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.
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:$$\frac {\partial f} {\partial y'}=A$$$$\dfrac {y'}{(1+y'^2)^{\frac {1}{2}}}=A$$...
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:
$$\delta J = \frac{\partial J}{\partial...
In the Classical Dynamics of Particles and Systems book, 5th Edition, by Stephen T. Thornton and Jerry B. Marion, page 220, the author derived Equation (6.67) from Equation (6.66) which is the following:
Equation (6.67):
$$\left(\frac{\partial f}{\partial y} − \ \frac{d}{dx}\frac{\partial...
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...
Imagining that an object spining around a spherical mass M has angular momentum that has z-component(θ=0) only, then
$$g_{μν}\frac{dx^μ}{dτ}\frac{dx^ν}{dτ}=(1-\frac{r_s}{r})c^2(\frac{dt}{dτ})^2-\frac{1}{1-\frac{r_s}{r}}(\frac{dr}{dτ})^2-r^2(\frac{dθ}{dτ})^2-r^2\sin^2θ(\frac{dφ}{dτ})^2=c^2$$...
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...
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...
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...
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.
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?
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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...
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!
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...
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:
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...
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...