What is Calculus of variations: Definition and 154 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 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. selim

    I How did Hamilton derive the characteristic function V in his essay?

    In Hamilton's "on a general method in dynamics", he starts with varying the function ##U## and writes the equation: $$\delta U=\sum m(\ddot x\delta x+\ddot y\delta y+\ddot z\delta z)$$ Then he defines ##T## to be: $$T=\frac{1}{2}\sum m (\dot x^2+\dot y^2+\dot z^2)$$ Then by ##dT=dU##, he...
  2. thalesofmiletus

    I Circular orbits aren't minimas of lagrangian for Kepler problem?

    My point of view is bases on CALCULUS OF VARIATIONS by I. M. GELFANDS. and V. FOMIN. I would assume that you are familiar with the topic. I've found this book online, If needed I can provide link, but for now I don't know if it's legal in this site, so I won't. (by the way, I'm new on this site...
  3. C

    I The Euler-Lagrange equation and the Beltrami identity

    This question is specifically about deriving the Beltrami identity. Just to give this question context I provide an example of a problem that is solved with Calculus of Variations: find the shape of a soap film that stretches between two coaxial rings. For the surface area the expression to be...
  4. E

    I Is My Calculus of Variations Approach Correct?

    I would like to use the Calculus of Variations to show the minimum path connecting two points is a straight line, but I wish to do it from scratch without using the pre-packaged general result, because I'm having some trouble following it. Points are ##(x_1,y_1),(x_2,y_2)##. And we are to...
  5. T

    A Can position and velocity vary independently in Hamilton's Principle?

    To carry out the machinery of Hamilton's Principle though the calculus of variations, we desire to vary the position and velocity, independently. We proceed by varying at action, and set the variation to zero (I will assume ONE generalized variable: q1) In the above, I can see how we vary...
  6. M

    Is this the correct way to find the Euler equation (strong form)?

    By the Euler's equation of the functional, we have ## J(\mathrm u)=\int ((\mathrm{u})^{2}+e^{\mathrm{u}}) \, dx ##. Then ## J(\mathrm{u}+\epsilon\eta)=\int ((\mathrm{u}'+\epsilon\eta')^{2}+e^{\mathrm{u}+\epsilon\eta}) \, dx=\int...
  7. O

    Modifying Euler-Lagrange equation to multivariable function

    I'm confused on how to derive the multidimensional generalization for a multivariable function. Everything makes sense here except the line, $$ \frac{\delta S}{\delta \psi} = \frac{\partial L}{\partial \psi} - \frac{d}{dx} \frac{\partial L}{\partial(\frac{\partial \psi}{\partial x})} -...
  8. Reuben_Leib

    I Help with Euler Lagrange equations: neighboring curves of the extremum

    I tried writing this out but I think there is a bug or something as its not always displaying the latex, so sorry for the image. I have gone through various sources and it seems that the reason for u being small varies. Sometimes it is needed because of the taylor expansion, this time (below) is...
  9. Reuben_Leib

    I Why does ##u## need to be small to represent the Taylor expansion

    Necessary condition for a curve to provide a weak extremum. Let ##x(t)## be the extremum curve. Let ##x=x(t,u) = x(t) + u\eta(t)## be the curve with variation in the neighbourhood of ##(\varepsilon,\varepsilon')##. Let $$I(u) = \int^b_aL(t,x(t,u),\dot{x}(t,u))dt = \int^b_aL(t,x(t) +...
  10. E

    I Solving the Boat Crossing a River Problem using the Calculus of Variations

    I was hoping to explore the Calculus of Variations. How do we prove by Calculus of Variations that the minimum time for boat crossing a river (perpendicular to the current for starters) with current ##v_r##, and boat velocity in still water ##v_b## that the path will be a straight line? I...
  11. T

    A Euler Lagrange and the Calculus of Variations

    Good Morning all Yesterday, as I was trying to formulate my confusion properly, I had a series of posts as I circled around the issue. I can now state it clearly: something is wrong :-) and I am so confused :-( Here is the issue: I formulate the Lagrangian for a simple mechanical system...
  12. Haorong Wu

    Classical Looking for simple materials for calculus of variations

    Hi, there. I have not systematically learned the calculus of variations. I would like to learn it now. Are there simple materials for the purpose of learning how to do the calculation in physics? No need for deeper consideration in mathematics. Is Mathematical methods for physicists by Arfken...
  13. M

    I Calculus of Variations on Kullback-Liebler Divergence

    Hi, This isn't a homework question, but a side task given in a machine learning class I am taking. Question: Using variational calculus, prove that one can minimize the KL-divergence by choosing ##q## to be equal to ##p##, given a fixed ##p##. Attempt: Unfortunately, I have never seen...
  14. K

    I Independent coordinates are dependent

    (This is not about independence of ##q##, ##\dot q##) A system has some holonomic constraints. Using them we can have a set of coordinates ##{q_i}##. Since any values for these coordinates is possible we say that these are independent coordinates. However the system will trace a path in the...
  15. K

    Calculus Best Calculus of variations (Sturm Liouville Theory) textbook?

    Hi, I have a course on calculus of variations and Sturm Liouville theory and was wondering if anyone had any good textbook suggestions? If they had questions and solutions it would be a bonus! I have put all the subtopics of the course below. Calculus of variations Variation subject to...
  16. T

    I Euler, Calculus of Variations and Mast on a ship

    From Wikipedia: "In 1727, [Euler] first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship." Does anyone know how he did this? Is there an on-line paper? (But what that is accessible with today's knowledge). And by...
  17. CrosisBH

    Maximizing the volume of a cylinder

    Note this is in our Lagrangian Mechanics section of Classical Mechanics, so I assume he wants us to use Calculus of Variations to solve it. The surface area is fixed, so that'll be the constraint. Maximizing volume, we need a functional to represent Volume. This was tricky, but my best guess for...
  18. J

    Derivation of Lagrangian in the calculus of variations

    Hello. In a chapter of a book I just read it is given that ##\frac {d} {d\epsilon}\left. L(q+\epsilon \psi) \right|_{\epsilon = 0} = \frac {\partial L} {\partial q} \psi ## While trying to get to this conclusion myself I've stumbled over some problem. First I apply the chain rule...
  19. sams

    A The δ Notation in Calculus of Variations

    On page 224 of the 5th edition of Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion, the authors introduced the ##δ## notation (in section 6.7). This notation is given by Equations (6.88) which are as follows: $$\delta J = \frac{\partial J}{\partial...
  20. sams

    A Question about Euler’s Equations when Auxiliary Conditions are Imposed

    In the Classical Dynamics of Particles and Systems book, 5th Edition, by Stephen T. Thornton and Jerry B. Marion, page 220, the author derived Equation (6.67) from Equation (6.66) which is the following: Equation (6.67): $$\left(\frac{\partial f}{\partial y} − \ \frac{d}{dx}\frac{\partial...
  21. mishima

    Calculus of Variations, Isoperimetric, given surface area max volume

    My volume integral is... $$\pi\int y^2 dx$$ My surface area integral is... $$2\pi\int y \sqrt {1+x'^2} dy$$I'm fairly sure the variable of integration on my volume and surface area integrals has to be the same, is that right? But when I change the variable in the surface area integral to...
  22. B

    Textbook for calculus of variations? Hamiltonian mechanics?

    I need to learn about Hamiltonian mechanics involving functional and functional derivative... Also, I need to learn about generalized real and imaginary Hamiltonian... I only learned the basics of Hamiltonian mechanics during undergrad and now those papers I read show very generalized version...
  23. Philip Koeck

    I Applying a constraint in the calculus of variations

    I have an analytical function F of the discrete variables ni, which are natural numbers. I also know that the sum of all ni is constant and equal to N. N also appears explicitly in F, but F is not a function of N. F exists in a coordinate system given by the ni only. Should I carry out the...
  24. T

    A Gateaux vs. Frechet in Calc.Variations

    Good Morning Could someone summarize the distinction between the Frechet and the Gateaux derivative with specific regard to the Calculus of Variations? Why use one or the other? Or both? Is one easier than the other? More precise? Or Accessible? I feel I understand the process, but with...
  25. redtree

    I Derivation of the Euler-Lagrangian

    I have a question about a very specific step in the derivation of Euler-Lagrangian. Sorry if it seems simple and trivial. I present the question in the course of the derivation. Given: \begin{equation} \begin{split} F &=\int_{x_a}^{x_b} g(f,f_x,x) dx \end{split} \end{equation} Thus...
  26. M

    A Calculus of Variations and Natural BCs

    Hi PF! Given a functional ##J[y]##, if the first variation is $$\delta J[y] = \int_D(ay+y'')y \, dV + \int_{\partial D} (y'+by)y\,dS$$ am I correct to think that when finding stationary points of ##J[y]##, I would solve ##ay+y''=0## on ##D## subject to boundary conditions, which would either...
  27. Q

    I Derivative of a Variation vs Variation of a Derivative

    When a classical field is varied so that ##\phi ^{'}=\phi +\delta \phi## the spatial partial derivatives of the field is often written $$\partial _{\mu }\phi ^{'}=\partial _{\mu }(\phi +\delta \phi )=\partial _{\mu }\phi +\partial _{\mu }\delta \phi $$. Often times the next step is to switch...
  28. aphirst

    A Derivative of argmin/argmax w.r.t. auxiliary parameter?

    As part of my work, I'm making use of the familiar properties of function minima/maxima in a way which I can't seem to find in the literature. I was hoping that by describing it here, someone else might recognise it and be able to point me to a citation. I think it's highly unlikely that I'm the...
  29. petterson

    A Maximization problem using Euler Lagrange

    Hi, I'm trying to solve the following problem ##\max_{f(x)} \int_{f^{-1}(0)}^0 (kx- \int_0^x f(u)du) f'(x) dx##. I have only little experience with calculus of variations - the problem resembles something like ## I(x) = \int_0^1 F(t, x(t), x'(t),x''(t))dt## but I don't know about the...
  30. J

    A Maximization Problem: Double Int. w/ C not Dependent on Integrals

    Consider a double integral $$K= \int_{-a}^a \int_{-b}^b \frac{B}{r_1(y,z)r_2^2(y,z)} \sin(kr_1+kr_2) \,dy\,dz$$ where $$r_1 =\sqrt{A^2+y^2+z^2}$$ $$r_2=\sqrt{B^2+(C-y)^2+z^2} $$ Now consider a function: $$C = C(a,b,k,A,B)$$ I want to find the function C such that K is maximized. In other...
  31. M

    A Eigenvalue Problem and the Calculus of Variations

    Hi PF! Given ##B u = \lambda A u## where ##A,B## are linear operators (matrices) and ##u## a function (vector) to be operated on with eigenvalue ##\lambda##, I read that the solution to this eigenvalue problem is equivalent to finding stationary values of ##(Bu,u)## subject to ##(Au,u)=1##...
  32. jamalkoiyess

    I Delta x in the derivation of Lagrange equation

    Hello PF, I was doing the derivation of the Lagrange equation of motion and had to do some calculus of variations. The first step in the derivation is to multiply the integral of ƒ(y(x),y'(x);x)dx from x1 to x2 by δ. and then by the chain rule we proceed. But I cannot understand why we are...
  33. JTC

    Difference between Hamiltonian and Lagrangian Mechanics

    Hello, I am trying to "integrate into my understanding" the difference between Hamiltonian and Lagrangian mechanics. In a nutshell: If Lagrange did all the work and formulated L = T - V, they why is Hamilton's name attached to the minimization principle? YES; I KNOW about Hamilton's Second...
  34. C

    A Derivation of Euler Lagrange, variations

    What is wrong with the simple localised geometric derivation of the Euler Lagrange equation. As opposed to the standard derivation that Lagrange provided. Sorry I haven't mastered writing mathematically using latex. I will have a look at this over the next few days. More clarification. I...
  35. S

    I Calculus of variations

    when inducing that the cycloid is the least time-taking course between the two points in the two dimension, we have to use calculus of variations. Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?
  36. Avatrin

    I Calculus of variations question

    Hey, I have a theorem I cannot prove. We have a function x^* that maximizes or minimizes the integral: \int^{t_1}_{t_0} F(t,x(t),\dot{x}(t))dt Our end point conditions are: x(t_0) = x_0, x(t_1) \geq x_1 I am told that x^* has to satisfy the Euler equation. That I can fully understand since...
  37. F

    I How to compute second-order variation of an action?

    Starting with the action for a free scalar field $$S[\phi]=\frac{1}{2}\int\;d^{4}x\left(\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)-m^{2}\phi^{2}(x)\right)=\int\;d^{4}x\mathcal{L}$$ Naively, if I expand this to second-order, I get $$S[\phi+\delta\phi]=S[\phi]+\int\;d^{4}x\frac{\delta...
  38. M

    Calculus of Variations: interesting substitution

    Homework Statement Find the externals of the functional $$\int\sqrt{x^2+y^2}\sqrt{1+y'^2}\,dx$$ Hint: use polar coordinates. Homework Equations ##x=r\cos\theta## ##y=r\sin\theta## The Attempt at a Solution Transforming the given functional where ##r=r(\theta)## yields...
  39. O

    A Frechet v Gateaux Derivative and the calculus of variations

    Good Morning Could someone please distinguish between the Frechet and Gateaux Derivatives and why one is better to use in the Calculus of Variations? In your response -- if you are so inclined -- please try to avoid the theoretical foundations of this distinction (as I can investigate that by...
  40. J

    I Extremal condition in calculus of variations, geometric

    Hi folks, I am a bit confused with the extreme condition used in the calculus of variations: δ = 0 I don't understand this rule to find extreme solutions (maximum or minimum) If in normal differential calculus we have a function y = y(x) and represent it graphically, you see that at the...
  41. A

    Calculus of Variations; Maximum enclosed area problem.

    The problem reads: "You are given a string of fixed length l with one end fastened at the origin O, and you are to place the string in the (x, y) plane with its other end on the x-axis in such a way as to maximise the area between the string and the x axis. Show that the required shape is a...
  42. P

    I Numerical Calculus of Variations

    I attempt to solve the brachistochrone problem numerically. I am using a direct method which considers the curve ##y(x)## as a Lagrange polynomial evaluated at fixed nodes ##x_i##, and the time functional as a multivariate function of the ##y_i##. The classical statement of the problem requires...
  43. A

    Calculus of Variations: Functional is product of 2 integrals

    Homework Statement Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the...
  44. O

    I What is a Functional? Definition & Uses

    In the calculus of variations, the integral itself is a "functional." It depends on the form of the function of the Lagrangian: q and q-dot But I have seen this word "functional" used elsewhere in different contexts. I have seen: "A functional is a real valued function on a vector space." I...
  45. MichPod

    Calculus Need calculus of variations book for a laymen

    While trying to study textbooks on analytical mechanics or QFT I realized that I simply cannot operate with variations of functions in the same way I can operate with derivatives and integrals. I have never learned calculus of variations in university and, frankly, I am not much interested in...
  46. F

    I What is the motivation for principle of stationary action

    Is the motivation for the action principle purely from empirical evidence, or theoretical arguments, or a mixture of the two? As I understand it, there was some empirical evidence from Fermat's observations in optics, i.e. that light follows the path of least time, notions of virtual work and...
  47. ShayanJ

    A Neumann boundary conditions in calculus of variations

    In calculus of variations, extremizing functionals is usually done with Dirichlet boundary conditions. But how will the calculations go on if Neumann boundary conditions are given? Can someone give a reference where this is discussed thoroughly? I searched but found nothing! Thanks
  48. G

    I Question on Calculus of variations formalism

    Hi all, I had a quick question regarding the formalism behind calculus of variations. In one-dimensional standard calc, we consider functions f:\mathbb{R}\to \mathbb{R} and define their derivatives using the conventional definition with the limit of the quotient of the change in the function...
  49. T

    A Solving polynomial coefficients to minimize square error

    Hi there, I'm working on a problem right now that relates to least squares error estimate for polynomial fitting. I'm aware of techniques (iterative formulas) for finding the coefficients of a polynomial that minimizes the square error from a data set. So for example, for a data set that I...
  50. A

    Bead sliding on a wire - calculus of variations

    I am asked to find the shape of a wire that will maximize the speed a sliding bead when it reaches the end point(Similar to the brachistochrone problem expect that the speed is to be maximized and not time minimized). But shouldn't the speed at the end be independent of the shape of the wire...
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