Typical Problem: Consider a definite integral that depends on an unknown function \(y(x)\), as well as its derivative \(y'(x)=\frac \right]. The mathematical techniques developed to solve this type of problem are collectively known as the calculus of variations. One example is finding the curve giving the shortest distance between two points - a straight line, of course, in Cartesian geometry (but can you prove it?) but less obvious if the two points lie on a curved surface (the problem of finding geodesics.) Many problems involve finding a function that maximizes or minimizes an integral expression. MATH0043 Handout: Fundamental lemma of the calculus of variations.The Euler-Lagrange Equation, or Euler’s Equation.I haven’t even begun to talk about more variables, higher order derivatives, or more functions but this is definitely a good introduction into this wonderful field of mathematics.MATH0043 §2: Calculus of Variations MATH0043 §2: Calculus of Variations Lagrangian mechanics is simply an application of calculus of variations and this alone should show why calculus of variations is so important. Notice any similarities between Euler’s equation and the Euler-Lagrange equation from The Lagrangian? Let me give you a hint. As a result, it seems that the bubbles would end up minimizing surface area by forming a revolved hyperbolic cosine or catenoid as shown below. Before getting into the integration, we simple make note of the fact that this is of the exact same form of the last integral so we know it has the same solution. The soap film will assume the shape of a catenoid. To calculate the surface area of any surface of revolution, we use the following integral. Find the shape of a soap film (i.e., minimal surface) which will fill two inverted conical funnels facing each other is known as Sinclair's soap film problem (Bliss 1925, p. Because it is a bubble, it would also try to minimize surface area. What shape would it form? Well, intuitively it would be symmetric around the axis from one ring center to the other. Imagine having a soap bubble connected by two rings. The integral we then wish to minimize is the following.Īgainst intuition, it seems the string forms the shape of a hyperbolic cosine. We present new quantitative estimates for the radially symmetric configuration concerning Serrin’s overdetermined problem for the torsional rigidity, Alexandrov’s Soap Bubble theorem, and other related problems. Assuming uniform mass density, we know this element has a mass so it has a potential energy. If we consider a small part of the string, we know it has length. A law of nature is that systems tend toward low potential energy and similarly a string would as well. Once again, we must determine the integral being minimized. The problem is determining the function that describes the shape of a string at rest when hung. However, the method I used required the consideration of many variables unlike calculus of variations. This is the equation for a cycloid, the solution to the Brachistochrone problem.Īs previously mentioned, I have approached this problem in a previous post: The Shape of a String. Once again, the -independent Euler equation is applied. This means the integral we wish to minimize is the following. From conservation of energy, we know the speed of the particle is. First let us determine the integral to be solved. Consider the time it takes to roll down a segment. It is simply one case of Euler’s equation. The Brachistochrone problem is the famous problem of finding the optimal track to roll a particle down such that it minimizes the time taken to do so. If we were to roll a ball down a track, what should the track’s shape be such that it minimizes the travel time between two points A and B. Plateau formulated a set of empirical rules, now known as Plateau’s Laws, for the formation of soap films: 1. As trivial as this may sound, it is not true in other matric spaces and it is important to know exactly how to calculate this minimal distance points in any metric. Mathematically, the problem falls within the ambit of the calculus of variations. We have essentially just proved the shortest distance between two points is a line. We know use the -independent Euler equation. What function takes the shortest path between two points and ? Well, the length of the function can be calculated by the following integral which is also the functional we wish to minimize. In fact, the drawn out results from the posts The Shape of a String and The Lagrangian are just two cases of the one equation. In the first part, we discussed the idea of a functional, what it means, and how to find its extrema using the calculus of variations. However, those equations don’t really capture how amazing and applicable calculus of variations really is so the following will be some examples of this.
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