile motion. New course video.

In this video we derive the equation of a minimum surface of revolution. New course video.

In this video we show the inverse square law gives an ellipse. New course video.

We do the same as the last video but for a direct distance law. New course video.

In this video we derive the shape of a fluid in rotational motion.

In this video I derive the slope of shape of minimum resistance. New course video.

In this video we derive Snell's law. New course video.

In this video we derive the Beltrami Identity. New course video.

In this video we look at the rules for performing partial differentiation of a functional. New course video.

In this video I derive the Euler Lagrange equation in terms of a total derivative. New course video.

What happens when the Euler Lagrange gives you zero ? New course video.

In this video I derive the Euler Lagrange using the modern accepted analytic approach. This approach signalled the beginning 'proper' or the calculus of variations. New course video.

We look at the fundamental leema of the calculus of variations.

First part of the derivation.

Second part of derivation.

Third part of derivation.

Fourth part of derivation.

Fifth part of derivation

Double pendulum in matlab

Double pendulum in Vpython

A final farewell.

Go deeper into the Calculus of Variations with the second course covering the second variation.

Traffic lights

Read about what's good
what should give you pause
and possible dealbreakers
Explores the Euler-Lagrange equation geometrically and analytically, which is fundamental in classical mechanics and field theory
Provides original derivations, some of which are not found in textbooks, offering a deeper understanding of the subject
Includes simulations using a graphical calculator, which can help visualize and understand abstract concepts in calculus of variations
Covers advanced topics such as the derivation and simulation of a double pendulum, which is a complex problem in mechanics
Derives equations of motion for various physical systems, such as a simple pendulum and projectile motion, demonstrating practical applications
Requires familiarity with calculus and differential equations, which may necessitate prior coursework for some learners

Save this course

Create your own learning path. Save this course to your list so you can find it easily later.
Save

Reviews summary

Deep calculus of variations understanding

According to learners, this course offers an incredibly deep and intuitive understanding of Calculus of Variations, going far beyond typical online resources. Students particularly appreciate the thorough, line-by-line derivations and the many well-chosen examples that illustrate concepts effectively. The instructor's passion for the subject is often highlighted as a significant positive. While the unique geometric approach is seen as brilliant by some for building a solid foundation, others initially found it confusing compared to the standard method covered later. Reviewers consistently note that the course is quite challenging and demanding, requiring a strong prior mathematical background in calculus and differential equations, but find it highly rewarding for serious students.
Intuitive but potentially confusing.
"The geometric derivation of the Euler-Lagrange equation in the beginning is brilliant and helps build a solid foundation."
"The unique geometric approach is interesting, though I found the standard analytical derivation later in the course easier to follow personally."
"The geometric derivation was confusing for me initially, but the analytical one was clearer."
"Sometimes the unique derivation methods felt unnecessarily complicated, but the standard one was also covered."
Downloadable slides and book are useful.
"PDFs of lectures are useful."
"The supplemental materials (PDFs) are very helpful."
"The course book, in order to give you a flavour of what is to come. You can download this pdf in colour or black and white for easy printing."
Applications illustrate concepts well.
"The examples are well-chosen and the derivations are clear and thorough, often going line-by-line."
"There are many examples covering different areas like physics and geometry."
"The examples illustrating different applications are great. I feel I finally grasp the intuition behind the Euler-Lagrange equation."
"Many examples helped solidify concepts."
Steps are clearly explained.
"The examples are well-chosen and the derivations are clear and thorough, often going line-by-line."
"I really appreciated how he derives everything from first principles and doesn't skip steps. The examples illustrating different applications are great."
"It goes into derivations more than most online resources I've seen... PDFs of lectures are useful."
"The derivations are presented beautifully. The course is challenging but incredibly rewarding."
Provides profound grasp of concepts.
"This course provides an incredibly deep and intuitive understanding of the Calculus of Variations. The instructor clearly has a passion for the subject..."
"The focus on deep understanding rather than just formulas is exactly what I needed. The derivation examples are fantastic."
"I feel I finally grasp the intuition behind the Euler-Lagrange equation. It's a challenging subject, but this course makes it accessible..."
"Masterpiece! The depth and clarity are unmatched. The derivations are presented beautifully. The course is challenging but incredibly rewarding."
Demanding course needs prerequisites.
"It's quite demanding, definitely not a quick overview. Requires a strong math background."
"Definitely requires significant prior knowledge in advanced calculus and differential equations. Not for the faint of heart."
"Be prepared for a lot of detailed math. This is for serious students."
"Covered the topic well... Requires a strong mathematical background."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Calculus of Variations with these activities:
Review Classical Mechanics
Strengthen your understanding of the underlying physics principles that the calculus of variations builds upon.
Show steps
  • Review Lagrangian and Hamiltonian formulations of classical mechanics.
  • Practice solving problems involving energy conservation and constraints.
Read 'Calculus of Variations' by Gelfand and Fomin
Gain a deeper understanding of the mathematical foundations and applications of the calculus of variations.
View Calculus of Variations (Dover Books on... on Amazon
Show steps
  • Read the first few chapters covering the basic concepts and theorems.
  • Work through the examples and exercises to solidify your understanding.
Simulate a Brachistochrone Curve
Apply the calculus of variations to a classic problem and visualize the solution.
Show steps
  • Derive the equation for the brachistochrone curve using the Euler-Lagrange equation.
  • Write a program to simulate the motion of an object along the brachistochrone curve and compare it to other paths.
  • Visualize the results using a plotting library.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Create a Video Explaining the Euler-Lagrange Equation
Solidify your understanding by explaining the core concept of the course to others.
Show steps
  • Research and gather information about the Euler-Lagrange equation.
  • Write a script that explains the equation in a clear and concise manner.
  • Record a video explaining the equation, including examples and visualizations.
  • Edit the video and upload it to a video-sharing platform.
Solve Variational Problems
Reinforce your understanding by solving a variety of problems involving the calculus of variations.
Show steps
  • Find practice problems in textbooks or online resources.
  • Solve the problems using the Euler-Lagrange equation and other techniques learned in the course.
  • Check your answers and review the solutions to identify any mistakes.
Read 'Methods of Mathematical Physics' by Courant and Hilbert
Expand your knowledge of the mathematical tools used in the calculus of variations.
View Methods of Mathematical Physics, Volume 1 on Amazon
Show steps
  • Focus on the chapters related to variational methods and their applications.
  • Work through the examples and exercises to solidify your understanding.
Contribute to a Physics Simulation Project
Apply your knowledge of the calculus of variations to a real-world problem and collaborate with other developers.
Show steps
  • Find an open-source physics simulation project that uses the calculus of variations.
  • Identify a bug or feature that you can contribute to the project.
  • Implement the fix or feature and submit a pull request.

Career center

Learners who complete Calculus of Variations will develop knowledge and skills that may be useful to these careers:
Applied Mathematician
An applied mathematician uses mathematical theories and techniques to solve practical problems in various fields. This course may be useful for an applied mathematician because it delves into the calculus of variations, providing an understanding of optimization and problem-solving techniques. The course may be useful for an applied mathematician. Applied mathematicians may find the derivation of equations from first principles useful for solving problems in science, engineering, and finance.
See salaries and explore the career path for Applied Mathematician
Physicist
Physicists study the fundamental laws of nature and develop theories to explain physical phenomena. This course may be useful to a physicist. With its focus on the calculus of variations and its applications to classical mechanics, this course is directly relevant to the work of a physicist. The course derives the Euler-Lagrange equation and explores various physical systems. A physicist who takes this course may gain a deeper understanding of theoretical physics and mathematical modeling.
See salaries and explore the career path for Physicist
Mathematical Modeler
Mathematical modelers create and analyze mathematical representations of real-world systems to understand and predict their behavior. The Calculus of Variations course may be useful to a mathematical modeler because it teaches the calculus of variations, covering its applications to various physical systems. The course may find the simulations helpful. A mathematical modeler may apply these skills to model and optimize complex problems.
See salaries and explore the career path for Mathematical Modeler
Research Scientist
Research scientists design and conduct experiments, analyze data, and publish findings to advance knowledge in a particular field. This course helps a research scientist, as it explores advanced mathematical concepts and problem-solving techniques in the calculus of variations. This course may be helpful for someone wanting to be a research scientist. The course emphasizes understanding the subject matter and deriving equations from first principles, providing a deeper understanding of the underlying phenomena. This course's detailed derivations and exploration of various examples may be useful.
See salaries and explore the career path for Research Scientist
Control Systems Engineer
Control systems engineers design, develop, and test systems that regulate and control dynamic processes. This course may be useful for a control systems engineer. The course emphasizes understanding the principles of the calculus of variations and its applications to dynamic systems. The course can provide a foundation for designing advanced control algorithms and optimizing system performance. The course's practical examples may be useful for control systems engineers.
See salaries and explore the career path for Control Systems Engineer
Simulation Specialist
Simulation specialists create computer models of systems to analyze and predict their behavior under different conditions. For Simulation Specialists, this course is directly relevant, as it focuses on the calculus of variations and its application to various physical systems. The course emphasizes understanding the subject matter through graphical simulations. It may be beneficial for a simulation specialist. The coverage of the Euler-Lagrange equation, combined with simulations of systems like the double pendulum, can deepen the understanding of a simulation specialist.
See salaries and explore the career path for Simulation Specialist
Robotics Engineer
Robotics engineers design, build, and program robots and robotic systems often integrating mechanical, electrical, and computer engineering principles. This course may be useful for a robotics engineer. With its thorough exploration of calculus of variations and its applications to dynamic systems, this course provides valuable knowledge for optimizing robot movement and control. The course's coverage of the Euler-Lagrange equation and simulations can inform the design of sophisticated robotic systems. The course’s double pendulum simulation may be attractive to a robotics engineer.
See salaries and explore the career path for Robotics Engineer
Aerospace Engineer
Aerospace engineers design, test, and supervise the manufacturing of aircraft and spacecraft. For an aerospace engineer, this course may be helpful because it offers a unique approach to understanding calculus of variations, deriving the Euler-Lagrange equation, and exploring its applications. Aerospace engineering often involves optimizing designs for efficiency and stability, and in this realm, understanding the calculus of variations is paramount. The course's exploration of geodesics, projectile motion, and problems involving minimums may deepen the understanding of an aerospace engineer.
See salaries and explore the career path for Aerospace Engineer
Mechanical Engineer
Mechanical engineers design, develop, and test mechanical and thermal devices, including tools, engines, and machines. The innovative approach to understanding the calculus of variations offered by this course may be useful for any mechanical engineer. This course derives the Euler-Lagrange equation and explores various mechanical systems like pendulums and objects falling under gravity. A mechanical engineer may gain a deeper understanding of mechanical systems through this course, which covers topics such as deriving equations of motion and analyzing system behavior. The simulations may be of particular benefit to a mechanical engineer.
See salaries and explore the career path for Mechanical Engineer
Quantitative Analyst
Quantitative analysts, often working in the financial industry, develop and implement mathematical models for pricing derivatives, managing risk, and making trading decisions. This course may be useful for a quantitative analyst. With its emphasis on the calculus of variations, especially the Euler-Lagrange equation, this course can help provide a strong mathematical foundation for quantitative modeling. The principles taught in this course can inform the development of sophisticated models for risk management and derivative pricing.
See salaries and explore the career path for Quantitative Analyst
Data Scientist
Data scientists analyze large datasets to extract meaningful insights and develop predictive models. This course may be useful for a data scientist. The course emphasizes understanding the underlying principles of calculus of variations, deriving equations, and applying them to real world challenges. The course's focus on simulations and understanding complex systems may deepen the mathematical foundation of a data scientist, particularly in areas involving optimization and modeling. The data scientist may better understand data related to dynamics and control systems.
See salaries and explore the career path for Data Scientist
Financial Modeler
Financial modelers construct mathematical models to forecast financial outcomes, assess risk, and support investment decisions. This course may be useful to a financial modeler. The course emphasizes a strong conceptual understanding of calculus of variations and its applications in optimizing various scenarios. This course's approach to problem solving may be applicable to financial scenarios. The course may help a financial modeler build a foundation for creating sophisticated models of financial systems.
See salaries and explore the career path for Financial Modeler
Econometrician
Econometricians use statistical methods to analyze economic data and test economic theories. Econometricians work with complex mathematical models of economic systems. This course may be useful for an econometrician. With its focus on understanding the calculus of variations and deriving fundamental equations from first principles, this course helps to explore optimization problems. The skills to analyze economic data, test economic theories, and forecast economic trends may be obtained in the course.
See salaries and explore the career path for Econometrician
Statistician
Statisticians collect, analyze, and interpret numerical data to identify trends and relationships. This course may be useful for a Statistician. The course emphasizes understanding the underlying mathematical principles of calculus of variations, deriving equations, and applying them to solve optimization problems. This may help a statistician. The course may provide deeper insights into data analysis, modeling, and prediction. Also the examples may be useful.
See salaries and explore the career path for Statistician
Actuary
Actuaries analyze statistical data to estimate the probability and financial impact of future events. This course may be useful for an actuary. The course emphasizes understanding the underlying mathematical principles of calculus of variations, deriving equations, and applying them to solve optimization problems. This course may provide a solid foundation for understanding and modeling uncertainties related to the future.
See salaries and explore the career path for Actuary

Reading list

We've selected two books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Calculus of Variations.
Cover image
Calculus of Variations
Save
Classic and comprehensive treatment of the calculus of variations. It provides a rigorous mathematical foundation and covers a wide range of applications. It is particularly useful for understanding the theoretical underpinnings of the subject and for solving advanced problems. This book is commonly used as a textbook in university courses.
Calculus of Variations (Dover Books on Mathematics)
Paperback
$$
Calculus of Variations (Dover Books on Mathematics)
Kindle Edition
$
Cover image
Methods of Mathematical Physics, Volume 1
Save
Provides a comprehensive treatment of mathematical physics, including the calculus of variations. While it more advanced text, it offers a deeper understanding of the mathematical tools and techniques used in the field. It is more valuable as additional reading for those seeking a more rigorous treatment of the subject. This book is commonly used as a reference by researchers and advanced students.
Methods of Mathematical Physics, Volume 1
Paperback
$$$$

Share

Help others find this course page by sharing it with your friends and followers:
Facebook
LinkedIn
Reddit
X
Link
Email

Similar courses

Similar courses are unavailable at this time. Please try again later.
Course image
Rating
4.6 Filled star Filled star Filled star Filled star Half star
Effort
6 total hours
Via
Udemy
Instructor
Ross McGowan
Language
English
Traffic lights
Read about what's good
what should give you pause
and possible dealbreakers
Explores the Euler-Lagrange equation geometrically and analytically, which is fundamental in classical mechanics and field theory
Provides original derivations, some of which are not found in textbooks, offering a deeper understanding of the subject
Includes simulations using a graphical calculator, which can help visualize and understand abstract concepts in calculus of variations
Covers advanced topics such as the derivation and simulation of a double pendulum, which is a complex problem in mechanics
Derives equations of motion for various physical systems, such as a simple pendulum and projectile motion, demonstrating practical applications
Requires familiarity with calculus and differential equations, which may necessitate prior coursework for some learners
Share this
Share to help others discover this course.
Facebook LinkedIn X Reddit Link Email
Begin learning today
Enroll now to gain full access to Calculus of Variations.
Enroll now
Save for later
Add this course to your list. Find it anytime.
Save
Our mission

OpenCourser helps millions of learners each year. People visit us to learn workspace skills, ace their exams, and nurture their curiosity.

Our extensive catalog contains over 50,000 courses and twice as many books. Browse by search, by topic, or even by career interests. We'll match you to the right resources quickly.

Find this site helpful? Tell a friend about us.

Affiliate disclosure

We're supported by our community of learners. When you purchase or subscribe to courses and programs or purchase books, we may earn a commission from our partners.

Your purchases help us maintain our catalog and keep our servers humming without ads.

Thank you for supporting OpenCourser.

© 2016 - 2025 OpenCourser