![]() ![]() ![]() Selecting which variable to use as the independent variable does not change the physics of a problem, but some selections can simplify the mathematics for obtaining an analytic solution. 5.4: Selection of the Independent Variable A wide selection of variables can be chosen as the independent variable for variational calculus.The Brachistochrone problem stimulated the development of the calculus of variations by John Bernoulli and Euler. 5.3: Applications of Euler’s Equation The Brachistochrone problem involves finding the path having the minimum transit time between two points.Variational calculus, developed for classical mechanics, now has become an essential approach to many other disciplines in science, engineering, economics, and medicine. 5.2: Euler’s Differential Equation The calculus of variations, presented here, underlies the powerful variational approaches that were developed for classical mechanics.He solved the brachistochrone problem which involves finding the path for which the transit time between two points is the shortest. ![]() 5.1: Introduction to the Calculus of Variations During the 18th century, Bernoulli, who was a student of Leibniz, developed the field of variational calculus which underlies the integral variational approach to mechanics. ![]() Thus, if you have trouble the first time seeing this, don't fret, others had trouble too the first time.\) On seeing it for the first time, it is a bit of a jump from ordinary calculus but after a couple of reads and scratching out the math on paper, I think it takes hold. Its Amazon URL is: Īnd, at the Amazon site you can delve into the table of contents and I believe a portion of the first chapter.įrom my own experience and that of others who have delved into this topic for the first time, especially early in the education process, is in understanding the Calculus of Variations used to derive the Euler-Lagrange equations. This book is "A Student's Guide to Lagrangians and Hamiltonians" by Patrick Hamill. It is presented at a level that anyone with Calculus (multi-variable at least to the degree of partial derivatives) can read and understand. However, there is a more complete book with more examples and even problems to solve that is specific to Lagrangians and Hamiltonians. I agree with other answers in that Susskind's book that parallels the lectures, "The Theoretical Minimum" is a good read. (Having said which, I'm not very familiar with Susskind's book.) These serve a dual purpose: they let you test your newly-gained skills to see how it works out in practice and, in doing so, they let you see how and why the new formulations are better or cleaner (or not). On the physics side, it will be helpful to have a small but well-refined workhorse set of physical systems on whose Newtonian mechanics you've worked with thoroughly - think harmonic oscillator, pendulums in 2D and 3D, Keplerian motion, and so on. The only really new tool you will need is the calculus of variations this is usually developed enough in analytical mechanics textbooks that you'll learn enough of it from there to keep you going, but it wouldn't hurt to have a read on it beforehand or parallel to the mechanics. On the mathematical side, you will likely need to fluent enough with the calculus of several real variables as well as comfortable with the associated geometrical manipulations. As such, you have two main types of prerequisites: Lagrangian and Hamiltonian mechanics are about taking a good look at the foundations of classical mechanics, and reformulating them in ways which are cleaner and provide nice insights, but which are still strictly equivalent to Newtonian mechanics. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |