What is Calculus of variation: Definition and 33 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. MatinSAR

    Geodesic on a sphere and on a plane in 2D

    I start with the 2D plane. Suppose y(x) is the curve that connects these two points. Its length is given by: $$S=\int_1^2 \, ds=\int_1^2 (1+y'^2)^{\frac {1}{2}} \, dx$$ Applying Euler's equation we get:$$\frac {\partial f} {\partial y'}=A$$$$\dfrac {y'}{(1+y'^2)^{\frac {1}{2}}}=A$$...
  2. Z

    On Landau vol.1 Pg.5 (Question about conclusion drawn by Landau)

    I understand that d/dv(L) = constant, and L is only dependent on v, but how do we get to the fact that v = constant?
  3. Pispi Choudhury

    Calculus Book suggestions and good lecture notes on the calculus of variation

    I need suggestions on books and good lecture notes on calculus of variation. I've previously studied vector calculus and multivariable calculus.
  4. 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...
  5. 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...
  6. Pushoam

    Using symmetry of action to find the constant of motion

    Homework Statement Homework EquationsThe Attempt at a Solution I need help in solving second part of this question. I put ## e^{i \alpha }\psi ## instead of ##\psi ## and got to see that the integrand doesn't change which means the given transformation is a symmetry of the given action. But...
  7. C

    I Gravitational DE(?) from Schwartzschild spacetime

    Imagining that an object spining around a spherical mass M has angular momentum that has z-component(θ=0) only, then $$g_{μν}\frac{dx^μ}{dτ}\frac{dx^ν}{dτ}=(1-\frac{r_s}{r})c^2(\frac{dt}{dτ})^2-\frac{1}{1-\frac{r_s}{r}}(\frac{dr}{dτ})^2-r^2(\frac{dθ}{dτ})^2-r^2\sin^2θ(\frac{dφ}{dτ})^2=c^2$$...
  8. K

    B Calculus of variation. Minimum surface

    so df/dy' is yy'/ √(1+y'^2) then we are supposed to do y' . [ yy'/ √(1+y'^2) ] - y√(1+y'^2) how does this bring equation 2 in the image ?
  9. D

    Calculus Calculus of variation textbook 'not under a single integral'

    I have to find functions that maximise certain criterea. The problem can however not be put "under a single integral", for example I've to find ##f(t)##, ##g(t)## that maximise: ## \int_0^{t_e}f(t)^2dt\int_0^{t_e}g(t)^2dt - (\int_0^{t_e}f(t)g(t)dt)^2 ## With ## -1 \leq f(t)\leq1## and ## -1...
  10. J

    Analysis Recommend me a calculus of variation book

    Hi! I have some trobles while studying the lagrange mechanics chapter of Marion's Classical Mechanics. There are some variation techniques in that book, but I only studied calculus and elementary linear algebra in my freshman year. I can't understand how partial derivation and delta notation...
  11. Noctisdark

    How to Change Variables in an Integral?

    Homework Statement Determine the Lagrange-Euler equation for the functional J[y] = ∫F(x,y,y')dx = ∫F(x(y),y,1/x')*x' dx = I[x(y)] Homework Equations The Lagrange-Euler equations ∂F/∂y = d(∂F/∂y')/dx The Attempt at a Solution I'm new to the subject, so I don't really know what to do, I've...
  12. J

    Good reference on multi-variable calculus of variation

    I am looking for a good and easy access reference on multi-variable calculus of variation with many examples and demonstrations. Although I have many books and references on the calculus of variation, most are focused on single-variable. Any advice will be appreciated.
  13. D

    MHB Calculus of Variation: Maximizing Volume & Min Area

    How do I set up the following problem? What geometric surface encloses the maximum volume with the minimum surface area?
  14. L

    Calculus of Variation: Extremum & Further Variances

    If for some functional ##I##, ##δI=0## where ##δ## is symbol for variation functional has extremum. For ##δ^2I>0## it is minimum, and for ##\delta^2I>0## it is maximum. What if ##δI=δ^2I=0##. Then I must go with finding further variations. And if ##δ^3I>0## is then that minimum? Or what?
  15. L

    Calculus of Variation: Extrema & Further Variations

    If for some functional ##I##, ##\delta I=0## where ##\delta## is symbol for variation functional has extremum. For ##\delta^2 I>0## it is minimum, and for ##\delta^2 I<0## it is maximum. What if ##\delta I=\delta^2 I=0##. Then I must go with finding further variations. And if ##\delta^3I>0## is...
  16. C

    Calculus of Variation - Shortest path on the surface of a sphere

    Refer to "2.jpg", it said that the shortest path on the surface of a sphere is Ay-Bx=z , which is a plane passing through the center of the sphere. I cannot really understand about this. Does it mean that the shortest path is a ring that connects two points with its center at the center of the...
  17. H

    Why are y and y' treated as independent in calculus of variation?

    In calculus of variation, we use Euler's equation to minimize the integral. e.g. ∫f{y,y' ;x}dx why we treat y and y' independent ?
  18. J

    Non-linear second order from calculus of variation I can't solve

    Hi, I derived the equation: 1+(y')^2-y y''-2y\left(1+(y')^2\right)^{3/2}=0 Letting y'=p and y''=p\frac{dp}{dy}, I obtain: \frac{dp}{dy}=\frac{1+p^2-2y(1+p^2)^{3/2}}{yp} I believe it's tractable in p because Mathematica gives a relatively simple answer: p=\begin{cases}\frac{i...
  19. Astrum

    Calculus of Variation - Classical Mechanics

    I'm reading Classical Mechanics (Taylor), and the 6th chapter is a basic introduction to calculus of variations. I'm super confused :confused: I've tried to go to other sources for an explanation, but they just make it even worse! So, let me see if I can get some help here...
  20. D

    Calculus of Variation on Local Regions of Function Space

    I am familiar with basic calculus of variations. For example, how to find a function that makes some integral functional stationary (Euler-Lagrange Equations). Or for example, how to perform that same problem but with some additional holonomic constraint or with some integral constraint. The...
  21. M

    Calculus of Variation Questions

    Hey guys. In my mechanics course, we have began discussing calculus of variations, and I don't really understand what's going on, entirely. Any help understanding would be great. Our professor gave us an easy problem, but I feel like I am just missing something. Homework Statement...
  22. L

    Is the Variation of a Functional in Calculus of Variation Correctly Calculated?

    Hi everyone! Here's my problem: Let's suppose that we have a functional I[f,g]=\int{L(f,\dot{f},g,\dot{g},x)\,dx}. Is it right to say that the variation of I whit respect to g (thus taking g\;\rightarrow\;g+\delta g) is \delta I=\int{[L(f,\dot{f},g+\delta g,\dot{g}+\delta \dot...
  23. G

    Calculus of Variation: Chain Rule and Formulation Proof

    I have a question about calculus of variation. does anybody here know a proof for the chain rule: \delta S= \frac{dS}{dx} \delta x and for the formulation: \delta S= p \delta x => \frac{dS}{dx}= p it would be totally sufficient, if anyone here knows(e.g. a weblink) where one could see this...
  24. G

    Calculus of Variation: "Help Me Understand a Step!

    hey I do not understand a step here! The integral is: \delta S(x,t)=-mc \int_a^b u_i d \delta x^i =0 and now they say one should do integration by parts, but I do not know how this should work here? Where are my two functions?As far as I see there is only the four-velocity and I do not how...
  25. P

    About calculus of variation and lagrangian formulation

    I was reading about the principle of least action and how to derive Newton's second out of it. at a certain point I didn't follow the calculations, so the author defines a variation in the path, x(t) \longrightarrow x'(t) = x(t) + a(t), a \ll x a(t_1) = a(t_2) = 0 Now, S...
  26. N

    Path of Light- Calculus of Variation

    Homework Statement Let y(x) represent the path of light through a variable transparent medium. The speed of light at some point (x,y) in the medium is a function of x alone and is written c(x). Write down an expression for the time T taken for the light to travel along some arbitrary path y(x)...
  27. D

    Minimizing Time with Calculus of Variation in Vertical Plane

    Im supposed to show that a ligth beam traveling in a vertical plane satisfies d^2z/dx^2=1/n(z) dn/dz[1+(dz/dx)^2]. Using calculus of variations to minimize the total time. The vertical plane got a refracting index n=n(z) there z is the vertical position and z=z(x) there x is the horisontal...
  28. quasar987

    What does the central thm of calculus of variation says?

    Based on the proof given in the action article of wiki (http://en.wikipedia.org/wiki/Action_%28physics%29#Euler-Lagrange_equations_for_the_action_integral), it would seems that the statement of the "central thm of calculus of...
  29. Z

    Exploring the Impact of Calculus of Variation in Modern Society

    can anyone help me? Desperate! What are the CURRENT development of calculus of variation that has impact on our society? Like in the area of engineering, I.T. etc?? And can anyone provide me a webby on that please thank you!
  30. Z

    What is the brachistochrone problem and how does it affect society?

    Hi all, I'm a maths and physics student from college. I have been asked by tutor to make a small report on the uses of calculus of variation which has an impact of society... I have been searching up and down but can't find anything specific or easy enough to understand on the internet.. I was...
  31. L

    Calculus of Variation (pls help )

    Calculus of Variation (pls help!) hello! can somebody explain to me what's calculus of variation?? and more importantly, how it is applied in everyday life (such as consumer's products, industries etc) ? :) thanks so much! :!) really really need help! :yuck:
  32. M

    How to Find Geodesics on a Curved Surface Using Calculus of Variations

    hi, we have just got to the point in my physics course where Newtons laws are now longer that easy to work with anymore and we are now beginning to reformulate those using variational methods, and I am a little confused on one of the problems. The shortest path between two points on a curved...
  33. M

    The math of physics - Calculus of Variation?

    We escape the problems of particle physics by exploring the higher dimensions of String theory. When we have questions about String theory, we jump to the higher dimensions of M-theory to answer them. And some have purposed to use the higher dimensions of F-theory to answer questions about...