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.
So I think the mass can only move in two "coordinates" the axis of which the mass is connected to ##k_1## and the axis connecting it to ##k_2##. Therefore, the D.O.F is 2. I don't understand what it the meaning of "variables of integration" What does it mean?
Apart from that, I attempted to...
While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ q_k}-(\frac d {dt}\frac {\partial\mathcal L}{\partial\dot q_k}))\delta q_k=0\tag{2}$$
However...
I'm confused on how to derive the multidimensional generalization for a multivariable function. Everything makes sense here except the line,
$$
\frac{\delta S}{\delta \psi} = \frac{\partial L}{\partial \psi} - \frac{d}{dx} \frac{\partial L}{\partial(\frac{\partial \psi}{\partial x})} -...
I tried writing this out but I think there is a bug or something as its not always displaying the latex, so sorry for the image.
I have gone through various sources and it seems that the reason for u being small varies. Sometimes it is needed because of the taylor expansion, this time (below) is...
Here is my epic fail at trying to derive the equation using Lagrange (this was my first time trying to use lagrangian mechanics except for when I memorized the derivation for a pendulum)
$$L = \frac{m \dot r^2}{2} - \frac{k q_1 q_2}{r}$$
$$\frac{\partial L}{\partial r} = \frac{k q_1 q_2}{r^2}$$...
Hi,
In my book I have and expression that I don't really understand.
Using the definition of action ##\delta S = \frac{\partial L}{\partial \dot{q}} \delta q |_{t_1}^{t_2} + \int_{t_1}^{t_2} (\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}) \delta q dt##
Where L...
Good morning, I'm not a student but I'm curious about physics.
I would like to calculate the equation of motion of a system using the Lagrangian mechanics. Suppose a particle subjected to some external forces.
From Wikipedia, I found two method:
1. using kinetic energy and generalized forces...
A V-shaped tube with a cross-section A contains a perfect liquid with mass density and length L plus and the angles between the horizontal plane and the tube arms as shown in the attached figure.
We displace the liquid from its equilibrium position with a distance and without any initial...
Lagrangian principle is easier to solve any kind of problem. But we always "forget" (not really. But we don't take it into account directly.) of Tension in a system when looking at Lagrangian. But some questions say to find Tension. Since we can get the equation of motion from Newton's 2nd law...
Hello everyone,
my question is, if there is a case, where you can't you Langrange (1 or 2) but only Newton to solve the equation of motion?
My guess is, that it might be, when we have no restrictions at all, so a totally free motion.
Does anybody know?
$$L = \frac {mv^2}{2} - mgy$$
It is clear that ##\dot{x}=\dot{\theta}L## and ##y=-Lcos \theta##. After substituting these two equations to Lagrange equation, we will get the answer by simply using this equation: $$\frac {d} {dt} \frac {∂L}{∂\dot{\theta}} - \frac {∂L}{∂\theta }= 0$$
But, What if...
Hello.
With the recent interest in the JWST orbiting at the L2 Lagrange point of the Earth - Moon system, I was wondering about the dynamics of the Pluto - Charon system. Specifically, the barycentre of that system.
This barycentre lies at a point in space between these two bodies. Does...
So we are finding the L2 Lagrange point, specifically the distance from the earth, or d in this instance. I have used the equation above and I have come out with 1.5 * 10^9 meters as d, or L2's distance from the earth. Can anyone verify this, is the equation correct and is my final distance...
So I understand the concept of lagrange multiplier but I fail at every single execise I encounter anyways.
Because you always end up with unsolvable equations of x^3yzb3gh + 37y^38x^3 + k^5x = 0
Anways here's my stupid attempt:
Instead of doing
$$grad(f) + \lambda grad(g) = 0$$
I solve
$$...
Good Morning all
Yesterday, as I was trying to formulate my confusion properly, I had a series of posts as I circled around the issue.
I can now state it clearly: something is wrong :-) and I am so confused :-(
Here is the issue:
I formulate the Lagrangian for a simple mechanical system...
I think I undeerstand Lagrangian mechanics but I have a question that will help to clarify some concepts.
Imagine I throw a pencil. For that I have 5 generalised coordinates (x,y,z and 2 rotational).
When I express Kinetic Energy (T) as:
$$T = 1/2m\dot{x^{2}}+1/2m\dot{y^{2}}+1/2m\dot{z^{2}} +...
Let ##g_{\mu \nu}(x)## be a time-independent metric. A photon following a curve ##\Gamma## has action\begin{align*}
I[x,e]= \dfrac{1}{2} \int_{\Gamma} e^{-1}(\lambda) g_{\mu \nu}(x)\dot{x}^{\mu} \dot{x}^{\nu} d\lambda
\end{align*}with ##e(\lambda)## an independent function of ##\lambda## (an...
Good Morning
I am "comfortable" with formulating Hamilton's Principle with a Lagrangian (KE - PE), conducting the calculus of variations and obtaining the Euler Lagrange Equations. Advanced mathematical theory, is beyond me.
I also have a minimal understanding of using Lagrange multipliers...
What is the delta-v requirements from each of the Earth-Moon lagrange points to landing on the lunar surface?
What would be the best software I could use to visualise and calculate that kind of thing?
Thanks.
Hey! :giggle:
Business operates on the basis of the production function $Q=25\cdot K^{1/3}\cdot L^{2/3}$ (where $L$ = units of work and $K$ = units of capital).
If the prices of inputs $K$ and $L$ are respectively $3$ euros and $6$ euros per unit, then find :
a) the optimal combination of...
Hi again
How much mass would a centaur need to have Lagrange points? (is a centaur of this size plausible?)
In the story I'm working on a massive centaur passes near Jupiter's 4th Lagrange point, such that the centaur's Lagrange point and Jupiter's overlap.
Could the centaur come at an angle...
In Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws (starting on page 291 here), the authors introduce the concept of a generator function G, where the transformation generated by G is given by (equation 12.29 on page 292 in the text)...
Problem statement : Let ##f\in C^\infty ([-1;1])## with ##f(1)=f(-1)=0## and ##\int_{-1}^1f(x)dx=1##
Which curve has the lowest (maximal) absolute slope ?
Attempt :
Trying to minimize ##f′(x)−\lambda f″(x)## with Lagrange multipliers but to find f not x ?
I got...
I was having a doubt about the Lagrangian mechanics. Possible we can derive the lagranges equations of by extremisation principle of action, that is assume we already guess what is the lagrangian of the systeme. I say that minimisation procedure rely on assume a lagrangian, and then show it...
In class our teacher told us that, if a Lagrangian contain ##\ddot{q_i}## (i.e., ##L(q_i, \dot{q_i}, \ddot{q_i}, t)##) the energy will be unbounded from below and it can take any lower values (in other words be unstable). In this type of systems can we show that the energy is conserved ? Or in...
Now $\sum_{i=0}^{10}(x_{i}+1) L _{10,i}(5) = (x_{0}+1) L _{10,0}(5) + (x_{1}+1) L _{10,1}(5) + ... + (x_{10}+1) L _{10,10}(5)$
Which I can further decompose into
$\frac{(x_{0}+1)(5-x_{1})(5-x_{2})...(5-x_{10})}{(x_{0}-x_{1})(x_{0}-x_{2})...(x_{0}-x_{10})} +...
Summary:: equation system of complex mechanism need to be done. Lagrange Formalism could be helpfull to do it but small errors could all destroy ..
Hi everyone,
I tried to find the equation system of the joined Mechanism (4 DoF). I think there is a mistake but I am not sure where is it . I...
l am italian student from Milan university, so sorry for my bad english.
l am studying lagrange meccanics. We are linearizating lagrange equations. Here l don't understand how you can expand A matrix, how the function f is derivable, how the inverse matrix A is expanded? l am expanding with q0...
Alright, so I did some progress and then I got stuck. After some time I went to check the solution. Up to some point, it's all well and good:
I understand everything that is happening up to the point where he takes the partial derivative of S wrt ρ(Γ). I don't understand how he gets the...
OK I've been stuck for a while in how to derive ##(1)##, so I better solve a simplified problem first:
We work with
Where
$$\mathscr{L} = \mathscr{L}(\phi_a (\vec x, t), \partial_{\mu} \phi_a (\vec x, t)) \tag{3}$$
And ##(3)## implies that ##\mathscr{L}(\vec x, t)##
We know that...
I found that f= x -2yz. To maximize f, I can first inspect the solutions to grad(F)=0. z=y=0 pops out, but I'm not sure what to do with the x-component equaling 1. Do we just include (x,0,0) as a solution? I think the problem wants specifics though, based on what I've seen previously from...
ƒ(x,y) = 3x + y
x² + 2y² ≤ 1
It is easy to find the maximum, the really problem is find the minimum, here is the system:
(3,1) = λ(2x,4y)
x² + 2y² ≤ 1
how to deal with the inequality?
I am new to Lagrangian mechanics and I am unable to comprehend why the Euler Lagrange equation works, and also what really is the significance of the lagrangian.
According to the book "Principles of Statistical Mechanics" by Amnon Katz, page 123, ##\alpha## must be such that ##\exp ( -\alpha N ) ## can be expanded in powers of ##\alpha## with only the first order term kept. Is this the necessary and sufficient condition for small deviations from...
Hello, I have been working on the three-dimensional topological massive gravity (I'm new to this field) and I already faced the first problem concerning the mathematics, after deriving the lagrangian from the action I had a problem in variating it
Here is the Lagrangian
The first variation...
Classical mechanics is based on conservation laws which represent the symmetries of spacetime. The lagrangian function L is a function of position and velocity while the hamiltonian is a function of position and momentum. The velocity and momentum descriptions are related by a legendre...
I started to understand how to apply Lagrange multiplier methods. But, for problem like this, I have difficulty to build the function to describe the volume that will be maximized. For the second question, I know from the example (in ML Boas) that ##V=8xyz## becase (x,y,z) is in the 1st octant...
I've problems understanding why the kinetic energy of the string is only
$$T_{string}=\frac{1}{2}m\dot{y} $$
Why the contribution of the string in the horizontal line isn't considered?
I'm having some trouble understanding the following proof (##a_{ik}## and ##b_{ik}## are constants)
Shouldn't it be ##a_{ik}q_iq_k - \frac 1 {\lambda} (b_{ik}q_iq_k-1)## ?
(Summation convention is used)
Thanks Ric
Hi there!
Kindly help me to solve the problem below.
A company is using frustum of a cone containers for their products. What are the dimensions of the least expensive container that can hold 300 cubic cm? Use Lagrange Multipliers to solve the problem.
Thanks.
From what I understand, constraint forces do no work because they are perpendicular to the allowed virtual displacements of the system. However, if you consider an unbalanced Atwood machine, in which both masses are accelerating in opposite directions, you'll find that the tension force of the...
A homogen box with the mass M rolls without sliding on two round wheels. The wheels with mass mass m are also homogen and roll without sliding, on top of the banked Surface. We use Gravitation g.
Find the accelration xM of the box
I don't know which solution is correct. i got 0.67 m for xM...
Exercise statement:
Given the action (note ##G_{ab}## is a symmetric matrix, i.e. ##G_{ba} = G_{ab}##):
$$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.
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...