Questions tagged [variational-calculus]
The variational-calculus tag has no usage guidance.
46
questions
0
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0answers
27 views
Using VariationalD on previously defined functions [closed]
Do you know whether there is a way to get around the following problem?
Say we define a very simple function:
f[t] = 2 y[t]
and want to use the ...
0
votes
0answers
57 views
NDSolve with Functions Depending on time and time dependent Functions
EDIT:
following the advice of using Raw Input Form, the equations have the following form
...
2
votes
1answer
37 views
How to compute functional derivative against vector in xAct, xTensor or xTras?
I want to compute the functional derivative against vectors.
For example, I have an object that looks like this
$R = h_{ijkl}a^i a^j a^k a^l$
I need to compute $\frac{\delta R}{\delta a^p}= 4 h_{...
0
votes
0answers
81 views
Minimal Surface of Revolution - Integrate Challenge
I'm trying to use Mathematica to go from equation (11) to equation (12) in this example of a Minimal Surface of Revolution. There is a MMA notebook on that link, but it only shows how to find the ...
2
votes
0answers
152 views
Solving for a minimum action path in mathematica
I am interested in computing a quantity that is mathematically defined as follows,
$$\phi(x_1,x_2) = \inf_{T>0} \inf_{\gamma \in C^{x_2}_{x_1 }(0,T)} \int_{0}^T L(\gamma, \dot{\gamma})\mathrm{d}t$$...
2
votes
1answer
137 views
$\min \int_{0}^{\infty} (a f'(x)^2+b f(x)^2+c |f(x)|) dx$
Can Mathematica solve
$\min_{f(x) s.t. f(0)=f_0} \int_{0}^{\infty} (a f'(x)^2+b f(x)^2+c |f(x)|$)dx ?
I tried
...
0
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0answers
38 views
Problems with the VariationalD (VariationalMethods package)
I'm facing the following problem: The VariationalD function doens't commute with the series expasion. Physically this means that if I compute the equation of motion starting with an Hamiltonian with a ...
0
votes
0answers
35 views
How to include `TimeConstraint` in `EulerEquations`? Is there any error here?
There is a complicated statement which takes a long time to compute.
IS THERE ANY ERROR IN THE ATTACHED PICTURE? If YES, how can I include TimeConstraint of ...
2
votes
0answers
72 views
How to remove the Riemann tensor derivatives using Bianchi identity?
For the third-order Lovelock gravity, after varying the Lagrangian versus the metric, I found some derivatives of the Riemann tensor which should not be appeared. How can I remove them? Maybe using ...
0
votes
0answers
23 views
How to get the boundary terms of a physical system using variational package?
I have a mechanical system as shown in the figure. I have extracted the kinetic and potential energy of the system. Using variational package I can able to get only the governing differential ...
0
votes
1answer
58 views
Could not able to set up a differential equations using EulerEquations command
I have a constructed the Lagrangian of a mechanical system by finding the kinetic and potential energy of the system. Then I tried using EulerEquations to set up a differential equation by taking the ...
1
vote
1answer
122 views
Increasing the time constraint on EulerEquations [closed]
How can I set the Infinity time constraint in EulerEquations?
Actually, there is an error I don't have any idea about. Can you ...
0
votes
1answer
36 views
0
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0answers
179 views
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
139 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 ...
11
votes
3answers
552 views
Minimum energy path of a potential energy surface
I have calculated the energies of a molecule by varying two parameters (Data points are here data). I want to find the minimum energy path from the point (180.0, 179.99, 21.132) to (124.5, 124.49, 0) ...
5
votes
2answers
286 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 - ...
2
votes
3answers
315 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 ...
1
vote
1answer
247 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 ...
0
votes
1answer
86 views
VariationalD of Nested Integral
I am trying to calculate the variational derivative with respect to f of the the nested integral below
$$ \mathcal{J}\left[f(x) \right] = \int\limits_a^b \text{d} x \,e^{-f(x)} \int\limits_a^x \text{...
2
votes
0answers
85 views
Variational derivative of a functional depending on multiple functions
I wanted to use the VariationalD command to calculate a list of variational derivatives.
A simple example is the following and it works perfectly fine.
...
0
votes
1answer
169 views
How to do variation of an integration of a function with consideration of limits?
I have a function dependent on w(x,t) which is integrated in an interval. After integration of this function, I want to apply variation followed by integral by parts. Integral by parts results in to ...
1
vote
0answers
115 views
Could I take a variation of a functional in Mathematica? [closed]
In finite element, the weak form can be derived by taking the variation of a energy functional as:
\begin{align}
L(u(x,y)) &= -\dfrac{1}{2}\int_{\Omega}(\dfrac{\partial{u}}{\partial{x}}\dfrac{...
0
votes
1answer
368 views
Drawing geodesic lines on the membrane (Plot3D)
The most common method of creating patterns on a membrane involves drawing geodesic lines on the membrane. Justification of the layout of the geodesic lines is beyond the scope of this text .
...
0
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0answers
107 views
VariationalD with respect to two-variable function of integral over one variable
I have a function $q(x,t)=u(x,t)+iv(x,t),$ some functions $p_n$ and I want to find the following functional derivative:
$$\frac{\delta}{\delta\overline{q}}\int p_ndt$$
Because Mathematica's ...
3
votes
1answer
357 views
Legendre Transform of a function of a 3-vector
I am trying to implement the Legendre transform of a function in Mathematica with the purpose of calculating the Hamiltonian of a system starting from the Lagrangian.
I have found this page which ...
3
votes
1answer
349 views
How to formulate the Hamiltonian equations of motion? Part II
I've recently asked a question on an issue I was facing with numerically integrating Hamiltonian equations of motion. I got a great answer.
Following on from this, I wanted to write a very similar ...
4
votes
1answer
215 views
Numerically Integrating a Hamiltonian but getting different results when compared with alternative equivalent equations
currently trying to perform an integration of the Hamiltonian for a Schwarzschild black hole. However, I'm coming up short.
The code is as follows:
...
2
votes
1answer
364 views
how to find a function that minimizes the other function with a known function
The problem that i am facing is that of energy minimization. I have a function called totalenergy which is sum of two functions the elastic energy and the surface ...
1
vote
0answers
530 views
how to find stationary point of a multivariable function [closed]
I am quite new, how can I find minumum, maximum or saddle point the following multivariable function by using mathematica
$f(x,y)= 2x^3 +6xy^2 −3y^3 −150x$
2
votes
0answers
117 views
Book Recommendations About Using Mathematica in Topics like Calculus of Variation and Optimization?
I want to use effectively Mathematica about topics like Calculus of variation and Optimization. Do you know books about it?
or what do you suggest me?
1
vote
0answers
168 views
How to find the extremals of the functional?
How to find the extremals of the functional by Wolfram Mathematica?
$$J(y(x))=\int_{0}^{1}\big(y(x)^3+y'(x)\big)dx$$
2
votes
0answers
106 views
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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2
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0answers
147 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
votes
0answers
67 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
votes
1answer
63 views
NVariationalBound
I am trying to use the NVariationalBound to find the eigenvalues and eigenfunctions.
NVariationalBound documentation example ...
2
votes
2answers
285 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 ...
6
votes
0answers
267 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\...
1
vote
1answer
174 views
1
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0answers
217 views
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:
...
8
votes
2answers
1k views
Lagrangian to Hamiltonian
I want to go from Lagrangian description to Hamiltonian one. Using the example given by Mathematica I do something like:
...
1
vote
1answer
230 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 ...
10
votes
2answers
1k views
Geodesics on a 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"
...
1
vote
0answers
264 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 ...
6
votes
1answer
347 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 ...
1
vote
0answers
470 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_{...
