Questions tagged [variational-calculus]

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Finding Hamiltonian from a perturbative Action

I am given a perturbative action $$\frac{S}{\mathcal{T}}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+7.11 c_0^3+35.3 c_0 c_1^2+4.66 c_0 \dot{c_0}{}^2+1.32 c_0 \dot{c_1}{}^2-7.57 \dot{c_0} ...
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43 views

How to use VarD to vary $\partial T$

I know that how the VarD works which is $ VarD[T, covd][expr] $. What about when I want to vary to a partial of a tensor ($\partial T)$? Let me say a simple question I wanted to test: $ L = \partial_{...
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  • 57
9 votes
2 answers
195 views

Methods of Numerically Finding Function Minimizing Functional

Say we have some functional like the following: $H = (\partial_yf(y))^2 -w(y) f(y)^2 +f(y)^4/2$. This is the functional for the Gross Pitaevskii equation. Lets say $w(y)$, the trapping potential in ...
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3 votes
2 answers
119 views

Roots, multivariable functions and Mathematica [closed]

Let $f(x,y)=(10 x^2 + 4 x y - 2 x + 4 y^2 - 4 y + 1)^2 (32 x^2 - 64 x y + 24 x + 40 y^2 - 28 y + 5)^2$ ...
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2 votes
1 answer
87 views

How to get numerical value of functional derivative at specific point

I have a functional in the form: $$F[s(x)]=\int_0^1(s(x)-a(x))^2\mathrm{d}x$$ where $a(x)$ is a parameter function, and I would like to find the functional derivative of $F$ with respect to $s$. ...
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0 answers
85 views

Is an analytic solution possible?

I have this problem: the parameters are T1, T2, R, Rthhot, Rthcold, and Z with feasible range >0. The variables are Rth and Rload. X and Y are functions of Rload and Rth given by the 2 equations: <...
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  • 11
8 votes
0 answers
212 views

Shortest Distance between two points on a 2D surface

I am working on an interesting problem, that consists of trying to compute the shortest distance between two points on a 2D surface. This 2D surface can be represented in a 3D space by the function $f(...
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  • 397
5 votes
1 answer
200 views

Geodesics on arbitrary 2D surfaces

Geodesics on 2D Function Surfaces I am trying to approximately find the shortets path between two point on a surface defined as z=F[x,y]. I am solving for X,Y as function t. However, when trying to ...
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0 votes
0 answers
36 views

Solving constraint optimization problem for different intervals using VariationalD

I have been trying to solve for a constraint optimization problem (which eventually yields to a Laplace distribution). Consider the following equation (1) \begin{equation*} J=\int^{\infty}_{-\infty} p(...
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1 vote
1 answer
70 views

Using VariationalD, getting derivative of Implicit Functions, derivative independent variables [closed]

Hey I am trying to solve this problem with wolfram: I figured using VariationalD was the way. However I can not figure out how this package works. It would be nice if someone could help me. I ...
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2 votes
2 answers
95 views

Minimization problem in proving Fermat's principle

I am trying to prove the Fermat's Principle of least distance, the problem statement goes: ...
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0 votes
2 answers
291 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 ...
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4 votes
2 answers
167 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 ...
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1 vote
2 answers
77 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: ...
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4 votes
1 answer
377 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}}{\...
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2 votes
1 answer
118 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: ...
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1 vote
1 answer
187 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. ...
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9 votes
1 answer
1k 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}...
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1 vote
0 answers
46 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\...
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4 votes
1 answer
411 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: ...
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0 votes
1 answer
110 views

How do I solve this variational problem?

I have to find the function $\rho(r)$ that extremizes the functional $F$: ...
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1 vote
2 answers
175 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 ...
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1 vote
1 answer
119 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 ...
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0 votes
1 answer
108 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 ...
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1 vote
1 answer
207 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 : ...
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2 votes
1 answer
210 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 ...
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0 votes
0 answers
144 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 ...
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3 votes
1 answer
592 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_{...
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0 votes
0 answers
117 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 ...
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2 votes
0 answers
250 views

Solving for a minimum action path in mathematica

I am interested in computing a quantity that is mathematically defined as follows, $$\phi(x_1,x_2) = \inf_{T>0} \inf_{\gamma \in C^{x_2}_{x_1 }(0,T)} \int_{0}^T L(\gamma, \dot{\gamma})\mathrm{d}t$$...
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1 vote
1 answer
158 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 ...
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  • 274
0 votes
0 answers
75 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 ...
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0 answers
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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 ...
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2 votes
0 answers
125 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 ...
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0 votes
1 answer
86 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 ...
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  • 1,649
1 vote
1 answer
416 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 ...
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0 votes
1 answer
55 views

Defining an expression to insert into NDSolve

Consider the follwoing Code: ...
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0 votes
0 answers
385 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 ...
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9 votes
1 answer
385 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 ...
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  • 101
11 votes
3 answers
1k 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) ...
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6 votes
2 answers
392 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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  • 309
2 votes
3 answers
921 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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1 vote
1 answer
493 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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  • 113
4 votes
1 answer
639 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 ...
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0 votes
1 answer
120 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{...
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2 votes
0 answers
163 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. ...
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  • 4,113
0 votes
1 answer
447 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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1 vote
0 answers
133 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{...
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  • 193
2 votes
1 answer
527 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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0 votes
0 answers
145 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 ...
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