# Tagged Questions

Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

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### Best Differential Equation, Partial Differential Equation and Calculus of Variations books?

Electrical Engineer here thinking of switching to physics. What are the best Differential Equation, Partial Differential Equation and Calculus of Variations books? Ideally they explain the topic ...
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### what is difference between variations of the work and virtual work

virtual work part https://en.wikipedia.org/wiki/Virtual_work I really want to know that both equations are same or not.(mathematically)(I think that they are the same.) Thanks for reading.
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### Pontryagin's maximum principle

So I've been doing some optimal control theory lately. It's really interesting but I've spent the whole day trying to wrap my head around pontryagin's maximum principle. There's a lot of mathematical ...
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Let $I:=[0, 1]$ and let $\alpha:I\times I\longrightarrow X$ be a family of smooth curves on a smooth manifold $X$. Let us think of $\alpha$ as a family of curves $\{\alpha_s: I\longrightarrow X\}_{s\... 1answer 64 views ### How would I solve integrals of derivatives? Solving integrals of values like$\int f \, \mathrm{d} f $is simple enough but how would I solve integrals of derivatives? For example consider the integral: $$\Delta F = \int^b_a \int \frac{\... 0answers 17 views ### Momentum Potential Term in Optimization Problem for Implicit Euler Solver I'm trying to understand the explanation of the implicit Euler solver (Section 3.1) set forth in this paper: Projective Dynamics: Fusing Constraint Projections for Fast Simulation For the purposes of ... 1answer 41 views ### Vainberg Theorem in measure theory In a lecture notes about Variational Methdos, I found the following theorem: THEOREM: Let (f_n) a sequence in L^{p}(\Omega) and f \in L^{p}(\Omega), such that f_{n} \rightarrow f in L^{p}(\... 0answers 14 views ### gradient of total variation norm in total variation denoising I am learning total variation denoising. The gradient of TV norm need calculated. From the link: http://www.numerical-tours.com/matlab/denoisingsimp_4_denoiseregul/ It says that the gradient is ... 2answers 64 views ### How to solve this functional problem? I am new to calculus of variations, till now I know how to get the extremal functions for a given functional using Euler-Lagrange equation. What if I have a functional but I am not looking for ... 1answer 32 views ### Question about calculus of variation. What is the difference between finding maxima or mimima i.e. critical point of a function and calculus of variation? 2answers 124 views ### Curve enclosing the maximum area the curve of fixed length l that joins the points (0,0) and (1,0) lies above the x-axis and encloses the maximum area between itself and the x-axis, is a segment of A straight line A ... 0answers 25 views ### More precise trail function in Rayleigh–Ritz method In order to obtain displacement field of an elasticity problem, say a plate structure, we approximate the solution using trigonometric series with unknown coefficients which satisfy the essential ... 1answer 46 views ### The plane curve of maximum average speed under constant gravitational force A line gives us the minimum distance from A to B. A cycloid gives us the minimum traveling time of a point mass from A to B (under constant gravitational acceleration g). What about the ... 1answer 40 views ### Minimum of F over Finite Perimeter Sets in \mathbb R^N Problem: Let G be a bounded Borel set. Let X be the set of finite perimeter sets in \mathbb R^N and F: X \to \mathbb R \cup \{+\infty\} defined as $F(E)= \begin{cases} Per(E) \hspace{1,... 0answers 33 views ### Derivative of one functional by another functional I have two functionals, F(h, \nabla h) and G(h, \nabla h). I'd like to calculate \frac{\delta F}{\delta G} Since functional derivatives also follow the chain rule, would I be correct in ... 1answer 48 views ### Find the Minimum of F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R). Let F: BV(\mathbb R) \to \mathbb R be a functional defined as: \[ F(u)= \int\limits_{-2}^{+2}|u(x) - \chi_{[0,2]}(x)|dx + |Du|(\mathbb R).$ Show that there is no minimum on W^{1,1}, but the ... 2answers 53 views ### Maximising \int_{0}^{T} v(t)\, dt, subject to constraints |v(t)| \leq a; v(0)=0; v(T)=0 Besides those constraints, we know nothing else about v(t). Interpreting the integral as the distance travelled by a particle, a little geometry tells us that the answer should be aT^{2}/4 ... 0answers 24 views ### Interplanetary Optimisation using a simulator with PyGMO or SciPy I am currently trying to use a N-body gravity simulator to model a spacecraft trajectory and using the simulator as a BlackBox to optimise the trajectory. I am thinking of using basin hopping/ ... 3answers 66 views ### Extremizing a functional subject to an equality constraint Question at hand is: Let y\in\cal C^2([0,\pi]) satisfying y(0)=y(\pi)=0 and \int_0^\pi y^2(x)dx=1 extremize the functional$$J(y)=\int_0^\pi\left(y'(x)\right)^2dx$$It's an MCQ, and ... 1answer 23 views ### Rings of same gravity center Using calculus of variations or otherwise, how do we find all non-circular ovals of loop length 2\pi in the plane with its center of gravity of arc at (0,0)? 2answers 26 views ### Variational Inequalities - What excatly does the definition say? Why are they useful? I am having issues understanding the definition of variational inequalities. We have the following definition: Given a set X \subset \mathcal{R}^n and a mapping F: X \rightarrow \mathcal{R}^n a ... 0answers 30 views ### Is this the correct way of using Variational Principle (Minimization Principle)? I am constructing a smooth function f(x)\equiv f(u(x),v(x)), such that u(x) and v(x) are some trial parameters. I have the following integral$$G=\int_{x_i}^{x_f} f(u(x),v(x)) \mathrm{d}x.$$My ... 0answers 19 views ### Is it possible to extend Jacobian, gradient, divergence and curl operators to the calculus of variations? In the calculus of variations one can extend the concept of the vector gradient to functionals or functions of the type \left(\mathrm{R} \rightarrow \mathrm{R}\right)\rightarrow \mathrm{R} by using ... 0answers 27 views ### Minimizing the functional \int (|\nabla u|^2- u^{2}V) on the Sobolev space H^1 I have a question about a function defined on a Banach space. Let \Omega be a bounded open subset of \mathbb{R}^{n} and V:\Omega \to [0,\infty] a bounded function on \Omega. Let H^{1}(\... 2answers 31 views ### Optimization inside integral I want maximize the integral$$\int_a^b \left( 2 cx y(x) - e y(x)^2 \right) \, \mathrm{d}x$$with respect to to y(x). If I discretize the problem, I get$$ \frac{b-a}{n}\sum_{i=1}^n 2c(i/n(b-a)+... 0answers 13 views ### What is the constant of integration for the functional antiderivative? Suppose I have the equation: $$Q = \varepsilon_0 \int_{\vec{s} \in \partial C} \vec{E} \cdot \hat{n} \, \mathrm{d} A$$ Then the functional derivative is: $$\oint_{\vec{s} \in \partial C}\frac{\... 0answers 19 views ### A curve in the first quadrant joins (0,0) and (1,0) and has a given area beneath it. Show that the shortest such curve is an arc of a circle. [closed] This is an isoperimetric problem that I am not sure how to approach. Any insight would be appreciated. Not sure how to find the function that I am trying to maximize or the constraint. 0answers 35 views ### finding curve along which a function extremizes via theory of calculus of variations [closed] Consider$$ I(y)= \int \limits _0 ^1 [y'(x)]^2dx \ +y(1)^2$$with y subsjected to the initial condition y(0)=1. Find the equation of curve along which y extremizes. 0answers 32 views ### Calculus of variation problem. The functional$$\int_{0}^{1}(1+x)(y')^{2}dx,y(0)=0,y(1)=1$$Possesses 1. Strong maxima. 2. Strong minima. 3. Weak maxima but not a strong maxima. 4. Weak minima but not a strong minima. ... 0answers 24 views ### Probability if variable has 15\% CV I have a relatively simple question, but I am not sure if I understand it right. I have estimated through my calculations the value X. X depends on many things, but one of them is Y and I know ... 0answers 26 views ### Intuition for second frechet derivative I am now used to thinking of the first derivative of a map between vector spaces f:V\to W in the "proper" Frechet sense, as being "the assignment to each point v of V of the linear map f'(v):V\... 0answers 94 views ### Determining whether the extremal problem has a weak minimum or strong minimum or both The extremal of the functional \int_{0}^{\alpha}{\left((y')^2 - y^2\right)dx} that passes through (0,0) and ({\alpha},0) has a weak minimum if {\alpha} < \pi strong minimum if {\alpha} ... 1answer 50 views ### Calculus of variations with inequality and non-integral constraints I have a question on solving an optimization problem with calculus of variations. I am attempting to maximize the functional$$ J[y] = \displaystyle\int_a^b F(x,y,y') \, \mathrm{d}x, \tag{1}$$... 1answer 41 views ### Solution of a sublinear elliptic problem. In a lecture notes, the author showed the problem \tag{P} \begin{cases} -\Delta u = |u|^{q-2}u \textrm{ in } \Omega, \\ u(x)= 0 \textrm{ in } \partial\Omega, \end{cases} where \Omega \... 2answers 123 views ### Proving that a sphere has a minimal surface to volume ratio using Calculus of Variations I know the problem is traditionally solved via the isoperimetric inequality, but I was hoping to solve it by minimizing a surface of revolution subject to a volume constraint. The surface area of a ... 0answers 14 views ### Euler-Lagrange Single function of single variable with higher derivatives Here is the page on Wikipedia: So it says the fixed boundary conditions for the function itself as well as for the first n-1 derivatives. You can fix the boundary points physically say y(a)=a' ... 1answer 28 views ### When the Euler-Lagrange equation reduces to 0=0 I've gotten the functional$$\int_a^b(y^2+2xyy')dx$$with Dirichlet boundary conditions. Applying the Euler-Lagrange equation I get:$$0=\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\... 1answer 105 views ### How do I determine a cricital point of an area functional? The orientated area$A(\gamma)$of a regular closed plane curve$(\gamma, \tau)$is defined as $$A(\gamma) :=\frac{1}{2}\int_{0}^\tau \det (\gamma,\gamma')$$ Now how can I determine the cricital ... 0answers 17 views ### Obscure first order approximation I don't understand this first order approximation from Gelfand, Fomin "Calculus of Variation": $$\int_{x_0}^{x_0 + \delta x_0} F(x, y + h, y'+h') dx \sim F(x,y,y')\big|_{x = x_0}\delta x_0$$ where$...
I am confused by one of the proof in the Calculus of Variations by Gelfand and Fomin. On page 9, we have Lemma: If $\alpha(x)$ is continuous on $[a,b]$, and if $\int_a^b \alpha(x)h(x)=0$ for every ...