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. A

    I Is the Hamiltonian always the total energy?

    I'm working on some classical mechanics and just got a question stated: Is the Hamiltonian for this system conserved? Is it the total energy? In my problem it was indeed the total energy and it was conserved but it got me thinking, isn't the Hamiltonian always the total energy of a system...
  2. S

    I Calculus of Variations Dependent variables and constraints

    If we have a function: \begin{equation} f(x,x',y,y',t) \end{equation} and we are trying to minimise this subject to a constraint of \begin{equation} g(x,x',y,y',t) \end{equation} Would we simply have a set of two euler lagrange equations for each dependent variable, here we have x and y...
  3. muscaria

    A Variation of Lagrangian w/r to canonical momenta

    Hi, I've been working through Cornelius Lanczos book "The Variational Principles of Mechanics" and there's something I'm having difficulty understanding on page 168 of the Dover edition (which is attached). After introducing the Legendre transformation and transforming the Lagrangian equations...
  4. M

    Other Best Book for Calculus of Variations

    Hi PF! What book do you recommend for studying the calculus of variations? I have a masters degree in mechanical engineering and undergrad in math (if that helps you decide fro a book thats's not beyond my level). Thanks! Josh
  5. samgrace

    Understanding the Role of Partial Derivatives in Calculus of Variations

    Hello, here is my problem.http://imgur.com/VAu2sXl'][/PLAIN] http://imgur.com/VAu2sXl My confusion lies in, why those particular partial derivatives are chosen to be acted upon the auxiliary function and then how they are put together to get the Euler-Lagrange equation? My guess is that it's...
  6. bananabandana

    Euler Lagrange Derivation (Taylor Series)

    Mod note: Moved from Homework section 1. Homework Statement Understand most of the derivation of the E-L just fine, but am confused about the fact that we can somehow Taylor expand ##L## in this way: $$ L\bigg[ y+\alpha\eta(x),y'+\alpha \eta^{'}(x),x\bigg] = L \bigg[ y, y',x\bigg] +...
  7. P

    Calculus of variations question

    Okay, so I've run into a rather weird functional that I am trying to optimize using calculus of variations. It is a functional of three functions of a single variable, with a constraint, but I can't figure out how to set up the Euler-Lagrange equation. The functional in question is (sorry it's...
  8. NihalRi

    Why do we need to imagine a varied path in the calculus of variations?

    I'm very new to this. So in the context of finding the shortest path the idea is that you imagine another path that starts and ends at the same point. The shortest path is a minima so you differentiate and find for what values the differential is zero. I don't understand why we need to imagine...
  9. M

    Calculus of variations with circular boundary conditions

    The Euler-Lagrange equations give a necessary condition for the action be extremal given some lagrangian which depends on some function to be varied over. The basic form assumes fixed endpoints for the function to be varied over, but we can extend to cases in which one or both endpoints are free...
  10. evinda

    MHB What is the significance of Calculus of Variations in Classical Mechanics?

    Hello! (Wave) Could you give me some information about the subject Calculus of variations? What is it about? What backround is needed?
  11. B3NR4Y

    Calculus of Variations (Geodesics on a Cone)

    Homework Statement Find the geodesics on the cone whose equation in cylindrical-polar coordinates is z = λρ [Let the required curve have the form φ=φ(ρ)] check your result for the case λ→0 Homework Equations \frac{\partial F}{\partial y} - \frac{d}{dx} (\frac{\partial F}{\partial y'}) = 0...
  12. hideelo

    Questioning an assumption in calculus of variations

    When deriving stationary points of a function defined by a 1-D integral (think lagranian mechanics, Fermat's priniciple, geodesics, etc) and arriving at the Euler Lagrange equation, there seems to me to be an unjustified assumption in the derivation. The derivations I have seen start with...
  13. S

    Why y, y' (derivative of y), x are independent?

    In calculus of variations when we solve Euler's equation we always do think of y, x and y' as independent variables. In thermodynamics we think that different potentials have totally different variables I don't understand why the slope of the function is not directly dependent on function itself.
  14. Hunter Bliss

    Calculus of Variations: Minimizing Fuel Consumption w/ v(t)

    Homework Statement (I'm learning all of this in German, so I apologize if something is translated incorrectly.) So last week we started calculus of variations, and I'm rather confused about how to approach the following problem: The fuel consumption of a vehicle per unit of time is expressed...
  15. S

    Calculus of Variations: Δ-Variation vs. δ- Variation

    Does anybody know what is the formal difference between the Δ -variation and the δ- variation is? They seem to be used interchangeably. I read somewhere that Δ = δ + Δt*(d/dt) but I have no idea how that is arrived at. I know that the δ- variation is employed in the calculus of variations and...
  16. C

    Functional derivative of normal function

    I can't convince myself whether the following functional derivative is trivial or not: ##\frac \delta {\delta \psi(x)} \big[ \partial_x \psi(x)\big],## where ##\partial_x## is a standard derivative with respect to ##x##. One could argue that ## \partial_x \psi(x) = \int dx' [\partial_{x'}...
  17. M

    Simple Symplectic Reduction Example

    Homework Statement I'm struggling to perform a symplectic reduction and don't really understand the process in general. I have a fairly solid understanding of differential equations but am just starting to explore differential geometry. Hopefully somebody will be able to walk me through this...
  18. I

    Calculus Gelfand & Fomin vs. Lanczos to learn Calculus of Variations

    I am learning the Lagrangian formalism from Landau & Lifshitz but I'm not very familiar with variational calculus. Landau assumes its knowledge and uses it directly. Although the equations look analogous to what you'd do with ordinary calculus, I'd like to understand the foundation and ideas...
  19. B

    Calculus of variations changing variables

    Homework Statement Hi I am given the functional I am asked to show that if and with an appropriate value for that Homework Equations [/B]The Attempt at a Solution So I get If I set then I get I think that it is correct but what about the factor of 2?
  20. M

    Calculus of Variations & Lagrange Multiplier in n-dimensions

    extremize $$S = \int \mathcal{L}(\mathbf{y}, \mathbf{y}', t) dt $$ subject to constraint $$g(\mathbf{y}, t) = 0 $$ We move away from the solution by $$y_i(t) = y_{i,0}(t) + \alpha n_i(t) $$ $$\delta S = \int \sum_i \left(\frac{\partial\mathcal{L} }{\partial y_i} - \frac{d}{dt} \frac{\partial...
  21. Last-cloud

    Obtain Equation Using Hamilton's Principle

    I want to obtain equation using Hamilton principle but I just couldn't figure it out; i have The kinetic energy : \begin{equation} E_{k}=\dfrac{1}{2}m_{z} \displaystyle\int\limits_{0}^{L}\ \left[ \left( \dfrac{\partial w(x,t)}{\partial t}\right)^{2}+\left( \dfrac{\partial v(x,t)}{\partial...
  22. R

    Papers on Calculus of Variations

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  23. C

    Deriving Calculus of Variations

    Hey I'm having an issue deriving the calculus of variations because the chain rule i use ends up different to the one in the textbook. Firstly I assume we have some function of 3 variables Y=y+alpha eta with grad Y'=y'+alpha eta' and x. Secondly we have an integral of this function over x and...
  24. T

    Resources for variational principle to solve coulomb problem in D dimension

    Hi, i have been struggling to find some good resources on variational principle , I have got an instructor in advanced quantum course who just have one rule for teaching students- "dig the Internet and I don't teach you anything".. So I digged a lot and came up with a lot reading but I need...
  25. D

    A question on Lagrangian dynamics

    Hi all, I've recently been asked for an explanation as to why the Lagrangian is a function of the positions and velocities of the particles constituting a physical system. What follows is my attempt to answer this question. I would be grateful if you could offer your thoughts on whether this is...
  26. J

    Optimizing y(x) for \int_a^b y^2(1+(y')^2) \, dx with given boundary conditions

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  27. J

    Calculus of variations applied to geodesics

    Homework Statement I'm working on a problem from my gravitation book. The question is the following: Given \begin{equation} \frac{D}{Ds} T^\mu = 0 \end{equation}, where \begin{equation} T^\mu \left(s,a\right) = \frac{\partial z^\mu}{\partial s} \end{equation} is the tangent vector to a...
  28. H

    What shape would produce the greatest electric field?

    Suppose you are given an incompressible material with a constant charge density. What shape would create the largest electric field at a given point in space? These seems like a calculus of variation problem, but I am wondering if there might be any clever trick. $$\vec E = \frac{\rho}{4 \pi...
  29. D

    Functionals and calculus of variations

    I have been studying calculus of variations and have been somewhat struggling to conceptualise why it is that we have functionals of the form I[y]= \int_{a}^{b} F\left(x,y,y' \right) dx in particular, why the integrand F\left(x,y,y' \right) is a function of both y and it's derivative y'? My...
  30. ShayanJ

    Calculus of variations and integrands containing second derivatives

    You know that the problem of calculus of variations is finding a y(x) for which \int_a^b L(x,y,y') dx is stationary. I want to know is it possible to solve this problem when L is a function of also y'' ? It happens e.g. in the variational method in quantum mechanics where we say that choosing...
  31. P

    Calculus of variations: multiple variables, functions of one variable

    Simply put, can you find the function which extremizes the integral J[f]=\iint L\left(x,y,f(x),f(y),f'(x),f'(y)\right) \,dx \,dy Where ##f## is the function to be extremized, and ##x## and ##y## are independent variables? A result seems possible by using the usual calculus of variation...
  32. kq6up

    Fermat's Principle with Calculus of Variations

    Homework Statement This is problem 6.3 in Taylor’s Classical Mechanics. It is in context of the calculus of variations. Consider a ray of light traveling in a vacuum from point P1to P2 by way of the point Q on a plane mirror, as in Figure 6.8. Show that Fermat's principle implies that, on the...
  33. M

    Calculus of variations with isoparametric constraint

    We seek stationary solutions to \int_{x_0}^{x_1} F(x, y, y')dx subject to the constraint \int_{x_0}^{x_1} G(x, y, y')dx = c where c is some constant. I have read that this can be solved by applying the Euler Lagrange equations to F(x, y, y') + \lambda G(x, y, y') and then finding the...
  34. J

    Good book on calculus of variations

    I am looking for a book/document (mainly free ones) about calculus of variations of practical nature, i.e. very little theory with many examples and solved problems based on physical applications. Any advice is appreciated.
  35. D

    Calculus of Variations: Nature of the Functional

    Let \normalsize S[y] = \int ^{a}_{b} f[y, \dot{y}, x] dx be the functional i want to minimize. Why does \normalsize f (inside the integral) take this specific form? Would i not be able to minimize the integral, \normalsize S , if f had any other form instead of f = f[x, y, \dot{y}] ?
  36. N

    A question on calculus of variations

    Homework Statement δ (∂x'^μ/∂x^β)=0 This equation is on my textbook. I don't quite understand. Where x'^μ is coordinate component. Homework Equations The Attempt at a Solution
  37. G

    Calculus of variations for known derivative on both extremes

    Minimizing a functional: When you know the values of the function y(x) on the boundary points y(x1) and y(x2), minimizing the functional ∫{L(x,y,y')} yields the Euler-Lagrange equation. How can you minimize the functional if, instead, you know the derivatives y'(x1) and y'(x2)? What if...
  38. V

    Where should I begin to eventually understand calculus of variations?

    I am a engineering undergraduate. And my classical mechanics module was all based on Newtonian mechanics, but I got very curious about the hamiltonian and lagrangian formulations and decided to read up on those. When I got to the principle of least action I couldn't understand much, mostly...
  39. V

    Calculus of Variations: Solving Differential Equations

    My first question is with regards to the "status" of calculus of variations. Because I read in wolfram alpha that it was a generalization of calculus? Is that right? Anyway; my main question has to do with the process of getting the answer you're looking for. Is every problem in calculus of...
  40. G

    Extremal condition calculus of variations

    if I have a functional with a Lagrangian L(t,x(t),y(t),x'(t),y'(t)), meaning two functions x and y of one parameter t. And want to solve the minimization problem $$ \int_0^t L dt $$ . Then I get necessary conditions to find extrema by getting the two Euler Lagrange equation $$ \frac{\partial...
  41. U

    MHB Calculus of variations with integral constraints

    http://img835.imageshack.us/img835/2079/minimise.jpg Both p(x,y) and q(x,y) are probability density functions, q(x,y) is an already known density function, my job is to minimise C[p,q] with respect to 3 conditions, they are listed in the red numbers, 1, 2, 3. Setting up the lagrange function...
  42. G

    Bad proof in Fomin's Calculus of Variations?

    I was just reading through the first few pages of Fomin's Calculus of Variations and I came across this proof, which really doesn't seem to prove the Lemma (I may be missing something though) could someone give me a second opinion and perhaps some clarification? It goes like this; If...
  43. A

    Calculus of variations for suspended rope

    So perhaps you know this classical problem: A rope is suspended between two endpoints x=±a. Find what function describing the shape of the rope that will minimize its potential energy. The example is worked through in my book but I have some questions: The solution assumes uniform linear...
  44. T

    Fundemental lemma of the calculus of variations

    Homework Statement Hi, I've been revising the calculus of variations and using the wiki entry on the euler lagrange equation (http://en.wikipedia.org/wiki/Euler-Lagrange_equation) as a reference. Scroll down and you'll see: Derivation of one-dimensional Euler–Lagrange equation. Expand this...
  45. C

    Proof of Lagrange multipliers method in calculus of variations.

    I have been reading a little about calculus of variations. I understand the basic method and it's proof. I also understand Lagrange multipliers with regular functions, ie since you are moving orthogonal to one gradient due to the constraint, unless you are also moving orthogonal to the other...
  46. G

    Shortest path to the Calculus of Variations

    Hello all, A friend of mine has recently developed an interest (rather, an obsession) with the Calculus of Variations. He's familiar with linear algebra and also with the contents of Spivak's "Calculus on Manifolds", and is now looking for the shortest path to Gelfand and Fomin's "Calculus of...
  47. M

    Calculus of variations problem

    I have a question about calculus of variations that is driving me absolutely nuts right now: I have followed the standard derivation of differential equations from the extrimization of a functional S = ∫(F(x,dx/dt,t)dt) By doing some manipulation involving an arbitrary perturbation to your...
  48. Telemachus

    Geodesic on a cone, calculus of variations

    I have to find the geodesics over a cone. I've used cylindrical coordinates. So, I've defined: x=r \cos\theta y=r \sin \theta z=Ar Then I've defined the arc lenght: ds^2=dr^2+r^2d\theta^2+A^2dr^2 So, the arclenght: ds=\int_{r_1}^{r_2}\sqrt { 1+A^2+r^2 \left ( \frac{d\theta}{dr}\right )^2...
  49. QuarkCharmer

    Common Prerequisites for the Calculus of Variations?

    I'm really interested in this subject. Would one be capable of learning this subject with a great working knowledge of Multi-var/Vector Calculus, ODE, Linear Algebra, and complex variables? What are some good books?
  50. J

    Calculus of variations: Euler-Lagrange

    This is from a past paper (from a lecturer I don't particularly understand) Homework Statement a) {4 marks} Find the Euler-Lagrange equations governing extrema of I subject to J=\text{constant} , whereI=\int_{t_1}^{t_2}\text{d}t \frac{1}{2}(x\dot{y}-y\dot{x})=\int f(t,x,y,\dot{x},\dot{y})...
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