# What is D'alembert: Definition and 50 Discussions

Jean-Baptiste le Rond d'Alembert (; French: [ʒɑ̃ batist lə ʁɔ̃ dalɑ̃bɛːʁ]; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the Encyclopédie. D'Alembert's formula for obtaining solutions to the wave equation is named after him. The wave equation is sometimes referred to as d'Alembert's equation, and the fundamental theorem of algebra is named after d'Alembert in French.

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1. ### Why is Hamilton's Principle assumed to be valid for non-holonomic systems?

I am using Nivaldo Lemos' "Analytical Mechanics" textbook and on section 2.4 (Hamilton's Principle in the Non-Holonomic Case) he uses Hamilton's Principle and Lagrange Multipliers to arrive at the Lagrange Equations for the non-holonomic case. I don't understand why it is assumed that the...
2. ### I 4D d'Alembert Green's function for linearised metric

Q. Calculate the linearised metric of a spherically symmetric body ##\epsilon M## at the origin. The energy momentum tensor is ##T_{ab} = \epsilon M \delta(\mathbf{r}) u_a u_b##. In the harmonic (de Donder) gauge ##\square \bar{h}_{ab} = -16\pi G \epsilon^{-1} T_{ab}## (proved in previous...
3. ### Wave Equation: d'Alembert solution -- semi-infinite string with a fixed end

Hi, I was trying to get some practice with the wave equation and am struggling to solve the problem below. I am unsure of how to proceed in this situation. My attempt: So we are told that the string is held at rest, so we only need to think about the displacement conditions for the wave...
4. ### I D'Alembert solution for the wave equation: question about the speeds

The solution for the wave equation with initial conditions $$u(x,0)=f(x)$$ and $$u_t(x,0)=g(x)$$ Is given for example on wikipedia : $$u(x,t)=(f(x+ct)+f(x-ct)+1/c*\int_{x-ct}^{x+ct}g(s)ds)/2$$ So a vibrating string, since there is no conditions on ##g## (like ##\sqrt{1-g(x)^2/c^2}##), could...

13. ### Partial differential wave (d'Alembert) solution check please

Homework Statement Homework Equations General wave solution y=f(x+ct)+g(x-ct) [/B] The Attempt at a Solution [/B] Graphical sketch
14. ### Conservation of energy: Crane raising a mass

Homework Statement Mechanical crane raises 225kg at a rate of 0.031m/s^2. from a rest to a speed of 0.5m/s over a distance of 4m. Frictional resistance is 112N. m1=225kg a1=0.031m/s^2 u1=0m/s v1=0.5m/s s=4m Fr=112N A. Work input from the motor B. Tension in the lifting cable C. Max power...
15. ### I D'Alembert's Solution – Domain of dependence

When considering the Wave equation subject to initial conditions as follows… …then D'Alembert's solution is given by (where c is wave speed): I'd like to understand physically how this formula allows us to know the value of u (where u is the height of the wave, say) at some point (x0,t0)...
16. ### I D'Alembert equation and Galilean transformation

The D'Alembert equation for the mechanical waves was written in 1750. It is not invariant under a Galilean transformation. Why nobody was shocked about this at the time? Why we had to wait more than a hundred years (Maxwell's equations) to discover that Galilean transformations are wrong...
17. ### B Proof / derivation of d'Alembert principle?

I can't find a derivation of d'Alembert principle. Wikipédia says there is no general proof of it. Same with stackexchange. I find it surprising so I thought I'd come here to check with you guys. D'Alembert principle has indeed no proof ?
18. ### Understanding Holonomic Constraints in Lagrangian Mechanics

Hi, I'm in the masters year of a theoretical physics course which begins this September. I'm reading the classical mechanics notes ahead of time, and I came across the idea of holonomic and non-holonomic constraints. I understand that in the case of a holonomic system, you can use the...
19. ### Calculating Tractive Force using two methods

1. A car accelerates 1800m down an incline of 1 in 4 at 0.4ms^2. The car has a mass of 4,000kg and the resistance to motion is 400N Determine: a) The Tractive effort required by using D'Alemberts principle b) The Tractive effort required by using the conservation of energyHomework Equations...
20. ### D'Alembert's Principle: Finding Velocity of Hammer Before Impact

Homework Statement i have a question on D'alemberts principle in which it asks me to find the velocity of a hammer immediately before impact with a pile the information i have been given is as follows: mass of hammer;300kg height of hammer;3.5m gravity to be taken as;9.81 mass of pile ;500kg...
21. ### Graphing solutions to PDEs at various times

Homework Statement Graph snapshots of the solution in the x-u plane for various times t if \begin{align*} f(x) = \begin{cases} & 3, \text{if } -4 \leq x \leq 0 \\ & 2, \text{if } 4 \leq x \leq 8 \\ & 0, \text{otherwise} \end{cases} \end{align*} Homework Equations Assuming that c=1 and g(x)...
22. ### D'Alembert solution of wave equation on semi infinite domain

Homework Statement Wave equation: ytt=yxx Initial conditions: Y(x,0) =f(x) = x (0 ≤ x < 1) 2.5(5-x) (1 ≤ x < 3) 0 (Otherwise) and yt(x,0) = 0 Boundary condition: y(0,t) =0 Semi infinite domain: 0 ≤ x < infinity Homework Equations d'Alembert solution...
23. ### D'Alembert operator is commute covariant derivative?

For example: [itex] D_α D_β D^β F_ab= D_β D^β D_α F_ab is true or not? Are there any books sources?
24. ### Retarded Green's Function for D'Alembertian

Hey All, I recently posted this in another area but was suggested to put it here instead. Here is my original post:
25. ### Retarded Green's Function for D'Alembertian

Homework Statement Hi all, I'm currently reviewing for a final and would like some help understanding a certain part of this particular problem: Determine the retarded Green's Function for the D'Alembertian operator ##D = \partial_s^2 - \Delta##, where ##\Delta \equiv \nabla \cdot \nabla## ...
26. ### D'alembert force and what the body "really" feels

hello everybody, I have a question regarding the fictitious force (d'alembert force) we usually add to an examined body in a noninertial reference system. As I understood from reading and leraning about this topic, this force is artificially added only to compensate for exploring this body in...
27. ### MATLAB Help please in matlab -- plotting a D'Alembert wave equation

Can someone please help me in plotting a D'Alembert wave equation solution in MATLAB? I am so confused as how to plot it in MATLAB I need to plot a graph like the one below
28. ### D'alembert solution for the semi-infinite string

Homework Statement Find the solution of the wave equation using d'Alembert solution. Homework Equations u(0,t)=0[/B] and u(x,0)=0 u_t(x,0)=\frac{x^2}{1+x^3}, \, x\geq0 u_t(x,0)=0, \, x<0 The Attempt at a Solution For a semi infinite string we have the solution u(x,t)=\frac{1}{2}\left(...
29. ### Question about d'Alembert operator

Hey guys, The expression \partial_{\mu}\partial^{\nu}\phi is equal to \Box \phi when \mu = \nu. However when they are not equal, is this operator 0? I'm just curious cos this sort of thing has turned up in a calculation of mine...if its 0 I'd be a very happy boy
30. ### Solving D'alembert Equation for Wave eq. u(4,1) & u(1,4)

Homework Statement For a wave equation, utt-uxx=0, 0<x< ∞ u(0,t)=t^2, t>0 u(x,0)=x^2, 0<x< ∞ ut(x,0)=6x, 0<x< ∞ evaulate u(4,1) and u(1,4) uxx is taking 2 derivatives in respective of x Homework Equations D'alembert's equation u=(f(x+ct)+f(x-ct))/2 + (1/ct)(∫g(s)ds...
31. ### D'Alembert Method to solve Diff Eqns

I understand why this is a good method, but in one of the problems I am trying I yield 4 unknown parameters in a second order differential equation. I believe I should only have 2. Let f(x) be a part of a homogeneous solution and and u(x) be some unknown function in x. Then a particular...
32. ### D'Alembert solution of wave equation with initial velocity given

Hi there, This is a problem concerning hyperbolic type partial differential equations. Currently I am studying the book of S. J. Farlow "Partial differential equations for scientists and engineers". The attached pages show my problems. Fig. 18.4 from case two (which starts in the lower part...
33. ### D'Alembert question - boundary conditions parts

Homework Statement I have a general wave equation on the half line utt-c2uxx=0 u(x,0)=α(x) ut(x,0)=β(x) and the boundary condition; ut(0,t)=cηux where α is α extended as an odd function to the real line (and same for β) I have to find the d'alembert solution for x>=0; and show that in...
34. ### Solve Problem Using D'Alembert's Method of Virtual Works

I need someone helps me to solve this problem using method of virtual works by d'alembert. The datas are: m1=2kg;m3=4kg;m=0,5kg;M=1.5kg;R=0.6;r=0,2m;\alpha=30°. One of the requests is to calculate the acceleration of the mass m1, the solution is: 3.90m/s^2. The wheel is formed by two coassial...
35. ### D'Alembert Problem for 1-D wave equation

[b]1. For the 1-D wave equation, the d’Alembert solution is u(t, x) = f (x + ct) + g(x − ct) where f , g are each a function of 1 variable. Suppose c = 1 and we know f (x) = x^2 and g(x) = cos 2x for x > 0. Find u(t, x) for al l t, x ≥ 0 if you are also given the BC: u ≡ 1 at x = 0...
36. ### Goldstein's derivation of E-L equations from D'Alembert

In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein arrives at the expression (equation 1.46) \mathbf{v}_i = \frac{d\mathbf{r}_i}{dt} = \sum_k \frac{\partial \mathbf{r}_i}{\partial q_k} \dot{q}_k + \frac{\partial \mathbf{r}_i}{\partial t} where \mathbf{r}_i...
37. ### PDEs- D'Alembert Solution of the Wave Equation

Homework Statement The question is here: http://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/assignments/probwave1solns.pdf It's a long question and I figured attaching the link here would be better. I need help with the question on page 4. when...
38. ### D'alembert solution Help (easy)

Hi I have been trying to figure out how this method works. Searched the forum but got no results. Have attached an example I am trying to do. If someone could show me it would be great. Or even show me a similar example worked out. Thanks in advance
39. ### Velocity in D'Alembert solution is the same as virtical velocity?

One dimensional wave equation: \frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2} Where c is the vertical velocity of the vibrating string. This will give D'Alembert solution of u(x,t) = \frac{1}{2}[f(x+ct) + G(x+ct)] + \frac{1}{2}[f(x+ct) + G(x+ct)] Where...
40. ### D'Alembert Ratio Test: Convergence test

Hi this is just a general question about using the ratio test for convergence. If I have to carry out the test to find out if something converges (and I don't need to find out if its absolutely converges, but just convergence), then can my answer to the test be negative? Or does the...
41. ### How Does Symmetry Affect the Solution to D'Alembert's Problem?

Homework Statement \frac{\partial u^2}{\partial t^2} = c^2 \frac{\partial u^2}{\partial x^2} \;\;,\;\; u(0,t)=u(L,t)=0 \;\;,\;\; u(x,0)=f(x) \;\;,\;\; \frac{\partial u}{\partial t}(x,0)=g(x) f(x) \;and\; g(x) \;are\; symmetric\; about\;\; x=\frac{L}{2} \;\Rightarrow f(L-x)=f(x)...
42. ### Solving Wave Equation with D'Alembert: Step by Step

Homework Statement Ok so hope someone will be able to help... I've used the D'Alembert method to solve the wave equation and have got that the general form should be y(x,t) = f(x+ct) + g(x-ct) Now I am also told that the time dependence at x=0 is sinusoidal.. that is, y(x,0) =...
43. ### D'Alembert solution with f(x)=0, g(x)=x

Homework Statement This is an example I copy from the book. The book showed the steps of solving and provide the answer. I don't understand the book at all. Below I show the question and the solution from the book. Then I am going to ask my question at the bottom. Question Use...
44. ### Why the book call f(x+ct) and f(x-ct) odd extension of D'Alembert Method?

For wave equation: \frac{\partial^2 u}{\partial t^2} \;=\; c^2\frac{\partial^2 u}{\partial x^2} \;\;,\;\; u(x,0)\; =\; f(x) \;\;,\;\; \frac{\partial u}{\partial t}(x,0) \;=\; g(x) D'Alembert Mothod: u(x,t)\; = \;\frac{1}{2} f(x\;-\;ct)\; +\; \frac{1}{2} f(x\;+\;ct)\; +\; \frac{1}{2c}...
45. ### Question regarding D'Alembert solution for one dimension wave equation

In am studying PDE and I have question about D'Alembert solution for one dimension wave equation. I am going to reference Wolfram: http://mathworld.wolfram.com/dAlembertsSolution.html 1) I want to verify the step of \frac{\partial y_0}{\partial t} of step (14) of the page...
46. ### D'alembert for semi-infinite string (on R-)

D'Alembert problem for semi infinite string on R- utt=c2 uxx ( -\infty<x<0) Initial condition: u(x,0)=f(x) ut(x,0)=g(x) Boundary condition: u(0,t)=0 please help me to solve it
47. ### D'Alembert's Principle: Is the Force Conservative?

Well,most of You know this principle, I just know 1 part of his work, which is that there is a force in an accelerated frame on a mass that equals MA and is in the opposite direction. so I have a question, is the force(fictitious or not) used in the accelerated frame ,conservative or not...
48. ### Using the d'Alembert operator for wave equations

1. Is there anywhere where I can find a derivation for the generalized inhomogeneous wave equation? I found a derivation of the 1D wave equation for an infinitesimal region of an elastic string here: http://www.math.ubc.ca/~feldman/apps/wave.pdf But I am looking for proof of the...
49. ### THE D'Alembert Solution for the 1D Wave Equation

Homework Statement I am looking at the derivation of the D'alembert equation, and I'm having trouble with understanding where the limits of integration come in. Homework Equations Given the 1-d wave equation: u_{tt} = c^2u_{xx} , with the general solution u(x,t)= \theta(x-ct) +...
50. ### What is the D'Alembert operator

I've seen two different textbooks write two different expressions for this, what is the proper D'Alembert Operator?