Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.
In 1766, on the recommendation of Swiss Leonhard Euler and French d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty years, producing volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1788–89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.
In 1787, at age 51, he moved from Berlin to Paris and became a member of the French Academy of Sciences. He remained in France until the end of his life. He was instrumental in the decimalisation in Revolutionary France, became the first professor of analysis at the École Polytechnique upon its opening in 1794, was a founding member of the Bureau des Longitudes, and became Senator in 1799.
Hi,
To calculate the intersection of two straight lines the cross product of the line vectors can be used, i.e. when the lines start in points p and q, and have direction vectors r and s, then if the cross product r x s is nonzero, the intersection point is q+us, and can be found from...
Homework Statement
Lagrange multipliers to find the maximum and minimum values of f(x,y) = 4x^3 + y^2 subject to the constraint 2x^2 + y^2 = 1. Find points of these extremum.
Homework Equations
The Attempt at a Solution
g(x,y)= 2x^2 + y^2 - 1
f(x,y)= 4x^3 + y^2
Gradient F=...
I am trying to understand an example from my textbook "applied finite element analysis" and in the variational calculus, Euler lagrange equation example I can't seem to understand the following derivation in one of its examples
∫((dT/dx)(d(δT)/dx))dx= ∫((dT/dx)δ(dT/dx))dx= ∫((1/2)δ(dT/dx)^2)dx...
I'm stuck on this problem for the past hour. I've tried solving for all variables and none of the steps I'm doing are getting me to the right answer.
Find the point on sphere x^2+y^2+z^2=25 farthest from point (1,1,-1).
My steps:
http://i.imgur.com/c5kUj9g.png
Correct Answer: x=...
Homework Statement
A particle of mass m moves over the cylindrical surface of radius R. The particle is connected to the origin O, located on the central point of the cylindrical surface axis, by a spring with spring constant k and length R. Ignore force of gravity.
a) State how many...
Homework Statement
Show that the largest rectangular box having a fixed surface area must be a cube.
Homework Equations
##V(x,y,z) = xyz##
##\sigma(x,y,z) = 2(xy + yz + zx) = C \in \mathbf{R}##
The Attempt at a Solution
As of this assignment, we haven't yet learned Lagrange...
I was checking the proof of this, when things came vague at one point.
It goes as follows, how to prove that Lagrange's equations hold in any coordinate system?
Answer:
Let q_{a} = q_{a}(x_{1},..., x_{3N}, t)
here the possibility of using a coordinate system that changes with time is...
Find the minimum value of \(\int_0^1y^{'2}dx\) subject to the conditions \(y(0) = y(1) = 0\) and \(\int_0^1y^2dx = 1\).
Let \(f = y^{'2}\) and \(h = y^2\).
Then
\begin{align*}
G[y(x)] &= \int_0^1[f - \lambda h]dx\\
&= \int_0^1\left[y^{'2} - \lambda y^2\right]dx
\end{align*}...
Given this \(F = p(x)y^{'2}-q(x)y^2+2f(x)y\). What would be the integral of \(f(x)\) and \(q(x)\)?
\begin{align*}
f(x) - q(x)y - \frac{d}{dx}\left[p(x)y'\right] &= 0\\
\frac{d}{dx}\left[p(x)y'\right] &= f(x) - q(x)y\\
y'p(x) &= \int f(x)dx - y\int q(x)dx
\end{align*}
Homework Statement
The problem asks to design a cantilever beam of a minimum weight consisting of 2 steps.
Given: total length (L), Force (F) at the end of the beam and allowable stress (σ)
Need to find the diameters D and d, the length of the smaller shoulder of the beam (x)...
Homework Statement
Do the Euler-Lagrange equations set to zero for each of the 3 orthogonal coordinates or do you sum them all equal to zero. Do the coordinates have to be orthogonal in order to write separate E-L equations? Or is there no such thing as non-orthogonal coordinates to analyze a...
If you have a craft parked at the Phobos-Mars Lagrange point, it's stable. I get that. But, what if you move, horizontally with respect to Mars? I assume you're no longer balanced out by Phobos' gravity, and will therefore fall towards Mars and end up in an elliptical orbit? If so, how far would...
Homework Statement
We have a three mass two strings system with:
m_1 string M string m_2
The end masses are not attached to anything but the springs, the system is at rest, and k is equal for both strings and m_1 and m_2 are equal. The distance between to m_1 and m_2, on both sides of M...
Here is the question:
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Please help with lagrange multipliers? - Yahoo! Answers
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Hi
we covered the Lagrange multiplier method in Lagrangian Mechanics and as far as I know, is the physical meaning behind this to be able to solve either some non-holonomic constraints or to get some information about the constraint forces. my problem is, i do not know the physical meaning of...
Hi
I wanted to know for which cases the Euler Lagrange equations are applicable?
1.) Imagine that we have a kinetic Energie T(q,q') and a potential that also depends on velocity V(q,q'). As far as i know the Euler Lagrange equations for a particle still hold in this case, is that true...
Lagrange Multiplier----to find out the dimensions when metal used min.
Homework Statement
I have a rectangular tank with a capacity of 1.0m^3. The tank is closed and the cover is made of metal half as thick as the sides and base. Find the dimensions of the tank for the total amount of metal...
Homework Statement
Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity ω about its vertical axis. Use cylindrical polar coordinates and let the equation of the parabola be ##z = kρ^{2}##. Write...
Homework Statement
Maximize C_{t} for any given expenditure level
\int_{0}^{1}P_{t}(i)C_{t}(i)di\equiv Z_{t} The Attempt at a Solution
The Lagrangian is given by:
L = \left(\int_{0}^{1}C_{t}(i)^{1-(1/\varepsilon)}di\right)^{\varepsilon/(\varepsilon-1)} - \lambda...
Homework Statement
This is not a homework problem but I would like to clarify my concern.
It is stated that a function can be written as such:
f(x) = \lim_{n \rightarrow ∞} \sum^{∞}_{k=0} f^{(k)} \frac{(x-x_{0})^k}{k!}
R_{n}=\int^{x}_{x_{0}} f^{(n+1)} (t) \frac{(x-t)^n}{n!} dt
They...
Homework Statement
Consider the function f(y,y',x) = 2yy' + 3x2y where y(x) = 3x4 - 2x +1. Compute ∂f/∂x and df/dx. Write both solutions of the variable x only.
Homework Equations
Euler Equation: ∂f/∂y - d/dx * ∂f/∂y' = 0
The Attempt at a Solution
Would I first just find...
Here is the question:
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Calc 3 Lagrange multiplier question? - Yahoo! Answers
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Homework Statement
Hi guys.
http://img189.imageshack.us/img189/5123/systemn.jpg
The image shows the situation. A pointlike particle of mass m is free to move without friction along a horizontal line. It is connected to a spring of constant k, which is connected to the origin O. A...
Problem stated: Let \(a_1, a_2, ... , a_n\) be \(n\) positive numbers. Find the maximum of
$$\sum_{i=1}^{n}a_ix_i$$ subject to the constraint $$\sum_{i=1}^{n}x_i^2=1$$.
I honestly have not much of an idea of how to go about solving this. If I use lagrange multipliers which I think I am supposed...
The problem given is to find the local extreme values of \(f(x,y)=x^2y\) on the line \(x+y=3\). I went through the system of equations with the partial derivatives of \(x\), \(y\), and \(\lambda\), and found two extreme points \((0,3)\) and \((2,1)\). Plugging that into the original function I...
Homework Statement
Consider the intersection of the elliptic paraboloid Z = X2+4Y2 , and the cylinder X2+Y2= 1. Use Lagrange multipliers to find the highest, and lowest points on the curve of intersection.Homework Equations
The gradient equations of both functions.The Attempt at a Solution
I...
Hello,
I have been trying to follow the start of these lecture notes I found online and I have having trouble understanding what is happening between two steps.
The notes I am looking at are located:
http://pillowlab.cps.utexas.edu/teaching/CompNeuro10/slides/slides16_EntropyMethods.pdf...
Homework Statement
Minimize f(X, Y, Z) = 2XY + 6YZ + 8XZ subject to the constraint XYZ = 12.
Homework Equations
The gradients of the equations, and XYZ = 12.
The Attempt at a Solution
I have the gradients for both of the equations.
∇f = <2Y + 8Z, 2X + 6Z, 6Y + 8X> ∇g = <...
Homework Statement
Here is the problem, the solution and my question (in red):
I'm guessing it was rejected because for the volume function, the dimensions cannot be negative? What if it was not volume and instead was just an arbitrary function. In that case you would not reject...
I am not sure how to do this one. Nothing I try goes anywhere.
Consider the two-body equation of motion in vector form
$$
\ddot{\mathbf{r}} = -\mu\frac{\mathbf{r}}{r^3}.
$$
Show that the $f$ and $g$ functions defined by
$$
\mathbf{r} = f\mathbf{r}_0 + g\mathbf{v}_0
$$
satisfy
$$
\ddot{f} =...
I think I solved it a week ago, but I didn't write down all the things and I want to be sure of doing the things right, plus the excersise of writing it here in latex helps me a loot (I wrote about 3 threads and didn't submited it because writing it here clarified me enough to find the answer...
Advanced Calculus of Several Variables, Edwards, problem II.4.1: Find the shortest distance from the point (1, 0) to a point of the parabola y^{2} = 4x.
This is the Lagrange multipliers chapter. There might be another way to solve this, but the only way I'm interested in right now is the...
Homework Statement
Find the geodesics on a sphere g(x,y,z)=x^{2}+y^{2}+z^{2}-1=0
arclength element ds=\sqrt{dx^{2}+dy^{2}+dz^{2}}
Homework Equations
f(x,y,z)=\sqrt{x'^{2}+y'^{2}+z'^{2}} where x'^{2} \text{means} \frac{dx^{2}}{ds^{2}} and not d^{2}x/ds^{2}
The Attempt at a...
Given the equations
$$
rv\cos\gamma - h = 0,\quad \frac{v^2}{2} - \frac{\mu}{r} + \frac{\mu}{2a} = 0
$$
I want to maximize gamma.
Do I have to solve for gamma in the first equation to use the method of Lagrange multipliers, or if not, how would I do this in the current form?
Homework Statement
Find extrema for f\left( x,y,z \right) ={ x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }
under the constraint g\left( x,y,z \right) ={ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=16Homework Equations
(1) \nabla f=\lambda \nabla g
(2) g\left( x,y,z \right) ={ x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=16...
Hi,
Homework Statement
I am trying to limit Lagrange's remainder on taylor expansion of ln(4/5) to be ≤ 1/1000.
Homework Equations
The Attempt at a Solution
I have tried using both ln(1+x), where x=-1/5 and x0(the center)=0, and ln(x), where x=4/5 and x0=1.
Every time I keep...
Homework Statement
Being a>0 and f:[a,b]--->R continuos and differentiable in (a,b), show that there exists a t ##\in## (a,b) such that:
## \frac{bf(a)-af(b)}{b-a}=f(t)-tf'(f)##
The Attempt at a Solution
For lagrange's theorem, we have:
## \frac{f(a)-f(b)}{b-a}= -f'(t) ##
thought i could...
My first post and I am new to forum etiquette, please go easy on me.
I have an exam in a few days and have a good grasp on the content. Historically the professor has been putting harder Lagrange questions on the exam than are available in my textbook and supplement books or that I can find...
Speaking of theorems, I have another question. I need to show, using Lagrange's theorem, that:
1.71<\sqrt{3}<1.75
By Lagrange's theorem I mean the one of:
f ' (c)=(f(b)-f(a)) / (b-a)
thanks !
Homework Statement
Homework Equations
Lagrance Multipliers.
The Attempt at a Solution
This is a pretty dumb question, and I feel a little embarassed asking but..
I know how to do the Lagrange part (I think). I'm assuming you maximize/minimize the distance, \sqrt{x^{2} +...
Homework Statement
Use Lagrange multipliers to find the eigenvalues and eigenvectors of the matrix
A=\begin{bmatrix}2 & 4\\4 & 8\end{bmatrix}
Homework Equations
...
The Attempt at a Solution
The book deals with this as an exercise. From what I understand, it says to consider...
Homework Statement
Find the max and min values of f(x,y,z) = x3 - y3 + 6z2 on the sphere x2 + y2 + z2 = 25.
Homework Equations
I will use λ to denote my Lagrange multipliers.
The Attempt at a Solution
So clearly there is no interior to examine since we are on the boundary of the sphere...
Consider a rectilinear rod (A-B) with negligible mass that is attached, without friction, to a vertical OZ axis. The rod rotates about that axis with a constant angular velocity ω and it maintains a angle α with OZ.
A particle of mass m moves about the rod and it is attracted by the...