What is Lagrange equation: Definition and 43 Discussions
Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle.
Lagrangian mechanics defines a mechanical system to be a pair
(
M
,
L
)
{\displaystyle (M,L)}
of a configuration space
M
{\displaystyle M}
and a smooth function
L
=
L
(
q
,
v
,
t
)
{\displaystyle L=L(q,v,t)}
called Lagrangian. By convention,
L
=
T
−
V
,
{\displaystyle L=T-V,}
where
T
{\displaystyle T}
and
V
{\displaystyle V}
are the kinetic and potential energy of the system, respectively. Here
q
∈
M
,
{\displaystyle q\in M,}
and
v
{\displaystyle v}
is the velocity vector at
q
{\displaystyle q}
(
v
{\displaystyle (v}
is tangential to
M
)
.
{\displaystyle M).}
(For those familiar with tangent bundles,
L
:
T
M
×
R
t
→
R
,
{\displaystyle L:TM\times \mathbb {R} _{t}\to \mathbb {R} ,}
and
v
∈
T
q
M
)
.
{\displaystyle v\in T_{q}M).}
Given the time instants
t
1
{\displaystyle t_{1}}
and
t
2
,
{\displaystyle t_{2},}
Lagrangian mechanics postulates that a smooth path
x
0
:
[
t
1
,
t
2
]
→
M
{\displaystyle x_{0}:[t_{1},t_{2}]\to M}
describes the time evolution of the given system if and only if
x
0
{\displaystyle x_{0}}
is a stationary point of the action functional
S
[
x
]
=
def
∫
t
1
t
2
L
(
x
(
t
)
,
x
˙
(
t
)
,
t
)
d
t
.
{\displaystyle {\cal {S}}[x]\,{\stackrel {\text{def}}{=}}\,\int _{t_{1}}^{t_{2}}L(x(t),{\dot {x}}(t),t)\,dt.}
If
M
{\displaystyle M}
is an open subset of
R
n
{\displaystyle \mathbb {R} ^{n}}
and
t
1
,
{\displaystyle t_{1},}
t
2
{\displaystyle t_{2}}
are finite, then the smooth path
x
0
{\displaystyle x_{0}}
is a stationary point of
S
{\displaystyle {\cal {S}}}
if all its directional derivatives at
x
0
{\displaystyle x_{0}}
vanish, i.e., for every smooth
{\displaystyle \delta {\cal {S}}\ {\stackrel {\text{def}}{=}}\ {\frac {d}{d\varepsilon }}{\Biggl |}_{\varepsilon =0}{\cal {S}}\left[x_{0}+\varepsilon \delta \right]=0.}
The function
δ
(
t
)
{\displaystyle \delta (t)}
on the right-hand side is called perturbation or virtual displacement. The directional derivative
δ
S
{\displaystyle \delta {\cal {S}}}
on the left is known as variation in physics and Gateaux derivative in Mathematics.
Lagrangian mechanics has been extended to allow for non-conservative forces.
Consider the above system, where both the wedge and the mass can move without friction. We want to get the equations of motion for the both of them using Lagrangian formalism, where the constraints in the solution sheet are given as:
$$y_2=0$$
$$\tan \alpha=\frac {y_1}{x_1-x_2}$$
However...
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...
I'm stuck in a problem of a spring mass system with a pendulum attached to it as showed in the figure below:
My goal is to find the movement equation for the mass, using Lagrangian dynamics.
If the spring moves, the wire will move the same amount. Therefore, we can write the x and y position...
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...
Hi
I've written D'Alembert's principle as you can see in the attached files. I computed the virtual work done by the weight and the elastic force (since the work done by the normal force is zero) and then I used the fundamental hypothesis, which states that the constraint forces can be written...
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.
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...
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 copy again the statement here:
So, I think I solved parts a to c but I don't get part d. I couldn't even start it because I don't understand how to set the problem.
I think it refers to some kind of motion like this one in the picture, so I'll have a maximum and a minimum r, and I can get...
Homework Statement
This could be a more general question about pendulums but I'll show it on an example.
We have a small body (mass m) hanging from a pendulum of length l.
The point where pendulum is hanged moves like this:
\xi = A\sin\Omega t, where A, \Omega = const. We have to find motion...
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
Show that for an arbitrary ideal holonomic system (n degrees of freedom)
\frac{1}{2} \frac{\partial \ddot T}{\partial\ddot q_j} - \frac{3}{2} \frac{\partial T}{\partial q_j} = Q_j
where T is kinetic energy and qj generalized coordinates.[/B]Homework Equations...
Homework Statement
For the following integral, find F and its partial derivatives and plug them into the Euler Lagrange equation
$$F(y,x,x')=y\sqrt{1+x'^2}\\$$
Homework Equations
Euler Lagrange equation : $$\frac{dF}{dx}-\frac{d}{dy}\frac{dF}{dx'}=0$$
The Attempt at a Solution...
am deriving lagrange's equation can anybody help me to understand this identity
the book says that he is using the chain rule for it but am not getting it
d/dt(∂x/∂q)
the identity is in the screen shot
thanks :)
With coordinates q en basis e ,textbooks use as line element :
ds=∑ ei*dqi But ei is a function of place, as one can see in deriving formulas for covariant derivative. Why don't they use as line element the correct:
ds=∑ (ei*dqi+dei*qi)
Same question in deriving covariant derivative,
Hi,
I have read this paper “Dynamic equations of motion for a 3-bar tensegrity based mobile robot” (1) and this one “Dynamic Simulation of Six-strut Tensegrity Robot Rolling”.
1) http://digital.csic.es/bitstream/10261/30336/1/Dynamic%20equations.pdf
I am trying to implement a tensegrity...
I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality
δ(dq/dt) = d(δq)/dt
Where q is some coordinate, and δf is the first variation in...
Homework Statement
Find the equations of motion for both r and \theta of
Homework Equations
My problem is taking the derivative wrt time of
and
\dfrac{\partial\mathcal{L}}{\partial\dot{r}}=m \dot{r} \left( 1 + \left( \dfrac{\partial H}{\partial r}\right)^2 \right)
The Attempt at a...
Homework Statement
Derive the equations of motion and show that the equation of motion for \phi implies that r^2\dot{\phi}=K where K is a constant
Homework Equations
Using cylindrical coordinates and z=\alpha r
The kinetic and potential energies are...
Hi! Does the Lagrange equation ONLY apply when the constraints are holonomic? What about the constraining forces acting on the system (i.e. normal force, or other perpendicular forces), do they make a system holonomic?
What about the Lagrange equation with the general force on the right hand...
I have a system with one generalized coordinate, x. In the potential energy part of the lagrangian, I have some constants multiplied by the absolute value of x. That is the only x dependence the lagrangian has, so when I take the partial derivative of the lagrangian with respect to x (to get the...
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...
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...
Homework Statement
The lagrange equations are obtained as in the picture. I am only showing the final part of the solution, where they consider the final case of x≠y≠z.
Homework Equations
The equation at the second paragraph is obtained by subtracting: (5.34 - 5.35).
The final equations are...
Hello all,
I am having some frustration understanding one derivation of the Euler Lagrange Equation. I think it most efficient if I provide a link to the derivation I am following (in wikipedia) and then highlight the portion that is giving me trouble.
The link is here
If you scroll...
I want to prove that Euler Lagrange equation and Einstein Field equation (and Geodesic equation) are the same thing so I made this calculation.
First, I modified Energy-momentum Tensor (talking about 2 dimension; space+time) :
T_{\mu\nu}=\begin{pmatrix} \nabla E& \dot{E}\\ \nabla p &...
Homework Statement
I am having trouble understanding how to apply Lagrange's equation. I will present a simplified version of one of my homework problems.
Imagine an inverted pendulum, consisting of a bar attached at a hinge at point A. At point A is a torsional spring with spring...
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...
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...
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...
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...
Euler Lagrange Equation : if y(x) is a curve which minimizes/maximizes the functional :
F\left[y(x)\right] = \int^{a}_{b} f(x,y(x),y'(x))dx
then, the following Euler Lagrange Differential Equation is true.
\frac{\partial}{\partial x} - \frac{d}{dx}(\frac{\partial f}{\partial y'})=0...
Dear everyone
can anyone help me with the euler lagrange equation which is stated in d'inverno chapter 11?
in equation (11.26) it is said that when we use the hilbert-einstein lagrangian we can have:
∂L/(∂g_(ab,cd) )=(g^(-1/2) )[(1/2)(g^ac g^bd+g^ad g^bc )-g^ab g^cd ]
haw can we derive...
Can someone kind of give me a step by step as to how you get the equations of motion for this problem?
http://www.enm.bris.ac.uk/teaching/projects/2002_03/ca9213/images/msp.jpg
the answer is this:
http://www.enm.bris.ac.uk/teaching/projects/2002_03/ca9213/msp.html
Though I am not quite...
Homework Statement
A uniform disk of mass M and radius a can roll along a rough horizontal rail. A particle of mass m is suspended from the center C of the disk by a light inextensible string b. The whole system moves in a vertical plane through a rail. Take as generalized coordinates x...
Could somebody explain to me how lagrange multipliers works in finding extrema of constrained functions? also, what is calculus of variations and lagrangian mechanics, and can somebody explain to me what the lagrangian function is and the euler-lagrange equation. And, i read something about...
[SOLVED] Euler Lagrange Equation
Hi there ,
I am missing a crucial point on the proof of Euler Lagrange equation , here is my question :
\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{df}{dy^{'}}\right)=0 (Euler-Lagrange equation)
If the function "f" doesn't depend on x explicitly...
Lagrange equation of motion
(from Marion 7-7)
A double pendulum consists of two simpe pendula, with one pendulum suspended from the bob of the other. If the two pendula have equal lenghts and have bobs of equal mass and if both pendula are confirned to move in the same plane, find...
A smooth wire is bent into the form of a helix the equations of which, in cylindrical coordinates, are z=a*beta and r=b , in which a and b are constants. The origin is a center of attractive force, , which varies directly as the distance, r. By means of Lagrange’s equations find the motion...