Questions tagged [euler-lagrange-equation]

In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.

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I have a Variation of Calculus, Euler Lagrange, Lagrange Multiplier problem and I don't know how to continue.

So I can't continue with my work. Don't know if I am just not seeing something under my nose or something. Given formula: $$I=\int\limits_0^{\pi/2}[(x'_2)^2-(x_2)^2]dt$$ with auxiliary constraints and ...
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What is the most generic way to write a Lagrangian quadratic in velocities?

I'm trying to generalize the expression for a Lagrangian to a manifold that doesn't posses a metric. To be more clear, when the configurations space has a metric, we write the lagrangian quadratic in ...
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Find curve that minimizes lenght, with integral constraint

I'm interested in finding the curve $q(t):[0,1] \rightarrow \mathbb{R}^+$ that satisfies the boundary conditions $q(0)=q(1)=0$, the integral condition $\int_0^1q(t)dt=a>0$, and that minimizes the ...
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Taking the partial derivative of both sides of an equation [duplicate]

I have this function: $G(x, y, z) = G(x, y, g(x,y))$ and the equation $$G = 0$$ I want to reach a specific equation: $$\frac{\partial{G}}{\partial{y}} + \frac{\partial{G}}{\partial{g}}*\frac{\partial{...
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Convert the pendulum differential equations of a second order into a first order system [closed]

I have this system of two differential equations of a second order. I got them from the Euler-Lagrange equations of double pendulum. I need to solve this using the Runge-Kutta numerical method, but my ...
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Euler-Lagrange equation, derivative calculation

Let we have $F(u)=\int_A\mathcal{L}(x,u,u')dx$, where $dx$ is Lebesgue measure, $A$ is open and bounded with regular boundary, $\mathcal{L}\in\mathcal{C}^2(A\times\mathbb{R}\times\mathbb{R}^n)$ and ...
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How can I find a function that minimizes this given cost function?

I am trying to find a function $f(x)$ that minimizes the following cost function $$E = \left(\int_{-p}^p{{x^2\mathcal{N}(x)}f(x)}dx-\epsilon\right)^2$$ With $\epsilon\geq 0$ and $\mathcal{N}(x)$ a ...
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Euler-Lagrange Variational Calculus extreme condition for generic functional

I need to do the following : I saw for a simpler case, but I didn't understand the passage indicated by an arrow in the image below . Why did the top integral become the bottom subtraction? Now, I ...
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How to solve $L(u,v)=\int_{-1}^8\sqrt{u'^2+v'^2}dx$

I am trying to solve this functional: $L(u,v)=\int_{-1}^8\sqrt{u'^2+v'^2}dx$ where the points $(u,v)$ on BC: $-1<x<8$ I use the Euler Lagrange formula and get two equations, considering $u$ and $...
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Extremals of $\int_0^\pi(y')^2dx$

I am trying to show that $\int_0^\pi(y')^2dx$ with $y(0) = 0$, $y(\pi) = 0$ has infinitely many extremals subject to the constraint $\int_0^\pi y^2 dx = \pi/2$. I know that the first-order necessary ...
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Constrained variational calculus: find extremums of $\int_0^\infty ay(t)^2 + by'(t)^2 + f(t)y(t) \ \text{d}t$ subject to $0 \leq y(t)\leq k$

I wish to find extremums of a functional $J[y]$ that is given by $$ J[y] =\int_0^\infty ay(t)^2 + by'(t)^2 + f(t)y(t) \ \text{d}t \hspace{0.5cm} \text{subject to} \hspace{0.5cm} 0 \leq y(t)\leq k $$ ...
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Solving the Euler Lagrange equation for the functional $L(y)=\int_{-1}^7\sqrt{1+y'^2} dx$

I want to solve this functional \begin{equation} L(y)=\int_{-1}^7\sqrt{1+y'^2} \ dx \end{equation} with IC: $y(0)=1,\ y(1)=2$ I start using the Euler Lagrange equation \begin{equation} \frac{d}{dx}\...
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Saddle Point Problem and Lagrangian

Let $X$, $M$ be Hilbert spaces. Consider two bilinear forms $a(\cdot,\cdot):X \times X\to \mathbb{R}$ and $b(\cdot,\cdot): X\times M\to \mathbb{R}$, and linear maps $f:X\to \mathbb{R}$, $g:M\to\mathbb{...
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Second-order Euler-Lagrange equations on homogeneous spaces

Consider a Lie Group $G$ with a Lie subgroup $H$, and the resulting Homogeneous space $M := G/H$ with canonical projection map $\pi$. Suppose we have a Lagrangian $L: TM \to \mathbb{R}$ and consider ...
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Formal definition of virtual displacement

I'm trying to read a book on classical mechanics, and I'm having a hard time trying to know what exactly is a virtual displacement, sometimes called a variation. In the Lagrangian "formalism"...
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Euler-Lagrange equation for the free scalar field

I'm looking at the Lagrangian density for the free scalar field: $ \mathcal{L} = \frac{1}{2} ( \partial_{\mu} \phi \partial^{\mu} \phi - m^2 \phi)$ and I'm trying to figure out how to write down the ...
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Finding the path of a light ray using differential geometry

Hi I am trying to solve a calculus of variations problem I would like to solve it using differential geometric approach but i am not sure. As an example how would I go back doing it for the example ...
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what to make out of it when Euler-Lagrange Equation is constantly zero

I'm new to Calculus of Variations, and I'm trying to apply it to a simple vector calculus problem. Let's consider finding a curve $C$ along which the work $W$ done by a given vector field $\textbf{F}$ ...
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Euler-Lagrange equation w.r.t. compact variation

I follow the lecture of Calculus of variations, the topic is Euler-Lagrange's equations. $$ \mathcal{L}: \ A\times\mathbb{R}\times\mathbb{R}^n\to \mathbb{R}, \ x\in A, \ u\in \mathbb{R}, \ \xi\in \...
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How to solve this functional $\int\int_\Omega \big((z_x)^2+(z_y)^2\big)dxdy$

Let $\Omega$ define the quadrant $0\leq x\leq L$, $0\leq y\leq L$. For $z=z(x,y)$ we want to solve Eulers equation for the functional: $\int\int_\Omega \big((z_x)^2+(z_y)^2\big)dxdy$ where $z=0$ along ...
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Why exactly do we write $\mathcal{L}(x(t),x'(t),t)$ instead of simply $\mathcal{L}(x(t),t)$?

I have many times seen Lagrangian written as $\mathcal{L}(x(t),x'(t),t)$. I undestand that this is a function of $x(t)$, $x'(t)$ and $t$. So it can theoretically look something like $$\mathcal{L}(x(t),...
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Can the geodesic equation be used to solve the Brachistochrone Problem?

Assume the initial condition is that a point mass starts at height $y_0$. After descending to height $y < y_0$, we know that its speed will be $v = \sqrt{2mg(y_0 - y)}$. Thus, the displacement ...
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Inverse function in an Euler-Lagrange Equation

I'm interested in minimizing the functional $I[f]=\int_{a}^{b} \mathcal{L}(x,f,f',f'',f^{-1},(f^{-1})', (f^{-1})'') dx$. Would it be allowable to consider that $f^{-1}$ is some function $g$ and apply ...
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Capturing $\infty$ endpoint and additional boundary conditions

I'm interested in minimizing the functional $I[f]=\int_{a}^{\infty} J(x,f(x),f'(x)) dx$ but the boundary conditions that I have on $f$ are a bit "weird": one endpoint is fixed in the ...
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Solving a system of second order ODEs with initial conditions.

Essentially, this is the tail end of a problem I have. The result of which gave two second order ODEs: $u'' = λv$ $v'' = λu$ Where $u(0) = 0$, $u(1) = 1$, $v(0) = 0$, $v(0) = 0$ and $\int_{0}^{1}uv \...
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Order of an Euler-Lagrange equation always even

If we look at the energy functional $$E(u) = \int_\Omega L(t, u(t), u'(t), \dots, u^{(n)}(t)) dt, $$ the Euler-Lagrange equation is defined as $$\frac{\partial L}{\partial u} + \sum_{k = 1}^{(n)} (-1)...
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Functional Poincaré Lemma for second-order PDE $\Delta u(x) + F(\nabla u(x))=0$?

Suppose I can write a second-order elliptic PDE for an unknown function $u:\mathbb R^n\to\mathbb R$ in the form $$\Delta u(x) + F(\nabla u(x))=0\qquad\forall x\in\mathbb R^n.$$ Under what conditions ...
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Euler-Lagrange equation - cannot find mistake

I have a problem at hand that I'm trying to solve with the Euler-Lagrange equation, but I think I made a mistake somewhere. Supposed I have two functions on two real variables, $u(x, t)$ and $v(x,t)$. ...
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Decide whether the extremal value of $I[y]$ is maximum or minimum

My question asks me to find $y(t)$ such that the integral \begin{equation} I[y] = \int_0^1 (dy/dx)^2 dx. \end{equation} is extremised subject to the constraint \begin{equation} \int x(dy/dx) dx = 0 \...
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Derivative of the action functional

I am self studying the book An Brief Introduction to Physics for Mathematicians. At the second page, the equation (1.6) says that the derivative of the action functional can be derived as $$ S'(x)(h) =...
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How to show that the extremal of a functional is not its minimizer?

In a set of practice problems I have a question that states: Show that the extremal for $E[y]=\int^1_0\cos^2(y'(x))dx$ with $y(0)=0, y(1)=1$ is not the minimizer over $D^1(0, 1)$. To be honest I don't ...
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Solving a calculus of variations problem is this term irrelevant?

I am trying to find the extremals for the following expression: $$L(x, y(x), y'(x)) = y^2 + y'^2 -2x\sin(x)$$ Since that last term does not depend on either $y$ nor $y'$ it should not matter since it ...
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Understanding Euler-Lagrange equation

So how much I understood it, it gives us the extremum of a functional. So we use it in isoperimetric problems too (just dealing with 2D problems). So there as we know we have to find the curve which ...
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Why does this term in the Euler-Lagrange equation vanish?

I am reading Evans, in the book it is stated that we define the following function $$i(\tau) = \int_U L(Du + \tau Dv, u + \tau v, x)dx$$ Then we compute: $$i'(\tau) = \int_U L_{p_i}(Du + \tau Dv, u + \...
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How to use Calculus of Variation with Lagrange Multipliers when the marginal conditions only depend on the multipliers?

I am trying to solve an optimization problem: $$\begin{aligned}&\max_{G(v)}\int_p^1 \frac{G(v)}{v} dv\\ s.t.& \int_0^1(1+\log u)G'(u)du=-\overline v<0\\ &\int_0^x(1-\frac{G(v)}{v})dv\...
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Linear odd order PDE equations as an Euler-Lagrange equation

If I have an energy of the form: $E(u) = \int_{\Omega}\|D^mu(x)\|^2\,dx$ then I get the Euler-Lagrange equations: $\Delta^m u(x) = 0$. Where $D^m$ is the $m$-dimensional tensor of partial derivatives ...
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How can this formula be reached from the Euler Lagrange equation?

In some lecture slides for a class I am taking, this is stated: I don't see how this is true. The E-L equation merely tells us that at extrema (i.e. the minimum, which is an attained infimum): $$\...
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Is there an intuition behind the Euler Lagrange equation?

I am taking calculus of variations at the moment and I am curious if there is a visualization of why the EL must be satisfied for all extremals. At first glance it's hard for me to relate: $$\frac{d}{...
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Differentiate a unknown function

I been reading The Theoretical Minimum, and I got to symmetries. A potential is defined as $$ V(q_1,q_2)=V(aq_1-b_2q_2) $$ and, using the Euler-Lagrangian equation $\dot{p_i}=\frac{\partial \mathcal{L}...
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1 vote
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Euler-Lagrange equation for a constant generalized coordinate

I have an action integral that I want to optimize of the form $$ S = \int_0^T \left( a(t)q^2 -2b(t)q + \gamma \dot{q}^2 \right) dt \,. $$ I know that in fact $\dot{q} = 0$, in which case the critical ...
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Lagrangian not solvable. Or is it?

Hey there helpful humans! I am stuck on trying to solve an equation to a variable H2. I derived the the equation by combining the different FOCs from the Lagrange. It's the third day and I am so ...
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1 vote
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Function whose minimum is independent of the input?

I have a question that asks me to show that the functional $$I(x) = \int^{x_2}_{x_1} (x^2 + 3y^2)y' + 2xydx$$ has an extremal that is independent of the choice of $y$ that joins two arbitrary points $(...
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1 vote
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Self-student variational calculus - the maxima and minima simplest problem

I am starting a self-taught study in Variational Calculus. The first thing I see in every book is the Euler-Lagrange equation. However, what comes to my mind as the first and simplest question of ...
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2 votes
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How to determine uniquely an extremal of a variational problem

This is a question from a competitive examination: Consider the functional $$ J(y) = \int_{a}^b (y+y^{'})dx $$ for admissible functions y, Then $J$ has: No extremals. Several extremals. $y(x)=e^{-x}$ ...
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Regularity theorem of minimizers

During my self study to the calculus of variations I come across this problem. Because of my search, I know what I wanted to do but I need some help to do them. The function $f:[-1,1] \times \mathbb R ...
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Euler-Lagrange equations from variational principle in 2D

I want to derive the Euler-Lagrange equations (ELE) from an action $$S=\int\int\mathcal{L}(x, y, z, \partial_y x, \partial_zx) dydz$$ where $x$ depends on two variables $x=x(y,z)$, imposing $\delta S=...
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1 vote
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Example of a Lagrangian satisfying all the conditions for existence and uniquess

I have been reading Evan's book on PDE's , namely the section on calculus of variations. The author then gives conditions so that we can have existence and uniqueness of a minimizer. Then later we ...
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4 votes
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Constraint in Lagrangian for mass spring system.

Suppose we have $p_1,...,p_n$ and we have springs attached to them. We know that the lagrangian is $$ L = T - U = \sum_{i=1}^n \frac{1}{2} m_i \dot{p}_i^2 - \sum_{(i,j) \in E} \frac{1}{2} k_{ij} \left(...
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Euler-Lagrange method example

I've been working on deriving dynamic equations of a "super articulated mechanical system (SAMS)" using the Euler-Lagrange method. I am probably making mistakes because I haven't done ...
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1 answer
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Why $u=1$ is not a minimizer of $I(u)=\int_{0}^{1}u^2(2-u)^2dx$?

I'm trying to find the minimizer of the following functional: $$I(u)=\int_{0}^{1}u^2(2-u)^2dx,$$ where $u\in Y,$ and $Y=\{u\in C([0,1]):u(0)=1,u(1)=1\}.$ The Euler-Lagrange equation of this question ...
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