Questions tagged [variational-calculus]
The variational-calculus tag has no usage guidance.
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How to differentiate with respect to a function
I have some complicated expression, let's call it $X_n$, for which I don't have a closed form, but I do have a recurrence relation. It is given in terms of some functions $u(x,y),$ $v(x,y),$ and ...
5
votes
1answer
69 views
Computing a functional derivative using the VariationalMethods package
To test the VariationalMethods` toolbox, I tried to compute $\frac{\delta I[y]}{\delta y(x)}$ for the following functional, assuming $w(x)$ is some well-behaved ...
5
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2answers
217 views
Black hole orbit in Mathematica
I am currently investigating the motion of a particle around a black hole. The Lagrangian for the system is
$$ \mathcal{L}=F(r)\dot{t}^2-\frac{\dot{r}^2}{F(r)}-(r\dot{\phi})^2 $$
Where $ F(r) = 1 - ...
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3answers
149 views
Use of variational operator, 𝛿 in Wolfram Mathematica (Version 11.0) [closed]
I have been working on a problem which requires the use of Energy Methods which is based on the variational principles. I am finding it hard to replicate my formulated equations in Mathematica.
How ...
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1answer
155 views
Does Mathematica understand the concept of infinitesimal increment?
Commonly, in textbooks, differential of differential increment vanishes, good example is how Euler-Lagrange equations are derived using the differential (from step 2 to step 3 here, you can see ...
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1answer
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VariationalD of Nested Integral
I am trying to calculate the variational derivative with respect to f of the the nested integral below
$$ \mathcal{J}\left[f(x) \right] = \int\limits_a^b \text{d} x \,e^{-f(x)} \int\limits_a^x \text{...
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0answers
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Variational derivative of a functional depending on multiple functions
I wanted to use the VariationalD command to calculate a list of variational derivatives.
A simple example is the following and it works perfectly fine.
...
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1answer
127 views
How to do variation of an integration of a function with consideration of limits?
I have a function dependent on w(x,t) which is integrated in an interval. After integration of this function, I want to apply variation followed by integral by parts. Integral by parts results in to ...
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0answers
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Could I take a variation of a functional in Mathematica? [closed]
In finite element, the weak form can be derived by taking the variation of a energy functional as:
\begin{align}
L(u(x,y)) &= -\dfrac{1}{2}\int_{\Omega}(\dfrac{\partial{u}}{\partial{x}}\dfrac{...
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1answer
275 views
Drawing geodesic lines on the membrane (Plot3D)
The most common method of creating patterns on a membrane involves drawing geodesic lines on the membrane. Justification of the layout of the geodesic lines is beyond the scope of this text .
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0answers
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VariationalD with respect to two-variable function of integral over one variable
I have a function $q(x,t)=u(x,t)+iv(x,t),$ some functions $p_n$ and I want to find the following functional derivative:
$$\frac{\delta}{\delta\overline{q}}\int p_ndt$$
Because Mathematica's ...
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1answer
273 views
Legendre Transform of a function of a 3-vector
I am trying to implement the Legendre transform of a function in Mathematica with the purpose of calculating the Hamiltonian of a system starting from the Lagrangian.
I have found this page which ...
3
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1answer
279 views
How to formulate the Hamiltonian equations of motion? Part II
I've recently asked a question on an issue I was facing with numerically integrating Hamiltonian equations of motion. I got a great answer.
Following on from this, I wanted to write a very similar ...
4
votes
1answer
189 views
Numerically Integrating a Hamiltonian but getting different results when compared with alternative equivalent equations
currently trying to perform an integration of the Hamiltonian for a Schwarzschild black hole. However, I'm coming up short.
The code is as follows:
...
2
votes
1answer
304 views
how to find a function that minimizes the other function with a known function
The problem that i am facing is that of energy minimization. I have a function called totalenergy which is sum of two functions the elastic energy and the surface ...
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0answers
319 views
how to find stationary point of a multivariable function [closed]
I am quite new, how can I find minumum, maximum or saddle point the following multivariable function by using mathematica
$f(x,y)= 2x^3 +6xy^2 −3y^3 −150x$
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0answers
94 views
Book Recommendations About Using Mathematica in Topics like Calculus of Variation and Optimization?
I want to use effectively Mathematica about topics like Calculus of variation and Optimization. Do you know books about it?
or what do you suggest me?
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0answers
149 views
How to find the extremals of the functional?
How to find the extremals of the functional by Wolfram Mathematica?
$$J(y(x))=\int_{0}^{1}\big(y(x)^3+y'(x)\big)dx$$
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0answers
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Get the parametric equations of the Cycloid
How to get the parametric equations of the cycloid from the following differential equation?
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0answers
126 views
Symbolic second variation (quadratic form matrix)
Mathematica has a calculus of variations package that can compute the first variational derivative symbolically, rather nicely.
Does anyone know if there is a way to compute the quadratic form matrix ...
0
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0answers
59 views
VariationalBound with and interpolated trial function
I am trying to solve a Sturm-Liouville problem using the VariationalBound function of Mathematica, as recomended in the wolfram mathworld site. The example given in the Mathematica documentation is ...
3
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1answer
55 views
NVariationalBound
I am trying to use the NVariationalBound to find the eigenvalues and eigenfunctions.
NVariationalBound documentation example ...
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1answer
200 views
Help to solve the brachystochrone problem using Euler equations
I'm trying to find the soltution of the brachystochrone problem. I'm was wondering if there is a way to solve it directly using Mathematica. The problem is the following:
I have to find the equation ...
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0answers
224 views
Variating the derivative of the metric using xTensor
I would like Mathematica to be able to use the following expression in a more complicated computation:
\begin{equation}
\tag{1}\frac{\delta{}g^{\alpha\beta}_{\;{},\gamma}\left(x\right)}{\delta{}g^{\mu\...
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1answer
135 views
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0answers
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Compute second functional derivative with VariationalD?
I have problems to compute the second functional derivative of a general function. The following lines generate the first derivative:
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2answers
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Lagrangian to Hamiltonian
I want to go from Lagrangian description to Hamiltonian one. Using the example given by Mathematica I do something like:
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1answer
206 views
NIntegrate (Helium Singlet and Triplet) [closed]
I have two integrals that I am calculating using nested NIntegrate. One is for the singlet helium atom and another for triplet. (here, I am calculating variationally the approximate energies and ...
8
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1answer
956 views
geodesics' on torus
Is it possible to draw geodesics between the points in a path on a torus - toroidal surface?
geodesics: generalization of the notion of a "straight line" to "curved spaces"
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0answers
250 views
Optimizing a functional using variational calculus
Here I am, trying to get the trajectory $(x(t),y(t))$ that will minimize the following Lagrangian (i.e. the integrand of the functional) between $(-1,-1)$ and $(-1,y_{eff})$ where $x_{eff}$ is defined ...
4
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1answer
285 views
Natural boundary conditions variational methods
I am working on a problem of the calculus of variations. From the Variational Methods package, I can very conveniently use EulerEquations to get stationarity ...
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0answers
438 views
Minimizing a functional takes forever
I need help with minimizing a functional in Mathematica.
I have a function $V(\xi)=\sum_{i=1}^\infty C_{3_i}J_0(\xi/C_i)+C_{4_i}Y_0(\xi/C_i),~~~\Sigma\leq\xi\leq\xi_1$, and want to find such $\{C_{...
