Calculus of variations Definition and 40 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.
(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...
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...
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...
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...
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...
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}...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
Homework Statement
Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=a
Homework 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...
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...
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...
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...
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...
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...
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...
Hello, here is my problem.http://imgur.com/VAu2sXl'][/PLAIN] [Broken]
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...
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] +...
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...
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.
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...
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'}...
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...
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...
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...
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...
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...
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...
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...