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Questions tagged [variational-calculus]

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5
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2answers
192 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 - ...
1
vote
3answers
128 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
139 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
76 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
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0answers
54 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
118 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
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0answers
99 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
248 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
75 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
257 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
259 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
171 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
295 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
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0answers
278 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
87 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
139 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
86 views

Get the parametric equations of the Cycloid

How to get the parametric equations of the cycloid from the following differential equation? ...
2
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0answers
117 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
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
votes
1answer
55 views

NVariationalBound

I am trying to use the NVariationalBound to find the eigenvalues and eigenfunctions. NVariationalBound documentation example ...
2
votes
1answer
191 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
202 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
131 views
1
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0answers
189 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
197 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
votes
1answer
921 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" ...
1
vote
0answers
243 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
votes
1answer
277 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
423 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_{...