Questions tagged [variational-calculus]
The variational-calculus tag has no usage guidance.
61
questions
2
votes
2answers
84 views
Minimization problem in proving Fermat's principle
I am trying to prove the Fermat's Principle of least distance, the problem statement goes:
...
0
votes
2answers
274 views
Plot of the derivative of a function
I have a function functionSL as a function of t (t<0) where I want to find the extremum ...
4
votes
1answer
96 views
Use of variational operator, 𝛿 in Mathematica
I am working in Hamilton's principle.
Part of deriving the equation of motion is to use the delta operator (𝛿) which can be operated just like a differential operator.
It is not the function ...
1
vote
2answers
56 views
Do not evaluate function composition in total derivative
How to achieve that Dt[] does not touch certain function compositions, but still applies all the other derivative transformations. Example:
...
4
votes
1answer
198 views
Finding a possible Lagrangian corresponding to a differential equation
A typical problem in the calculus of variations is to extremize a functional
$$ J[y]=\int f(x,y,y') \, \mathrm{d} x.$$
This usually involves solving the Euler-Lagrange equation
$$\frac{\mathrm{d}}{\...
2
votes
1answer
93 views
How do I find the equations of motions for a relativistic particle?
I have it working for the Lagrangian of a classical particle in a gravitational potential:
...
1
vote
1answer
120 views
How to use the variational method to solve this problem
I see this mechanical problem here.
I want to solve this problem with the variational method. The Lagrangian of this system is obtained by subtracting potential energy from kinetic energy.
...
asked Jun 6 '20 at 9:07
A little mouse on the pampas
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9
votes
1answer
536 views
Euler-Lagrange equation with a damping term
We know that there is a function VariationalMethods`EulerEquations readily handling the Euler-Lagrange equation of the "standard" form
$$
\frac{\partial L}...
1
vote
0answers
37 views
Tackling variational problem with constraints [closed]
I need to minimize (preferably symbolically) functional with constraints:
$$
F = \int_{0}^{\pi/2} f(x)cos{x}dx\,\\
1.1 \geq f(x) \geq 0\\
2.\int_{0}^{\pi/2} f(x)dx=Const\\
3. f(0) =1\\
4.f(\pi/2) = 0\...
4
votes
1answer
307 views
Find geodesics given two points
Suppose I want to find the geodesic on a paraboloid going through two points located on the paraboloid, using Mathematica 12.
I chose the paraboloid parametrization as follows:
...
0
votes
1answer
76 views
How do I solve this variational problem?
I have to find the function $\rho(r)$ that extremizes the functional $F$:
...
1
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2answers
149 views
How to find the variational result of this functional according to the definition of textbook
I see this variational problem here.
The functional is :
$$J(y)=y^{2}(x_{0})+\int_{x_{0}}^{x_{1}}(xy+y'^{2}) dx$$
In the textbook, the result of finding the functional variation according to the ...
asked Feb 20 '20 at 23:24
A little mouse on the pampas
5,1602 gold badges9 silver badges27 bronze badges
1
vote
1answer
94 views
How to find the variation of this functional according to the definition of Lagrange
The functional is :
$$J(y)=y^{2}(x_{0})+\int_{x_{0}}^{x_{1}}(xy+y'^{2}) dx$$
In the textbook, the result of finding the functional variation according to the functional variation defined by Lagrange ...
0
votes
1answer
98 views
How to find Euler equation of complex function by the textbook definition
In the help document, we know that EulerEquations[y[x] Sqrt[1 + Derivative[1][y][x]^2], y[x], x] can solve the following Euler ...
asked Feb 17 '20 at 2:39
A little mouse on the pampas
5,1602 gold badges9 silver badges27 bronze badges
1
vote
1answer
152 views
How to solve the differential equation of a Brachistochrone Problem
This is a differential equation to solve the most brachistochrone line, but it can't find the exact analytical solution :
...
asked Feb 14 '20 at 0:49
A little mouse on the pampas
5,1602 gold badges9 silver badges27 bronze badges
2
votes
1answer
172 views
How to interpolate {{xi,yi},… ,{xn,yn}} with minimum bending energy?
Thin Plate Splines are explained here, and a Mathematica implementation of that for real samples { { x1, y2, z1 }, { x2, y2, z2 }, ... , { xn, yn, zn } } is ...
0
votes
0answers
103 views
NDSolve with Functions Depending on time and time dependent Functions
EDIT:
following the advice of using Raw Input Form, the equations have the following form
...
2
votes
1answer
313 views
How to compute functional derivative against vector in xAct, xTensor or xTras?
I want to compute the functional derivative against vectors.
For example, I have an object that looks like this
$R = h_{ijkl}a^i a^j a^k a^l$
I need to compute $\frac{\delta R}{\delta a^p}= 4 h_{...
0
votes
0answers
99 views
Minimal Surface of Revolution - Integrate Challenge
I'm trying to use Mathematica to go from equation (11) to equation (12) in this example of a Minimal Surface of Revolution. There is a MMA notebook on that link, but it only shows how to find the ...
2
votes
0answers
197 views
Solving for a minimum action path in mathematica
I am interested in computing a quantity that is mathematically defined as follows,
$$\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$$...
1
vote
1answer
151 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
votes
0answers
51 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
43 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
100 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
1answer
74 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
296 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
42 views
0
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0answers
334 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 ...
6
votes
1answer
253 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
826 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
333 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 $$
Where $ F(r) = 1 - ...
2
votes
3answers
656 views
Use of variational operator, 𝛿 in Wolfram Mathematica (Version 11.0) [closed]
I have been working on a problem which requires the use of Energy Methods which is based on the variational principles. I am finding it hard to replicate my formulated equations in Mathematica.
How ...
1
vote
1answer
387 views
Does Mathematica understand the concept of infinitesimal increment?
Commonly, in textbooks, differential of differential increment vanishes, good example is how Euler-Lagrange equations are derived using the differential (from step 2 to step 3 here, you can see ...
4
votes
1answer
423 views
How to take derivative of a expression with respect to a function?
I want to program the Euler-Lagrange equations for continuous systems.
But in this formulation I have to compute the derivative of the Lagrangian with respect to all the derivatives of the generalized ...
0
votes
1answer
100 views
VariationalD of Nested Integral
I am trying to calculate the variational derivative with respect to f of the the nested integral below
$$ \mathcal{J}\left[f(x) \right] = \int\limits_a^b \text{d} x \,e^{-f(x)} \int\limits_a^x \text{...
2
votes
0answers
130 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
326 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
124 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{...
2
votes
1answer
476 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
votes
0answers
129 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
518 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 ...
4
votes
1answer
425 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 ...
5
votes
1answer
246 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
416 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
1k 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
$f(x,y)= 2x^3 +6xy^2 −3y^3 −150x$
3
votes
0answers
133 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
211 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
120 views
Get the parametric equations of the Cycloid
How to get the parametric equations of the cycloid from the following differential equation?
...
3
votes
0answers
179 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
72 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 ...
