Questions tagged [euler-lagrange-equation]

Question 1

Suppose I have the ellipsoid $$ \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1 $$ And I have two points on this surface, $P_1 = (x_1, y_1, z_1)$, and $P_2= (x_2, y_2, z_2) $. I am ...

Question 2

Starting with $$ I(u) := \int_{-1}^{1}{\left(x^2(u’)^2+x(u’)^3+(u’)^6\right)} {\rm d} x,$$ with boundary conditions $u(-1)=u(1)=0$ and using the Euler-Lagrange equation: $$\begin{align}\frac{d}{dx}\...

Question 3

Given the following functional with fixed parameters $a>0$ and $b>1$, $$F[x;u,u']: = \int_{1}^b\left(x^3-\frac{ax}{u^2}\right)(u')^2~\mathrm{d}x$$ Suppose the space of functions satisfy $u(1) = ...

Question 4

Having started to read Giaquinta/Hildebrandt "Calculus of Variations I", 2 ed. 2004, in order to get a better mathematical foundation for my understanding of the Hamilton principle of least ...

Question 5

This is a follow-up to the recent question of mine. Let $X$ be a $n$-dimensional configuration space. Consider a Lagrangian $L:TX\to\mathbb R$. A coordinate-free formulation of Euler-Lagrange ...

Question 6

Consider a Lagrangian $L=L(x, v, t)$ which is a smooth convex function from $TX\times\mathbb E$ to $\mathbb R$, where $X$ is a configuration space. The Euler-Lagrange equations read $$\frac{\partial L}...

Question 7

I am reading about the derivation of Euler-Lagrange equation. I don't understand why for $f[y(x)+\eta(x)\alpha, y'(x)+\eta'(x)\alpha,x]$ that $$\frac{\partial f}{\partial\alpha} = \eta\frac{\partial f}...

Question 8

In the context of Lagrangian Field Theory I am reading Quantum Fields and Strings: A Course for Mathematicians, Volume 1, Part 1, Classical Field Theory, chapter 2, wherein they denote $\mathcal{F}$ ...

Question 9

For the integral $$I =\int_a^b F(y', y, x) \, \mathrm dx$$ I’ve seen the requiremts expressed for the Euler Lagrange equation expressed in 2 different ways, but I do not see how they are equivalent. **...

Question 10

I am considering a Calculus of Variation problem about minimizing $I(x)=\int_{0}^{1} \phi(x(t),x'(t)) dt$, where $x(t)$ is in the space of absolutely continuous function on $[0,1]$. When the integrand ...

Question 11

I'm not currently a student; the material is being practiced purely for learning and independent research. I'm currently reading through Gelfand and Fomin's Calculus of Variations. Note that Gelfand ...

Question 12

This problem is adapted from Gelfand and Fomin's Calculus of Variations (Chapter 1, Problem 18). I'm not currently a student; the material is being practiced purely for learning and independent ...

Question 13

This problem is adapted from Gelfand and Fomin's Calculus of Variations (Chapter 1, Problem 18). I'm not currently a student; the material is being practiced purely for learning and independent ...

Question 14

The problem I am considering is the following: \begin{align*} &\max_{p = (p_1, p_2,\cdots, p_N)} \int_{v} \sum_{i=1}^{N} v_i p_i(v) dv - \int_{v} \sum_{i=1}^{N} \frac{ |...

Question 15

Consider the action integral $$ S[x] \;=\; \int_{t_0}^{t_1} L\bigl(x(t),\,\dot{x}(t),\,t\bigr)\,\mathrm{d}t, $$ where $L$ is the Lagrangian (for example, $T - V$ in mechanics). To compute the ...

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Questions tagged [euler-lagrange-equation]

Finding the minimum distance path on the surface of an ellipsoid [duplicate]

Minimise $\int_{-1}^{1}{\left (x^2(u’)^2+x(u’)^3+(u’)^6\right)}dx$ with $u(-1)=u(1)=0$

Does the functional have a minimizer?

Computing the first variation of a variational integral

Obtaining coordinate Euler-Lagrange equations from coordinate free Euler-Lagrange equations

Understanding the meaning of terms in Euler-Lagrange equation

Help in understanding a step in partial derivative of $f$

Leibniz rule for $\delta$ and $d$ on the Diffeological space $\mathcal{F} \times M$ in Lagrangian Field Theory

Requirements for Euler Lagrange Equations

Continuity of Lagrangian Multiplier function

Is there a simple, straightforward way to shift the Euler equation (and the function $F$) between coordinate systems?

What curve minimizes this integral when the values of $y$ are not specified at the end points?

Find the general solution of the Euler equation corresponding to the functional $\omega(y) = \int \left(f(x) \sqrt{1 + y'(x)} \right)dx$

Calculus of variation problem with vector-valued function and constraints

The integration by parts step in the derivation of the Euler–Lagrange equation

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