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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.

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. Reference: Wikipedia.

It was developed by Swiss-Russian mathematician Leonhard Euler and French-Italian mathematician Joseph-Louis Lagrange in the 1750s.

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Defining the Functions for Derivation of Euler-Lagrange Equation

I'm having trouble wrapping my head around the functions involved in the derivation of the Euler-Lagrange equation. Although, as a sidenote, I'm deriving it by trying to prove that the shortest path between two points ($x_a$ and $x_b$) on a plane…
mouldyfart
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Euler-Lagrange of $ L(t,x^i,\dot{x}^i) = B(t,x^i,\dot{x}^i) + A_j(x^i)\dot{x}^j $

If $E(L)$ is the Euler-Lagrange of $L$ and $$ L(t,x^i,\dot{x}^i) = B(t,x^i,\dot{x}^i) + A_j(x^i)\dot{x}^j $$ and $$ \frac{\partial A_j}{\partial x^j} = \frac{\partial A_j}{\partial x^i} $$ show that $$ E(L) = E(B) $$ for $n=1$. Firstly, am I right…
strider
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Dealing with total derivative in Euler Equation

I am trying to solve this: $I = \int_0^1(x^2-y^2-(y')^2)$ using the euler equation: $\frac{d}{dx}[\frac{\partial F}{\partial y'}]-\frac{\partial F}{\partial y} =0$ and find the function y(x). So, I have: $\frac{d}{dx}[2y']-2y=0$. How do I deal with…
CINA
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Deriving the Euler-Lagrange equations for a loaded and clamped plate.

This is a homework question so giving the full answer is not the intention. Rather, I am looking for a hint. I am asked to minimise the functional $F[u] = \int_\Omega\: \frac{1}{2} E(x,y) \left( \Delta u\right)^2 + q(x,y) u \: d\Omega$. Here,…
Daimonie
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solve Euler Lagrange equation when $f(u,z)=u^2+z^2$

Write down the Euler-Lagrange equation for the following problem: $$ \min { \int_0^1 f(u(x),u'(x))dx; \quad u(0)=0, u(1)=1 } $$ Solve this Euler-Lagrange equation when $f(u,z) = z^2+u^2$ I have gotten the Euler-Lagrange into the…
Sam Houston
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Confusion over equation of motion from lagrangian

I'm currently reading Topological Solitons by Manton & Sutcliffe and am having a bit of trouble with deriving an equation of motion. Suppose $M$ is a smooth manifold of dimension $D$ and $\mathbf{q}(t)=(q^1(t),...,q^D(t))$ is a smooth trajectory on…
Adam Rafferty
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Shortest Path Between 2 Points Wiki Proof

This is a very simple question. I'm stuck on one line in the Wiki proof for proving the shortest distance between two points is a line using the Euler-Lagrange equation. The proof can be found hereunder the "Example" heading. Basically, I follow…
k12345
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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 the contour $\partial\Omega$, under the…
Luthier415Hz
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Any hint to solve Euler–Lagrange equation with Lagrangian $L=\sqrt{\frac{\dot{x}}{1-x^{2}}+(1-x^2)\dot{y}}$?

I encounter it in a textbook with no solution. I calculated it and got a very complicated equation and cannot derive a decoupled first order set of equations from them. I would appreciate any hint about it. Thanks in advance.
YiPing
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A Lagrangian that is a vector... Can I apply the Euler Lagrangian equations to get equations of motion?

In physics, one only sees Lagrangian that are scalar-valued: $$ L:t,q,\dot{q}\to \mathbb{R} $$ whose integral over time is the action $$ S=\int_0^tL(t,q,\dot{q})dt $$ In my own amusement research I have ended up with a Lagrangian that is vector…
Anon21
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How to show that two Euler-Lagrange-equations have the same solution?

Let $\phi:\mathbb R^n \to \mathbb R$ be a twice continuously differentiable function and let $$L_{\phi}(t,x,\dot x) =\nabla\phi(x)^T\dot x = \sum_i\frac{\partial \phi}{\partial x_i}(x_1,..,x_n)\dot x_i.$$ Let $L:[a,b] \times \mathbb R^n \times…
Tesla
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Trouble with the proof for Nielsen's form of Lagrange's equation.

I just cannot understand how the step transition happened. Please please help.
PCeltide
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Obtaining system dynamic equation using Lagrangian 's approach?

I have the following problem related to my thesis which is a 3dof mechanical system. In this system, there are two masses ($m_1,m_2$) that are linked with weightless two links ($l_1,l_2$). Two joints have consisted of spring-damper systems which can…
Piyumal Samarathunga
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Can an integrable system not have a lagrangian?

It is known that some systems are non-integrable, but have a Lagrangian, e.g. the 3-body orbit problem. Now considering the converse, is there an example of an integrable system that does not have a Lagrangian (does not need to be a mechanical…
Feel My Black Hole
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euler-lagrange equation expansion

Euler-Lagrange equation $$\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial y'} = 0$$ Can also be written as $$f'_y-f''_{xy'}-f''_{yy'}y'-f''_{y'y'}y''=0$$ In my book it is provided as a self-evident fact. It is not evident to me…
user3600124
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