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.
Homework Statement
A cannonball is heated with with temperature distribution T(x,y,z)=60(y2+z2-x2).
The cannonball is a sphere of 1 ft with it's center at the origin
a) Where are the max and min temperatures in the cannonball, and where do they occur?Homework Equations
\nablaf=λ\nablag
Where...
Hello there,
I am interested in the following matter.
Given an ODE, can one always find a functional F such that the ODE is its Euler Lagrange equation?
I am thinking at the following concrete case.
I have the ODE y' = a y
I would like a functional given by the intergral over a...
Homework Statement
I'm given that the function f(x) is n times differentiable over an interval I and that there exists a polynomial Q(x) of degree less than or equal to n s.t.
\left|f(x) - Q(x)\right| \leq K\left|x - a\right|^{n+1}
for a constant K and for a \in I
I am to show that Q(x)...
Homework Statement
Prove I=T1+T2+...+Tk
Where Ti=pi(T)
Homework Equations
T is kxk
pi(x)=(x-c1)...(x-ck) is the minimal polynomial of T.
pi=\pii(x)/\pii(ci)
\pii=\pi(x)/(x-ci)
To evaluate these functions at a matrix, simply let ci=ciI
The Attempt at a Solution
From lagrange interpolation...
My question is simple is every classical mechanics problem which is solvable by Lagrangian & Hamiltonian methods also solvable by Newtonian methods of forces and torques?
And why does it seem that LH make solutions to be a lot more easier than Newtonian methods, and is it always this way?
In Lagrangian mechanics, can anyone show how to find the extrema of he action functional if you have more constraints than degrees of freedom (for example if the constraints are nonholonomic) using Lagrange Multipliers?
I've looked everywhere for this (books, papers, websites etc.) but none...
Homework Statement
Find the minimum and maximum values of the function subject to the given constraint
f(x,y) = x^2 + y^2, 2x + 3y = 6
Homework Equations
\nablaf, \nablag
The Attempt at a Solution
After doing all the calculation, x value and y value came out to be...
Hi
Homework Statement
Look at the drawing. Furthermore I have a constant acceleration \vec g = -g \hat y
I shall find the Lagrange function and find the equation of motion afterwards.Homework Equations
Lagrange/ Euler function and eqauation
The Attempt at a Solution
I found out the...
I'm in a bit of a hurry, so this isn't going to be very pretty.
Homework Statement
Maximize: V(l,d) = pi * (0.5*d)^2 * l
Subject to: l + 3.5d = 84 -> C(l, d) = l + 3.5d - 84Homework Equations
∇V(l,d) = λ ∇C(l,d)
The Attempt at a Solution
∇V(l,d) = 0.5*pi*d
∇C(l,d) = 0How do I find the...
Homework Statement
Find the maximum and minimum values of f(x,y) = x5y3 on the circle defined by x2 + y2 = 10. Do the same for the disc x2 + y2 ≤ 10.
The Attempt at a Solution
for the first part, if I call the circle g(x,y) defined by x2 + y2 = 10
I need to now define some F(x,y,λ) =...
Hello
Can any1 recommend a book that will show the derivation of the Euler-Lagrange equation.
(I am learning in the context of cosmology ie. to extremise the interval).
Ideally the derivation would be as simple/fundamental as possible - my maths is not up to scratch!
Homework Statement
"Vary the following actions and write down the Euler-Lagrange equations of motion."
Homework Equations
S =\int dt q
The Attempt at a Solution
Someone said there is a weird trick required to solve this but he couldn't remember. If you just vary normally you get \delta...
Homework Statement
Use the method of Lagrange multipliers to find the maximum and minimum values of the function
f(x, y) = x + y2
subject to the constraint g(x,y) = 2x2 + y2 - 1
Homework Equations
none
The Attempt at a Solution
We need to find \nablaf = λ\nablag
Hence...
Hello everyone, i have 2 problems in my multivariable calculus homework that i can't solve . Please help me out, thank you so much
1/f(x,y)= [(x^2) -2y]^(0.5)
a) Find directional derivatives of f at (2,-6) in the direction of <-4,3>
b) Find equation of the tangent plane to the function...
Homework Statement
Use Lagrange multipliers to find the max and min values of the function subject to the given constraints:
f(x1,x2,...,xn) = x1 + x2 + ... + xn
constraint: (x1)^2 + (x2)^2 + ... (xn)^2 = 1
The Attempt at a Solution
fo x1 to xn values, x must equal 1/sqrt(n) in...
Homework Statement
Use Lagrange multipliers to find the max and min values of the function subject to the given constraints:
f(x,y,z)= x2y2z2
constraint: x2 + y2 + z2 = 1
Homework Equations
∇f = ∇g * λ
fx = gx * λ
fy = gy * λ
fz = gz * λ
The Attempt at a Solution
i can't solve...
Homework Statement
Use Lagange Multipliers to find the max and min values of the function subject to the given constraint(s). f(x,y)=exp(xy) ; constraint: x^3 + y^3 = 16
Homework Equations
\nablaf = \nablag * \lambda
fx = gx * \lambda
fy = gy * \lambda
The Attempt at a Solution...
Homework Statement
I am trying to find the min and max values of f(x,y)=2x^2 + 3y^2 subject to xy=5.
Homework Equations
f(x,y)=2x^2 + 3y^2 subject to xy=5
\mathbf\nablaf=(4x, 6y)
\mathbf\nablag=(y,x)
The Attempt at a Solution
When I go through the calculations, I end up with two critical...
Homework Statement
Use Lagrange multipliers to ¯nd the maximum and mini-
mum value(s), if they exist, of
f(x; y; z) = x^2 -2y + 2z^2
subject to the constraint x^2+y^2+z^2
Homework Equations
The Attempt at a Solution
Basically after I find the gradient of the functions I get this.
2x=2x lamda...
Use lagrange multipliers to find max/min of
f(x,y)=x^2+6y
subject to
x^2-y^2=5
grad f =λgrad g
2x=2xλ, λ=1
6=-2yλ, λ=-3/y
1=-3/y, y=-3
x^2-(-3)^2=x^2-9=5
x^2=14
x=+/-√14
two points are √14, -3 and -√14, -3
plugging both points into f(x,y) gives me the same answer. now what?
Homework Statement
Find the polynomial p(x) of degree 20 satisfying:
p(-10) =p(-9) = p(-8) = ...=p(-1) = 0
p(0) = 1
p(1) = p(2) = p(3) = ...p(10) = 0
Homework Equations
L(x) := \sum_{j=0}^{k} y_j \ell_j(x)
The Attempt at a Solution
i tried using the formula above:
a =...
My textbook is using Lagrange multipliers in a way I'm not familiar with.
F(w,λ)=wCwT-λ(wuT-1)
Why is the first order necessary condition?:
2wC-λu=0
Is it because:
\nablaF=2wC-λu
Why does \nablaF equal this?
Many thanks!
Edit: C is a covariance matrix
Homework Statement
In Classical mechanics 2 i have an assignment based on the Euler Lagrange method and i cannot seem to grasp the concept, even with all the internet resources i can find as well as my two textbooks which have a chapter on it. (Boass (Mathematical methods in teh physical...
Homework Statement
Use Charpits equations to solve 4u\frac{\partial u}{\partial x} = (\frac{\partial u}{\partial x})^2
where u=1 on the line x+2y=2
Homework Equations
The Attempt at a Solution
from the charpit equations i get
\frac{dx}{dt} = 4u
\frac{dy}{dt} = -1...
Homework Statement I made this up, so I am not even sure if there is a solution
Let's say I have to find values for which these two inequality hold x^2 + y^5 + z = 6 and 8xy + z^9 \sin(x) + 2yx \leq 200And by Lagrange Multipliers that
\nabla f = \mu \nabla g
So can I let f = 8xy + z^9 \sin(x)...
Dear all,
I have an optimization problem with boundary conditions, the type that is usually solved with Lagrange multipliers. But the (many) variables my function depends on can take only the values 0 and 1. Does anyone know how to apply Lagrange multipliers in this case?
I am a...
so i know for example that d/dt (∂L/∂x*i) = ∂L/∂xi for cartesian coordinates, where xi is the ith coordinate in Rn and x*i is the derivative of the ith coordinate xi with respect to time. L represents the lagrangian.
so using an arbitrary change of coordinates, qi = qi(x1, x2, ..., xn)
i...
1. Assume we have function V(x,y,z) = 2x2y2z = 8xyz and we wish to maximise this function subject to the constraint x^2+Y^2+z^2=9. Find the value of V at which the max occurs
2. Function: V(x,y,z) = 2x2y2z = 8xyz
Constraint: x^2+Y^2+z^2=9
3. So far I have gone
Φ= 8xyz +...
A friend and I were debating the solution to this problem, seen below, and cannot solve it without using Lagrange equations but it is suppose to have a solution that is super simple; but we didn't see it.
Anyway, it is a old qualifier question from the Univ. of Wisconsin (open record so...
So I need to find the min and max values of f(x,y,z) = x^2 + 2y^2 + 3z^2 given the constraints x + y + z = 1 and x - y + 2z =2. I've gotten as far as (2x, 4y, 6z) = (u,u,u) + (m,-m,2m). I'm stuck trying to solve this system of equations. Any hints?
Hello, i have done the following code in c++ which gives me a result but :
1) I compute the product "prod*=(y-x[i])/(x[k]-x[i])" which has variable "y" inside ,but the result i take is just a number.(i want the result to be for example in this form "p(x)=1.02*x^3+2*x^2..."
I can't...
Homework Statement
L = - \Sigma x,y (P(x,y) log P(x,y)) + \lambda \Sigmay (P(x,y) - q(x))
This is the Lagrangian. I need to maximize the first term in the sum with respect to P(x,y), subject to the constraint in the second term.
The first term is a sum over all possible values of x,y...
Homework Statement
I seem to be struggling a bit to understand how my prof solved this problem...I think it might be my diminishing system of equation skills, so forgive me if this doesn't belong in the calc section.
Use Lagrange multipliers to find all extrema of the function subject to...
Homework Statement
Prove that for every pair of numbers x and h, \left|sin\left(x+h\right)-\left(sinx+hcosx\right)\right|\leq\frac{h^{2}}{2}
The Attempt at a Solution
Let f(x)= \left|sin\left(x+h\right)-\left(sinx-hcosx\right)\right|?
and then to center the taylor polynomial around 0...
In a exercise says:
Find max a min of f=-x^2+y^2 abaut the ellipse x^2+4y^2=4
i tried -2x=\lambda 2x
2y=\lambda 8y
x^2+4y^2-4=0
then \lambda =-1 or \lambda =\frac{1}{4} , but, ¿how i find x,y?
URGENT - Lagrange Multipliers
Homework Statement
:confused:
Using the method of lagrange multipliers prove the formula for the distance from a point (a,b,c) to a plane Ax + By + Cz = DThe Attempt at a Solution
Using the equation of the form;
H(x,y,z,L) = (x-a)^2 + (y -b)^2 +(z-c)^2 + L(Ax...
Homework Statement
Maximize f(x,y,z)=x^{2}+y^{2}+z^{2} with constraint x^{4}+y^{4}+z^{4}=1 using Lagrange multipliers
The Attempt at a Solution
I've got the setup as:
\Lambda(x,y,z,\lambda)=x^{2}+y^{2}+z^{2}+\lambdax^{4}+\lambday^{4}+\lambdaz^{4}+\lambda
I solve for all partials nice...
Physics gurus: I understood from Newton's Law that a 2 bodies would rotate around their common center of mass. Should one body disappear (Harry Potter invoked here), the other would go flying off at a tangent... like a 'David's Sling" releasing a missile. The mass of the bodies was crucial to...
Let f(x,y)= -2x^2-2xy+y^2+2 Use Lagrange multipliers to find the minimum of f subject to the constraint 4x-y = 6
∂F / ∂x =.....
i got -4x-2y+2y but i coming out as wrong what am i missing
∂F/ ∂Y= ...
The function f achieves its minimum, subject to the given constraint, where
x =...
Homework Statement
Im supposed to use the lagrange error bound to find a bound for the error when approximating ln(1.5) with a third degree taylor polynomial about x=0, where f(x)=ln(1+x)
Homework Equations
Lagrange error bound
m/(n+1)! abs(x-a)^n+1, m=f(n+1)(c)
The...
A rectangular solid of maximum volume is to be cut from a solid sphere of radius r. Determine the dimension of the solid so formed and its volume?
I have defined my function F(l,b,h) as lbh, but i don't know how to define my constraint condition from my question
1. Problem Statement:
Use Lagrange multipliers to find the volume of the largest box with faces parallel to the coordinate system that can be inscribed in the ellipsoid: 6x2 + y2 + 3z2 = 2
2. Homework Equations :
f(x,y,z) = \lambdag(x,y,z)
3. Attempt at a solution
f(x,y,z) is the...
Homework Statement
Find the extreme values of the function f(x,y,z) = xy + z^2 in
the set S:= { y\geq x, x^2+y^2+z^2=4 }
Homework Equations
The Attempt at a Solution
Ok, so This is clearly a lagrange multiplier question. Geometrically, I can see that the region that is the constraint is...
Find the maximum value of f(x,y,z) = 5xyz subject to the constraint of [PLAIN]http://www3.wolframalpha.com/Calculate/MSP/MSP9619f6019f3fia87i60000567g3gb3dhi833if?MSPStoreType=image/gif&s=6&w=126&h=20.
I know to find the partial derivatives of the function and the constraint. Then, set up...
1. Use the method of Lagrange multipliers to nd the minimum value of
the function:
f(x,y,z) = xy + 2xz + 2yz
subject to the constraint xyz = 32.
I understand the method how Lagranges Multipliers is donw done but seem to have got stuck solving the Simultaneous Equations involving the...
Homework Statement
Find the point closest to the origin on the line of intersection of the planes y + 2z = 12 and x + y = 6Homework Equations
\nuf = \lambda\nug1 +\mu\nug2
f = x2+y2+z2
g1: y + 2z = 12
g2: x + y = 6
There are supposed to be gradients on all of those, whether or not LaTeX...
Hey all, this is my first post, so I apologize in advance if data are missing/format is strange/etc.
I'm working with lagrange multipliers, and I can get to the answer about half the time. The problem is, I'm not really sure how to deal with things when the multiplier equation becomes...
Hello there,
I was wondering if anybody could indicate me a reference with regards to the following problem.
In general, the Euler - Lagrange equation can be used to find a necessary condition for a smooth function to be a minimizer.
Can the Euler - Lagrange approach be enriched to cover...
Homework Statement
Find the minimum of f(x,y) = x^2 + y^2 subject to the constraint g(x,y) = xy-3 = 0
Homework Equations
delF = lambda * delG
The Attempt at a Solution
Okay, after lecture, reviewing the chapter and looking at some online information, this is what I have so far...
Hello there,
I am dealing with the functional (http://en.wikipedia.org/wiki/First_variation)
J = integral of (y . dy/dx) dx
When trying to compute the Euler Lagrange eqaution I notice this reduces to a tautology, i.e.
dy/dx - dy/dx = 0
How could I proceed for finding the y(x) that...