What is Variational calculus: Definition and 36 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's 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 had some several questions about variational calculus, but seems like I can't get an answer on math stackexchange. Takes huge time. Hopefully, this topic discussion can help me resolve some of the worries I have.
Assume ##y(x)## is a true path and we do perturbation as ##y(x) + \epsilon...
I had an interesting thought and it just might be me, but I'm looking forward to hearing your thoughts, but not like the thoughts - "yeah, Landau is messy, complicated, don't read that, e.t.c". Just think about what you think about my thoughts.
Landau first starts to mention the variational...
Let us consider an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. The solution of the field ##c## is given by the expression ##f(a,b)##. On taking into account the relations obtained from the solutions for ##a## and ##b##, we find that ##f(a,b)=0##. If the...
If a Lagrangian has the fields ##a##, ##b## and ##c## whose equations of motion are denoted by ##E_a, E_b## and ##E_c## respectively, then if
\begin{align}
E_a=f_1(a,b,c)\,E_b+f_2(a,b,c)\,E_c
\end{align}
where ##f_1## and ##f_2## are some functions of the fields, if ##E_b## and ##E_c## are...
Having two null vectors with $$n^{a} l_{a}=-1, \\ g_{ab}=-(l_{a}n_{b}+n_{a}l_{b}),\\ n^{a}\nabla_{a}n^{b}=0$$ gives $$\nabla_{a}n_{b}=\kappa n_{a}n_{b},\\ \nabla_{a}n^{a}=0,\\ \nabla_{a}l_{b}=-\kappa n_{a}l_{b},\\ \nabla_{a}l^{a}=\kappa$$.
How to show that under the variation of the null...
Suppose we have an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. We denote the variation of ##S## wrt to a given field, say ##a##, i.e. ##\frac{\delta S}{\delta a}##, by ##E_a##.
Then ##S## is gauge invariant when
$$\delta S = \delta a E_a + \delta b E_b...
Taking the variation w.r.t f(x) of the integral over some x domain of F[f(x), f'(x), df(x)/dt], why doesn't df(x)/dt need to be taken a variational derivative and is treated as if it were constant?
Principle of stationary action allows us to find equations of motion if we plug appropriate lagrangian into Euler - Lagrange equation. In classical mechanics, this is the difference in kinetic and potential energy of the system.
However, how did Lagrange came to the idea that matter behaves...
Lagrangian mechanics is built upon calculus of variation. This means that we want to find out function which is a stationary point of particular function (functional) which in Lagrangian mechanics is called the action.
To know what this function is, action needs to be defined first. Action is...
We've a functional
##J(\alpha)=\int_{x_{1}}^{x_{2}} f\left\{y(\alpha, x), y^{\prime}(\alpha, x) ; x\right\} d x##
It's derivative with respect to the parameter ##\alpha## is given in textbook Thornton Marion as ##\frac{\partial J}{\partial \alpha}##
Shouldn't it have been ##\frac{d J}{d...
Homework Statement
Question:
If ##r_+ \neq r_{++}## and ## g(r=r_+) \neq g(r=r_{++}) ##
When is it fulfilled that ## d g (r=r_+) \neq d g (r=r_+) ## ?
Homework Equations
##r_+ \neq r_{++}##
## g(r=r_+) \neq g(r=r_{++}) ##
The Attempt at a Solution
I tried computing ## dg(r_+) =...
Hello PF, I’m going through a book called “A First Course in the Calculus of Variations.” I can’t remember who the author is at the moment, I’ll post it later. Anyway, I’m having trouble with one part: suppose we have a function ##y (x)## that gives a continuous polygonal curve from ##x = a## to...
my first post having just joined!
Problem statement - what curve describes the surface of a rotating liquid? Stirring my cup of coffee years ago sparked this thought.
Question - is the way to solve this problem to use variational calculus, or fluid dynamics? I have always thought the former but...
I'm reading about the Principle of Least Action.
As a prelude to it, we look at the functional
J(x)=\int f(y(x),y'(x);x) dx
where the limits of integration are x_1 and x_2.
We want to find the function y(x) that gives the functional an extremum.
Now, to do this, we write any possible function...
I am trying to understand an example from my textbook "applied finite element analysis" and in the variational calculus, Euler lagrange equation example I can't seem to understand the following derivation in one of its examples
∫((dT/dx)(d(δT)/dx))dx= ∫((dT/dx)δ(dT/dx))dx= ∫((1/2)δ(dT/dx)^2)dx...
Homework Statement
Momentum, involving variational calculus.
\psi^{\dagger}
\bar{\psi}
matrices (\alpha, \beta)
Homework Equations
i \bar{\psi} \gamma^0 = i\psi^{\dagger}
The Attempt at a SolutionAm I to understand, that the momentum P=i\psi^{\dagger}, concentrating on the right hand...
Homework Statement
The Riemann metric on \{z\in C: |z|<1\} is defined as dx^2+dy^2\over 1-(x^2+y^2). I wish to show that the geodesics are diameters. Please help! Homework Equations
As above. And I suspect the Euler Lagrange equations.The Attempt at a Solution
I have tried using the Euler...
Homework Statement
A curve is enclosing constant area P. By means of variational calculus show, that the curve with minimal arc length is a circle,Homework Equations
The Attempt at a Solution
F(t)= \int_{x_{1}}^{x_{2}}\sqrt{1+(f^{'})^{2}}dt
G(t)= \int_{x_{1}}^{x_{2}}f(t)=const
If i use...
I asked this question in here:
https://nrich.maths.org/discus/messages/7601/151442.html?1310911861
(post 4), and still haven't got any reply, does someone know how to get to g'(\epsilon) in the text cause I get something different with another terms plus the terms in the text.
I guess i'll...
Consider a line of length L=\frac{\pi}{2}a. We want to put small particles of lead (total mass of all particles M) in order that the line is hang in a circular arc. Both ends are at the same height. Show that the mass distribution needs to be
\rho(y)=\frac{M}{2}\frac{a}{y^2}
This exercise...
Homework Statement
A mountain is modeled by z=f(x,y) which is a known function.
a) What are the differential equations for x(t) and y(t) that minimize the distance between 2 points.
b) If
z=f(x,y)=(\sin^2 2\pi x)(\sin^2 2\pi y)
Solve the equations.
Homework Equations
The...
Homework Statement
Homework Equations
\frac{\delta{F}}{\delta{y}} - \frac{d}{dx}\frac{\delta{F}}{\delta{y'}} = 0The Attempt at a Solution
I'm having trouble setting this one up. If I let the functional be
F(x,x',y) = n(y)\sqrt{1+(x')^2}
Applying the LE equation I obtain...
Hello everyone,
Edit: title should read gradient of a functional. Oh well...
I need a bit of help regarding variational calculus. I am not really very good at advanced calculus and only took basic calculus in high school and am having a bit of difficulty with a particular problem.
I have a...
hello, this is my first post in this forum, i hope i find some help from you.
i wanted to know how to calculate uncertainty of measurments (in experiments) starting from the variational calculus. any books please.
thank you in advance
I am taking a Quantum Mechanics course this semester and the professor started off by showing us how Newtonian Mechanics lead to Lagrangian Mechanics, then Hamiltonian...etc...until we got to the Quantum Mechanical wave functions.
This was all done in a quantitative sense, using variational...
Homework Statement
Hi, I am in a variational calculus class and am working on a homework and need a bit of help with two of the problems.
the first one is to prove that the functional:
J(u)= \int (A u'^2 + B u u' + C u^2 + D u' + E u) dx where A,B,C,D,E are constants and A \neq 0 has no...
Homework Statement
Consider the cylinder S in R3 defined by the equation x^2+y^2=a^2
(a). The points A=(a,0,0) \: and \: B = (a \cos{\theta}, a \sin{\theta}, b) both lie on S. Find the geodesics joining them.
(b). Find 2 different extremals of the length functional joining A=(a,0,0)...
Hello all :)
Homework Statement
I'm trying to understand the fundamentals of General Relativity, but alas, I seem to be unable to grasp the fundamentals of variational calculus.
Specifically, I'd like to prove the following relation for the square root of the negated determinant of the...
I'm supposed to find the shortest path between the points (0,-1,0) and (0,1,0) on the conical surface z=1-\sqrt {{x}^{2}+{y}^{2}}
So the constraint equation is:
g \left( x,y,z \right) =1-\sqrt {{x}^{2}+{y}^{2}}-z=0
And the function to be minimized is...
the primal problem was:
min (x^T)Px
i found g(r) and the partial derivative of g(r) w.r.t. x to be: x=-1/2(P^-1)(A^T)r
i have found the dual problem to be:
max -1/4(r^T)A(P^(-1))(A^T)r - (b^T)r
subject to r>= 0
I am told to find x* and r* (which i think is just x and r):
i have not...
1. Suppose f:R^{NxN} is defined by f(x,y) = \varphi(x) + \varphi*(Ax+y) where \varphi\epsilon\Gamma(R^{N}) and A\epsilonR^{NxN} is skew-symmetric.
Prove that f*(y,x) = f(x,y)
3. The Attempt at a Solution
Information I know:
Skew-symmetric : A*=-A
f* computation: f*(y) =...
I was just going through the derivation of the euler - lagrange equation that rests on the proof of the fundamental lemma of the calculus of variations which states the following:
If a function g(x) vanishes at the endpoints i.e g(a) = 0, g(b) = 0 and is continuously differentiable within...
[SOLVED] Variational calculus with bounded derivative constraints
After learning about the calculus of variations and optimal control for a bit this semester, I've decided to tackle a "simple" (in the words of my professor) problem meant to illustrate a simplified example of highway...
Homework Statement
I am trying to find a geodesic with Euler-Lagrange equation by varying the function
ds/d\tau = \sqrt{\dot{x} + \dot{y}}
EDIT: it should be ds/d\tau = \sqrt{\dot{x}^2 + \dot{y}^2}
where tau is a parametrization and the dot means a tau derivative.
However, when I plug...
Hi,
I have a general question about variational calculus (VC). I know the standard derivation of Euler-Lagrange equations and I´m able to use them. Nevertheless I think what I generally read cannot be the whole truth.
Generally if f:M->R (M: arbitrary topological space, R: real numbers)...
Could someone please direct me to a good web page or comment on the main difference between the euler lagrange eqn and lagranges eqn of motion. I'm struggling to differentiate between the two...
Also, I'm struggling to grasp the concept of Lagrange density - when does one introduce this into...