What is Variational calculus: Definition and 36 Discussions

The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.
Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

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  1. G

    I Help with some confusions about variational calculus

    I had some several questions about variational calculus, but seems like I can't get an answer on math stackexchange. Takes huge time. Hopefully, this topic discussion can help me resolve some of the worries I have. Assume ##y(x)## is a true path and we do perturbation as ##y(x) + \epsilon...
  2. G

    I Using reference frames for derivation of Lagrangian

    I had an interesting thought and it just might be me, but I'm looking forward to hearing your thoughts, but not like the thoughts - "yeah, Landau is messy, complicated, don't read that, e.t.c". Just think about what you think about my thoughts. Landau first starts to mention the variational...
  3. Baela

    A If the solution of a field vanishes on-shell does it mean anything?

    Let us consider an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. The solution of the field ##c## is given by the expression ##f(a,b)##. On taking into account the relations obtained from the solutions for ##a## and ##b##, we find that ##f(a,b)=0##. If the...
  4. Baela

    A What does it mean when the eom of a field is trivially satisfied?

    If a Lagrangian has the fields ##a##, ##b## and ##c## whose equations of motion are denoted by ##E_a, E_b## and ##E_c## respectively, then if \begin{align} E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c \end{align} where ##f_1## and ##f_2## are some functions of the fields, if ##E_b## and ##E_c## are...
  5. sarriiss

    A Preserving Covariant Derivatives of Null Vectors Under Variation

    Having two null vectors with $$n^{a} l_{a}=-1, \\ g_{ab}=-(l_{a}n_{b}+n_{a}l_{b}),\\ n^{a}\nabla_{a}n^{b}=0$$ gives $$\nabla_{a}n_{b}=\kappa n_{a}n_{b},\\ \nabla_{a}n^{a}=0,\\ \nabla_{a}l_{b}=-\kappa n_{a}l_{b},\\ \nabla_{a}l^{a}=\kappa$$. How to show that under the variation of the null...
  6. Baela

    A Basic Question about Gauge Transformations

    Suppose we have an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. We denote the variation of ##S## wrt to a given field, say ##a##, i.e. ##\frac{\delta S}{\delta a}##, by ##E_a##. Then ##S## is gauge invariant when $$\delta S = \delta a E_a + \delta b E_b...
  7. F

    I Time derivatives in variational calculus

    Taking the variation w.r.t f(x) of the integral over some x domain of F[f(x), f'(x), df(x)/dt], why doesn't df(x)/dt need to be taken a variational derivative and is treated as if it were constant?
  8. Dario56

    I Principle of Stationary Action - Intuition

    Principle of stationary action allows us to find equations of motion if we plug appropriate lagrangian into Euler - Lagrange equation. In classical mechanics, this is the difference in kinetic and potential energy of the system. However, how did Lagrange came to the idea that matter behaves...
  9. Dario56

    I Action in Lagrangian Mechanics

    Lagrangian mechanics is built upon calculus of variation. This means that we want to find out function which is a stationary point of particular function (functional) which in Lagrangian mechanics is called the action. To know what this function is, action needs to be defined first. Action is...
  10. K

    I Basic question about variational calculus

    We've a functional ##J(\alpha)=\int_{x_{1}}^{x_{2}} f\left\{y(\alpha, x), y^{\prime}(\alpha, x) ; x\right\} d x## It's derivative with respect to the parameter ##\alpha## is given in textbook Thornton Marion as ##\frac{\partial J}{\partial \alpha}## Shouldn't it have been ##\frac{d J}{d...
  11. P

    Variational Calculus: When Is dg(r=r+) ≠ dg(r=r++)?

    Homework Statement Question: If ##r_+ \neq r_{++}## and ## g(r=r_+) \neq g(r=r_{++}) ## When is it fulfilled that ## d g (r=r_+) \neq d g (r=r_+) ## ? Homework Equations ##r_+ \neq r_{++}## ## g(r=r_+) \neq g(r=r_{++}) ## The Attempt at a Solution I tried computing ## dg(r_+) =...
  12. Pencilvester

    I Euler’s approach to variational calculus

    Hello PF, I’m going through a book called “A First Course in the Calculus of Variations.” I can’t remember who the author is at the moment, I’ll post it later. Anyway, I’m having trouble with one part: suppose we have a function ##y (x)## that gives a continuous polygonal curve from ##x = a## to...
  13. L

    Variational calculus or fluid dynamics for fluid rotating in a cup

    my first post having just joined! Problem statement - what curve describes the surface of a rotating liquid? Stirring my cup of coffee years ago sparked this thought. Question - is the way to solve this problem to use variational calculus, or fluid dynamics? I have always thought the former but...
  14. L

    Variational calculus in particle dynamics

    I'm reading about the Principle of Least Action. As a prelude to it, we look at the functional J(x)=\int f(y(x),y'(x);x) dx where the limits of integration are x_1 and x_2. We want to find the function y(x) that gives the functional an extremum. Now, to do this, we write any possible function...
  15. H

    Variational calculus Euler lagrange Equation

    I am trying to understand an example from my textbook "applied finite element analysis" and in the variational calculus, Euler lagrange equation example I can't seem to understand the following derivation in one of its examples ∫((dT/dx)(d(δT)/dx))dx= ∫((dT/dx)δ(dT/dx))dx= ∫((1/2)δ(dT/dx)^2)dx...
  16. H

    Variational Calculus and momentum

    Homework Statement Momentum, involving variational calculus. \psi^{\dagger} \bar{\psi} matrices (\alpha, \beta) Homework Equations i \bar{\psi} \gamma^0 = i\psi^{\dagger} The Attempt at a SolutionAm I to understand, that the momentum P=i\psi^{\dagger}, concentrating on the right hand...
  17. M

    Show Geodesics are Diameters on Riemann Metric | Variational Calculus Homework

    Homework Statement The Riemann metric on \{z\in C: |z|<1\} is defined as dx^2+dy^2\over 1-(x^2+y^2). I wish to show that the geodesics are diameters. Please help! Homework Equations As above. And I suspect the Euler Lagrange equations.The Attempt at a Solution I have tried using the Euler...
  18. Z

    Variational Calculus - Minimal Arc lenght for given surface

    Homework Statement A curve is enclosing constant area P. By means of variational calculus show, that the curve with minimal arc length is a circle,Homework Equations The Attempt at a Solution F(t)= \int_{x_{1}}^{x_{2}}\sqrt{1+(f^{'})^{2}}dt G(t)= \int_{x_{1}}^{x_{2}}f(t)=const If i use...
  19. MathematicalPhysicist

    Variational Calculus question.

    I asked this question in here: https://nrich.maths.org/discus/messages/7601/151442.html?1310911861 (post 4), and still haven't got any reply, does someone know how to get to g'(\epsilon) in the text cause I get something different with another terms plus the terms in the text. I guess i'll...
  20. O

    Variational Calculus - variable density line

    Consider a line of length L=\frac{\pi}{2}a. We want to put small particles of lead (total mass of all particles M) in order that the line is hang in a circular arc. Both ends are at the same height. Show that the mass distribution needs to be \rho(y)=\frac{M}{2}\frac{a}{y^2} This exercise...
  21. W

    Variational Calculus to minimize distance on a generalize 3D surface

    Homework Statement A mountain is modeled by z=f(x,y) which is a known function. a) What are the differential equations for x(t) and y(t) that minimize the distance between 2 points. b) If z=f(x,y)=(\sin^2 2\pi x)(\sin^2 2\pi y) Solve the equations. Homework Equations The...
  22. P

    Light in an optical fiber - variational calculus

    Homework Statement Homework Equations \frac{\delta{F}}{\delta{y}} - \frac{d}{dx}\frac{\delta{F}}{\delta{y'}} = 0The Attempt at a Solution I'm having trouble setting this one up. If I let the functional be F(x,x',y) = n(y)\sqrt{1+(x')^2} Applying the LE equation I obtain...
  23. P

    Variational Calculus help: minimising a function

    Hello everyone, Edit: title should read gradient of a functional. Oh well... I need a bit of help regarding variational calculus. I am not really very good at advanced calculus and only took basic calculus in high school and am having a bit of difficulty with a particular problem. I have a...
  24. T

    Uncertainty calculation and variational calculus

    hello, this is my first post in this forum, i hope i find some help from you. i wanted to know how to calculate uncertainty of measurments (in experiments) starting from the variational calculus. any books please. thank you in advance
  25. A

    Exploring Quantum Mechanics with Variational Calculus

    I am taking a Quantum Mechanics course this semester and the professor started off by showing us how Newtonian Mechanics lead to Lagrangian Mechanics, then Hamiltonian...etc...until we got to the Quantum Mechanical wave functions. This was all done in a quantitative sense, using variational...
  26. L

    Variational Calculus - Proving a functional has no broken extremals?

    Homework Statement Hi, I am in a variational calculus class and am working on a homework and need a bit of help with two of the problems. the first one is to prove that the functional: J(u)= \int (A u'^2 + B u u' + C u^2 + D u' + E u) dx where A,B,C,D,E are constants and A \neq 0 has no...
  27. L

    Variational Calculus : Geodesics w/ Constraints

    Homework Statement Consider the cylinder S in R3 defined by the equation x^2+y^2=a^2 (a). The points A=(a,0,0) \: and \: B = (a \cos{\theta}, a \sin{\theta}, b) both lie on S. Find the geodesics joining them. (b). Find 2 different extremals of the length functional joining A=(a,0,0)...
  28. W

    Variation of the metric's determinant [General Relativity, Variational Calculus]

    Hello all :) Homework Statement I'm trying to understand the fundamentals of General Relativity, but alas, I seem to be unable to grasp the fundamentals of variational calculus. Specifically, I'd like to prove the following relation for the square root of the negated determinant of the...
  29. D

    Shortest path on a conical surface (Variational Calculus)

    I'm supposed to find the shortest path between the points (0,-1,0) and (0,1,0) on the conical surface z=1-\sqrt {{x}^{2}+{y}^{2}} So the constraint equation is: g \left( x,y,z \right) =1-\sqrt {{x}^{2}+{y}^{2}}-z=0 And the function to be minimized is...
  30. B

    Variational calculus - dual problem

    the primal problem was: min (x^T)Px i found g(r) and the partial derivative of g(r) w.r.t. x to be: x=-1/2(P^-1)(A^T)r i have found the dual problem to be: max -1/4(r^T)A(P^(-1))(A^T)r - (b^T)r subject to r>= 0 I am told to find x* and r* (which i think is just x and r): i have not...
  31. J

    Variational calculus proof

    1. Suppose f:R^{NxN} is defined by f(x,y) = \varphi(x) + \varphi*(Ax+y) where \varphi\epsilon\Gamma(R^{N}) and A\epsilonR^{NxN} is skew-symmetric. Prove that f*(y,x) = f(x,y) 3. The Attempt at a Solution Information I know: Skew-symmetric : A*=-A f* computation: f*(y) =...
  32. A

    Fundamental Lemma of Variational Calculus

    I was just going through the derivation of the euler - lagrange equation that rests on the proof of the fundamental lemma of the calculus of variations which states the following: If a function g(x) vanishes at the endpoints i.e g(a) = 0, g(b) = 0 and is continuously differentiable within...
  33. R

    Variational calculus with bounded derivative constraints

    [SOLVED] Variational calculus with bounded derivative constraints After learning about the calculus of variations and optimal control for a bit this semester, I've decided to tackle a "simple" (in the words of my professor) problem meant to illustrate a simplified example of highway...
  34. E

    Variational Calculus: Finding a Geodesic with EL Equation

    Homework Statement I am trying to find a geodesic with Euler-Lagrange equation by varying the function ds/d\tau = \sqrt{\dot{x} + \dot{y}} EDIT: it should be ds/d\tau = \sqrt{\dot{x}^2 + \dot{y}^2} where tau is a parametrization and the dot means a tau derivative. However, when I plug...
  35. I

    Rigurous treatment of variational calculus

    Hi, I have a general question about variational calculus (VC). I know the standard derivation of Euler-Lagrange equations and I´m able to use them. Nevertheless I think what I generally read cannot be the whole truth. Generally if f:M->R (M: arbitrary topological space, R: real numbers)...
  36. B

    Variational Calculus: Euler-Lagrange vs. Lagrange's Equation of Motion

    Could someone please direct me to a good web page or comment on the main difference between the euler lagrange eqn and lagranges eqn of motion. I'm struggling to differentiate between the two... Also, I'm struggling to grasp the concept of Lagrange density - when does one introduce this into...