Lagrange Definition and 542 Threads

Exercise statement: Given the action (note ##G_{ab}## is a symmetric matrix, i.e. ##G_{ba} = G_{ab}##): $$S = \int dt \Big( \sum_{ab} G_{ab} \dot q^a\dot q^b-V(q)\Big)$$ Show (using Euler Lagrange's equation) that the following equation holds: $$\ddot q^d +...

In the situation described in the problem, the mass is moving on a horizontal circular path with constant velocity. Wouldn’t this make L and U both constant? Then the Lagrange equation would give 0 = 0, which isn’t what I’m looking for. Any help would be appreciated.

I couldn't even get the position vector. Help!

The final answer should have a negative b^2⋅r(dot)^2⋅r term but I have no idea how that term would become negative. Also I know for a fact that my Lagrangian is correct.

I start out by substituting rcos(Θ) and rsin(Θ) for x and y respectively. This gives me z=(b/2)r^2. The Lagrangian of this system is (1/2)m(rdot^2+r^2⋅Θdot^2+zdot^2)-mgz. (rdot and such is the time derivative of said variable). I then find the time derivative of z, giving me zdot=br⋅rdot and...

Using Lagrange multiplier ##\lambda## (only one is needed) the integral to minimize becomes $$\int_{\tau_1}^{\tau_2} (y + \lambda) \sqrt{{x'}^2+{y'}^2} d \tau = \int_{\tau_1}^{\tau_2} F(x, x', y, y', \lambda, \tau) d\tau $$ Using E-L equations: $$\frac {\partial F}{\partial x} - \frac d {d \tau}...

Hey, I solved a problem about a double pendulum and got 2 euler-lagrange equations: 1) x''+y''+g/r*x=0 2) x''+y'' +g/r*y=0 (where x is actually a tetha and y=phi) the '' stand for the 2nd derivation after t, so you can see the basic harmonic oscillator equation with a term x'' or y'' that...

Talking about the Jupiter Lagrange points at 60 degrees only. Hard to imagine a scenario where an asteroid comes from outside or inside the orbit of Jupiter and stops at a Lagrange point. That's like tossing a cone on a table and trying to make it end up standing on its nose. Or make the nose...

The error ##e_{n}(y)## for ##\frac{1}{1-y}## is given by ##\frac{1}{(1-c)^{n+2}}y^{n+1}##. It follows that ##\frac{1}{1+y^2}=t_n(-y^2)+e_n(-y^2)## where ##t_n(y)## is the Taylor polynomial of ##\frac{1}{1-y}##. Taking the definite integral from 0 to ##x## on both sides yields that...

Homework Statement I'd like to derive the equations of motion for a system with Lagrange density $$\mathcal{L}= \frac{1}{2}\partial_\mu\phi\partial^\mu\phi,$$ for ##\phi:\mathcal{M}\to \mathbb{R}## a real scalar field. Homework Equations $$\frac{\partial...

Assumptions: 200+ years from now Asteroids have been moved to all Lagrange points, and at least 90, 180, and 270 degrees on Earth's' orbit for mining, and shielding humans and equipment Tech to acceleration/decelerate at 1 gravity without need to carry fuel. (My main fiction.) Direct line of...

I'm a little confused when we transition from the Lagrangian of a point particle to consider Lagrangian of objects with dimension bigger then zero. For instance, the Lagrangian of a string is the sum of the Lagrangians for each infinitesimal part of the object. which means we are integrating...

How to derive Nielsen equation from Lagrange equation

Homework Statement This could be a more general question about pendulums but I'll show it on an example. We have a small body (mass m) hanging from a pendulum of length l. The point where pendulum is hanged moves like this: \xi = A\sin\Omega t, where A, \Omega = const. We have to find motion...

At an L1 LaGrangian point between two bodies, one could - materials science notwithstanding - pit two of Newton's Laws (LM3,UG) against each other to provide thruster-free stationkeeping. Would it be feasible to use that to launch free from the system ? either spit out like a watermelon seed...

I want to prove Cauchy–Schwarz' inequality, in Dirac notation, ##\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2##, using the Lagrange multiplier method, minimizing ##\left|\left<\psi\middle|\phi\right>\right|^2## subject to the...

Homework Statement maximize f(a,d,h,p)= (4a+3d+3h+c1)c2 *(2+0.01*floor(50+0.0001p)) subject to the constraint 1439a+427d+9259+912/5*h=k. This is not a homework problem but it may as well be: it comes from a game, the function f represents damage as a function of 4 stats and the constraint...

Homework Statement Find the local extreme values of ƒ(x, y) = x2y on the line x + y = 3 Homework Equations ∇ƒ = λ∇g The Attempt at a Solution 2yxi+x^2j = λi + λj [2yx=λ] [x^2=λ] [x+y=3] [2yx=x^2] & [(2y)+y=3] [2y=x] & [y=1]...

My question is related to the book: Classical Mechanics by Taylor. Section 7.8 So, In the book Taylor is trying to derive the conservation of momentum and energy from Lagrange's equation. I understood everything, but I am struggling with the concept and the following equation...

In Section 7.6 - Equivalence of Lagrange's and Newton's Equations in the Classical Dynamics of Particles and Systems book by Thornton and Marion, pages 255 and 256, introduces the following transformation from the xi-coordinates to the generalized coordinates qj in Equation (7.99): My...

Homework Statement This is a bit unusual, I don't know whether I should post it here or math forum tbh. When I was doing numerical method home work, I am required to do perform both of these interpolation on a set of 4 data points. It turns out that the result of these 2 methods always agrees...

Given the following expressions: and that ## \bf{B}_s = \nabla \times \bf{A}_s ## how does one solve for the following expressions given in (12) and (13)? I've attempted doing so and derive the following expressions (where the hat indicates a unit vector): ## bV = \bf{ \hat{V}} \cdot...

Hi, I'm trying to solve the following problem ##\max_{f(x)} \int_{f^{-1}(0)}^0 (kx- \int_0^x f(u)du) f'(x) dx##. I have only little experience with calculus of variations - the problem resembles something like ## I(x) = \int_0^1 F(t, x(t), x'(t),x''(t))dt## but I don't know about the...

I was trying to Extrapolate Eulers formula , after deriing the basic form I wanted to prove: ∂F/∂y - d(∂F/∂yx)/dx +d[SUP]2[/SUP](∂F/∂yxx)/dx2 = 0 Here is my attempt but I get different answers: J(y) = ∫abF(x,yx,y,yxx)dx δ(ε) = J(y+εη(x)) y = yt+εη(x) ∂y/∂ε = η(x) ∂yx/∂ε = η⋅(x)...

When we take some subgroup ##H## in ##G##. And form cosets ##g_1H, g_2H,...,g_{n}H##. Is ##H## also coset ##eH##, where ##e## is neutral? So do we have here ##n## or ##n+1## cosets?

Hello PF, I was doing the derivation of the Lagrange equation of motion and had to do some calculus of variations. The first step in the derivation is to multiply the integral of ƒ(y(x),y'(x);x)dx from x1 to x2 by δ. and then by the chain rule we proceed. But I cannot understand why we are...

Homework Statement I have an issue with understanding the idea of generalised momentum for the Lagrangian. For a central force problem, the Lagrangian is given by, $$L = \frac{1}{2}m(\dot{r} ^2 + p^2 \dot{\phi ^2}) - U(r)$$ with ##r## being radial distance. The angular momentum is then...

Homework Statement The skier is skiing without friction down the mountain, being all the time in a specified plane. The skier's altitude y(x) is described as a certain defined function of parameter x, which stands for the horizontal distance of the skier from the initial position. The skier is...

Say we have a Lagrange function with one multiplier a times a constrain. I minimize and solve the system to find a. I now add another constrain to the same system multiplied by the constant b. Is the value of a the same or can it change?

What is wrong with the simple localised geometric derivation of the Euler Lagrange equation. As opposed to the standard derivation that Lagrange provided. Sorry I haven't mastered writing mathematically using latex. I will have a look at this over the next few days. More clarification. I...

Hi PF! When minimizing some fraction ##f(x)/g(x)## can we use Lagrange multipliers and say we are trying to optimize ##f## subject to the constraint ##g=1##? Thanks

Hi Does anyone know what the straight line distances of L4 and L5 are from Earth? Many thanks.

Hi, I've derived the equation of motion for a regular single pendulum, but do not know how to solve the differential equation. I have the following: r2θ''2=mg(cosθ-rsinθ)

All the needed formulas are here http://hepweb.ucsd.edu/ph110b/110b_notes/node36.html I consider the following case $$p_\psi\ne 0,\quad p_\phi/p_\psi\in (\cos\theta_2,\cos\theta_1)$$ this case corresponds to the middle picture in the bottom of the cited page. I can not prove that the time...

Homework Statement Show that for an arbitrary ideal holonomic system (n degrees of freedom) \frac{1}{2} \frac{\partial \ddot T}{\partial\ddot q_j} - \frac{3}{2} \frac{\partial T}{\partial q_j} = Q_j where T is kinetic energy and qj generalized coordinates.[/B]Homework Equations...

Use Lagrange Multipliers to find the individual extrema, assuming that x and y are positive. Maximize: f (x, y) = e^(xy) Constraint: x^2 + y^2 = 8 My Work: I decided to rewrite the constraint as x^2 + y^2 without the constant 8 as originally given. g (x, y) = x^2 + y^2 I found the gradient...

Use Lagrange Multipliers to find the individual extrema, assuming that x and y are positive. Maximize: f (x, y) = sqrt {6 - x^2 - y^2} Constraint: x + y - 2 = 0 My Work: I first decided to rewrite the constraint as g (x, y) = x + y without the constant -2 as originally given. I found the...

Two questions about Lagrange points. (1) According to Wikipedia, "libration is a perceived oscillating motion of orbiting bodies relative to each other," whereas the Lagrange points are, with respect to two bodies, null points for a (real or hypothetical) third body with respect to the sum...

Homework Statement Homework Equations Partials for main equation equal the respective partials of the constraints with their multipliers The Attempt at a Solution Basically I am checking to see if this is correct I am pretty sure that 25/3 is the minimum but I am not sure how to find...

Homework Statement For the following integral, find F and its partial derivatives and plug them into the Euler Lagrange equation $$F(y,x,x')=y\sqrt{1+x'^2}\\$$ Homework Equations Euler Lagrange equation : $$\frac{dF}{dx}-\frac{d}{dy}\frac{dF}{dx'}=0$$ The Attempt at a Solution...

Problem: Express the quadratic form: q=x1x2+x1x3+x2x3 in canonical form using Lagrange's Method/Algorithm Attempt: Not really applicable in this case due to the nature of my question The answer is as follows: Using the change of variables: x1=y1+y2 x2=y1-y2 x3=y3 Indeed you get...

Hi, I have a question about lagrange multiplier Let's say we are given with the following constraints Σ{Ni}=N and Σ{NiEi}=total energy. N and total energy are constants by definition. if we take the derivative with respect to Nj, ∂Σ{Ni}/∂Nj=∂N/∂Nj where i=j, ∂Σ{Ni}/∂Nj=1 and ∂N/∂Nj = 0...

Hello, I have to find the density of probability which gives the maximum of the entropy with the following constraint\bar{x} = \int x\rho(x)dx \int \rho(x) dx = 1 the entropy is : S = -\int \rho(x) ln(\rho(x)) dx L = -\int \rho(x) ln(\rho(x)) dx - \lambda_1 ( \int \rho(x) dx -1 ) -...

am deriving lagrange's equation can anybody help me to understand this identity the book says that he is using the chain rule for it but am not getting it d/dt(∂x/∂q) the identity is in the screen shot thanks :)

Homework Statement Julia plans to make a cylindrical vase in which the bottom of the vase is 0.3 cm thick and the curved, lateral part of the vase is to be 0.2 cm thick. If the vase needs to have a volume of 1 liter, what should its dimensions be to minimize its weight? Homework Equations...

Given a question like this: Findhe maximum and minimum of http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/eq0043M.gif[PLAIN]http://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers_files/empty.gif subject to the constraint...

This book should introduce me to Lagrangian and Hamiltonian Mechanics and slowly teach me how to do problems. I know about Goldstein's Classical Mechanics, but don't know how do I approach the book.

I'm basically trying to understand the 2-D case of the catenary cable problem. The 1-D case is pretty straightforward, you have a functional of the shape of a cable with a constraint for length and gravity, and you get the explicit function of the shape of a cable. But if you imagine a square...

I'm nearly at the end of this derivation but totally stuck so I'd appreciate a nudge in the right direction Consider a set of N identical but distinguishable particles in a system of energy E. These particles are to be placed in energy levels ##E_i## for ##i = 1, 2 .. r##. Assume that we have...

Hi, I have (probably) a fundamental problem understanding something related critical points and Lagrange multipliers. As we know, if a function assumes an extreme value in an interior point of some open set, then the gradient of the function is 0. Now, when dealing with constraint...