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,
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...
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...
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:
$$T = 1/2m\dot{x^{2}}+1/2m\dot{y^{2}}+1/2m\dot{z^{2}} +...
In Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws (starting on page 291 here), the authors introduce the concept of a generator function G, where the transformation generated by G is given by (equation 12.29 on page 292 in the text)...
From what I understand, constraint forces do no work because they are perpendicular to the allowed virtual displacements of the system. However, if you consider an unbalanced Atwood machine, in which both masses are accelerating in opposite directions, you'll find that the tension force of the...
Exercise statement:
Given the action (note ##G_{ab}## is a symmetric matrix, i.e. ##G_{ba} = G_{ab}##):
$$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 +...
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...
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
$$\frac{\partial...
Assumptions:
200+ years from now
Asteroids have been moved to all Lagrange points, and at least 90, 180, and 270 degrees on Earth's' orbit for mining, and shielding humans and equipment
Tech to acceleration/decelerate at 1 gravity without need to carry fuel. (My main fiction.)
Direct line of...
At an L1 LaGrangian point between two bodies, one could - materials science notwithstanding - pit two of Newton's Laws (LM3,UG) against each other to provide thruster-free stationkeeping.
Would it be feasible to use that to launch free from the system ? either spit out like a watermelon seed...
My question is related to the book: Classical Mechanics by Taylor. Section 7.8
So, In the book Taylor is trying to derive the conservation of momentum and energy from Lagrange's equation. I understood everything, but I am struggling with the concept and the following equation...
In Section 7.6 - Equivalence of Lagrange's and Newton's Equations in the Classical Dynamics of Particles and Systems book by Thornton and Marion, pages 255 and 256, introduces the following transformation from the xi-coordinates to the generalized coordinates qj in Equation (7.99):
My...
Hello PF,
I was doing the derivation of the Lagrange equation of motion and had to do some calculus of variations.
The first step in the derivation is to multiply the integral of ƒ(y(x),y'(x);x)dx from x1 to x2 by δ.
and then by the chain rule we proceed. But I cannot understand why we are...
Homework Statement
I have an issue with understanding the idea of generalised momentum for the Lagrangian.
For a central force problem, the Lagrangian is given by,
$$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...
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...
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...
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.
Hey guys, looking to get some advice on satellite navigation. Can anyone recommend a nice textbook covering Newton, Kepler, Lagrange etc and their contributions to orbital motion. Also any textbooks on MATLAB or Maple examples of orbits - relativistic or Newtonian it doesn't matter. Not put off...
Hello,
When doing a little internet search today on generalized coordinates I stumbled on this document:
http://people.duke.edu/~hpgavin/cee541/LagrangesEqns.pdf
If you are willing, would you be so kind as to open it up and look at the top of (numbered) page 6?
OK, so the very existence of...
Yes, that is a serious title for the thread.
Could someone please define GENERALIZED COORDINATES?
In other words (and with a thread title like that, I damn well better be sure there are other words )
I understand variational methods, Lagrange, Hamilton, (and all that).
I understand the...
Homework Statement
I am going to paste the problem word for word, so you can have all the exact information that I have:
You’re part of a team that’s designing a rocket for a specific mission. The thrust (force) produced by the rocket’s engine will give it an acceleration of a feet per second...
Homework Statement
We've got a line element ds^2 = f(x) du^2 + dx^2 From that we should find the geodesic equation
Homework Equations
Line Element:
ds^2 = dq^j g_{jk} dq^k
Geodesic Equation:
\ddot{q}^j = -\Gamma_{km}^j \dot{q}^k \dot{q}^m
Christoffel Symbol:
\Gamma_{km}^j = \frac{g^{jl}}{2}...
Homework Statement
In this exercise, we are given a discrete Lagrangian which looks like this: http://imgur.com/TL0P61r. We have to minimize the discrete S with fixed point r_i and r_f and find the the discrete equations of motions.
In the second part we should derive a discrete trajectory for...
Hi all
I am facing a problem and I hope that you can give me a hand. Here I describe the situation
I am working on a digitizer that can detect the pen position by measuring the antennae energy that are placed in a grid fashion. To get the x coordinate of the pen I measure the energy of three...
Homework Statement
Im trying to understand the Legendre transform from Lagrange to Hamiltonian but I don't get it. This pdf was good but when compared to wolfram alphas example they're slightly different even when accounting for variables. I think one of them is wrong. I trust wolfram over the...
Hi all,
I'm working on a project to control the angles of a beam(purple) with a quadcopter(orange),see figure below. The angles for both the ground-beam and beam-quadcopter will be measured with joysticks, so only roll and pitch angles will be measured and the yaw rotation is fixed.
To obtain...
Hello everyone! I'm currently trying to plot the effective potential for Sun-Jupiter system, to show the lagrangian points in this system. I've converted to a system of units where G=1, m_sun+m_jupiter=1 and R=1, whereby I get the following equation describing the effective potential of a third...
Here is what we know from virtual work:
$$
\delta W=\sum_{i=1}^N{\vec F_i\cdot\delta\vec r_{i}}
$$
Where ##N## is the number of bodies in the system. I am considering a quadcopter, modeled as a rigid body so it is just one body and we have:
$$
\delta W=\vec F\cdot\delta\vec r
$$
My question...
www.youtube.com/watch?v=oW4jM0smS_E
That's the video I'm referencing in particular, but 1 and 3 are necessary prereqs if you're new to the matter (as I am).
He goes through and derives the product rule and power rule for polynomials using algebra.
My question is this: why don't we teach...
Hello Seniors,
I have done BSc in Physics but couldn't take lectures of Classical Mechanics. I am Almost blind in this subject. Since it's a core course in Physics, so i need your help to understand the basics in this course. If anyone of you have any helping material/notes/slides etc which...
Hi!
I need to figure out the Lagrange Equation for a rod or nail swinging from a horizontal plane. The thing is, that while it is swinging back and forth, the while nail is moving along the X axis as well. I was thinking to use 1/2mv^2+(1/2)Iø^2 . Any help would be appreciated!
Thanks.
Homework Statement
[/B]
A uniform solid ball of mass m rolls without slipping down a right angled wedge of mass M and angle θ from the horizontal, which itself can slide without friction on a horizontal floor. Find the acceleration of the ball relative to the wedge.
2. The attempt at a...
Homework Statement
A simple pendulum of length ##b## and bob with mass ##m## is attached to a massless support
moving vertically upward with constant acceleration ##a##. Determine (a) the
equations of motion and (b) the period for small oscillations.
2. Formulas
##U = mgh##
##T = (1/2)mv^2...
Hi! I've been trying to find the equation of motion for the simple pendulum using x as the generalized coordinate (instead of the angle), but I haven't been able to get the right solution...
Homework Statement
The data is as usual, mass m, length l and gravity g. The X,Y axes origin can be...