"Euler-Lagrange Differential Equation." From MathWorld-A Wolfram Web Resource. Referenced on Wolfram|Alpha Euler-Lagrange Differential Integral and the Euler Equations." §3.1 in Methods Variational Principles of Mechanics, 4th ed. The calculus of variations has a wide range of applications in physics, engineering, applied and pure mathematics, and is intimately connected to partial dierential equations. Reading, MA: Addison-Wesley, p. 44, 1980. The calculus of variations is a eld of mathematics concerned with minimizing (or maximizing) functionals (that is, real-valued functions whose inputs are functions). Mathematical Methods for Physicists, 3rd ed. A lot of popular ML/statistics techniques have the word "variational", which they get because they are somehow related to variational calculus.References Arfken, G. Variational Problems, Vietnamese-German University.Īs you have probably guessed, this is the primary reason I'm interested in this area of mathematics. "Advanced Variational Methods In Mechanics", Chapter 1: Variational Calculus Overview, University of Colorado at Boulder What is the practical difference between a differential and a derivative?, Arturo Magidin, Math.Stack Exchange. Previous Posts: Lagrange Multipliers, Max Entropy Distributionsĭirectional Derivatives, Paul Dawkins, Paul's Online Math Notes. I hope this post helps all the non-mathematicians and non-physicists Isn't too much of a stretch (at least when you're not trying to prove thingsįormally!). Parts (multivariable calculus, Lagrange multipliers etc.) the actual topic (variational calculus) As with most things, once you know enough about the individual Type of problem when searching for "variational calculus"), it also has manyĪpplications in statistics and machine learning (you can expect a future post Not onlyĭoes it have a myriad of applications in physical domains (it's the most common Variational calculus is an important topic that we missed out on. Of a distribution with support along the real line, we should assume the distributionįor those of us who aren't math or physics majors ( * cough * computer engineers), So by the principle of maximum entropy, if we only know the mean and variance Let's take another look at the total differential in Equation 2 again, but re-write Let's see how we can intuitively build it up from the same multivariable Something analogous to a derivative unsurprisingly called a functional derivative. We also want to be able to find the extrema of functionals. Regular calculus, whose premier application is finding minima and maxima, Now it's finally time to do something useful with functionals! As with You can differentiate it to get \(\frac_b\)) Intuition and explanation, it's actually not too difficult, enjoy!įrom multivariable functions to functionalsĬonsider a regular scalar function \(F(y)\), it maps a single value \(y\) to a Get to an application that relates back to probability. Regular calculus with a bunch of examples along the way. Notes to give some intuition on how we arrive at variational calculus from The most intuitive explanation I've read. Going to use (at least for the first part) is heavily based upon Svetitsky's Want to find a density that optimizes an objective 1. In fact, not only is it an exceedingly powerful alternative approach to the intuitive Newtonian approach in. This variational approach is both elegant and beautiful, and has withstood the rigors of experimental confirmation. You can also imagine, it's also used in machine learning/statistics where you The calculus of variations provides the mathematics required to determine the path that minimizes the action integral. It's used extensively in physics problems such asįinding the minimum energy path a particle takes under certain conditions. This one deals with functions of functions and how to This post is going to describe a specialized type of calculus calledĪnalogous to the usual methods of calculus that we learn in university,
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