What is Lagrangian: Definition and 1000 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.
Homework Statement
A meter stick stands on a frictionless surface and leans against a frictionless wall as shown. It is released to fall when it makes an angle of 1 degree from the vertical. Use Lagrange and Euler to find how long it takes the stick to fall to the ground.
The Attempt...
The Hamiltonian operator in quantum field theory (of Klein Gordon Lagrangian) is
H=\frac{1}{2} \int \frac{d^3p}{(2 \pi)^3} \omega_{\vec{p}} a_{\vec{p}}^\dagger a_{\vec{p}} after normal ordering
Now we construct energy eigenstates by acting on the vacuum |0 \rangle with a_{\vec{p}}^\dagger...
Homework Statement
In a shallow layer of water, the velocity of water in the z direction may be ignored and is therefore (\dot{x},\dot{y}). We can define the Lagrangian coordinates such that the depth of water h is satisfied by the relations
Given that h = \frac{1}{\alpha} and \alpha =...
Homework Statement
A cylinder of mass m and radius R rolls without slipping down a wedge of mass M. The wedge slides on a frictionless horizontal surface. The angle between the wedge's hypotenuse & longest leg (which lies on the frictionless ground) is beta. The wedge's hypotenuse DOES have...
Hello,
is it possible to write a Lagrangian (L) for a simple RC (or RL) circuit?
Normally L = kinetic - potential energy, but how would you write
this for an RC circuit?
thanks!
Please teach me this problem:
It seem that following Haag's theorem there not exist quantized equation of motion for interacting fields.So I don't understand how to know the form of interacting Lagrangian has form of product of fields(example Lagrangian of Fermi field interacting with...
1. A rigid straight uniform bar of mass m and length l is attached by a frictionless hinge
at one end to a fixed wall so that it can move in a vertical plane. At a distance a from
the hinge it is supported by a spring of stiffness constant k, as shown in the figure
Ignoring gravitational...
Hi all
I've found a way to include dissipation in the kinetic energy of the lagrangian for simple systems and I want to know if its ok to do this. My understanding is that dissipation is typically included using the Rayleigh dissipation function which is separate from the Lagrangian.
The...
Ive been doing some research on the title concepts...
And would love it if someone could answer some questions because I can't seem to find the answer anywhere.
1) How was the lagrangian found? I know its kind of defined, and there are other lagrangians- but is there an idea behind it or was...
Hey,
can somebody show me how to derive that the lagrangian in classical phyics is L=T-V
i have seen this formula so many times, but i have no idea where it really comes from?
Homework Statement
http://www.facebook.com/photo.php?fbid=1612917330439&set=a.1250823718325.2039017.1461460506&ref=fbx_album"
The Attempt at a Solution
I don't want to ask about the full solution for this thing, but only one thing: in that expression, which is the kinetic energy and...
Homework Statement
A combination of masses along the z-axis is separated by a distance 'a' with middle mass at origin. The potential is
V = \frac{1}{2}kx^2 .
What is the force of constraint using Lagrange multiplier?
Homework Equations
L = T - V + \lambda f
The Attempt at a...
Verify that the Lagrangian density
L= \frac{1}{2} \partial_\mu \phi_a \partial^\mu \phi_a - \frac{1}{2} m^2 \phi_a{}^2
for a triplet of real fields \phi_a (a=1,2,3) is invariant under the infinitesimal SO(3) rotation by \theta
\phi_a \rightarrow \phi_a + \theta \epsilon_{abc} n_b \phi_c...
Homework Statement
I am trying to get an equation of motion for the following (seemingly simple) setup. You place on a rod on a pivot. The rod's centre of mass is precisely over the pivot. Think of balancing a ruler horizontally on your finger. Gravity, of course acts downward.
The...
I have to create a simulation of the pendulum shown in the .pdf at the bottom of the page. The 3 rods are free to rotate around their pivots in a plane. The two edge rods are connected as close to their edges as possible. There is no friction.
Unfortunately my equations of motion are spitting...
I'm probably making a mistake, but looking at the free field lagrangian for QED
\mathcal{L} \propto (-F^{\mu\nu}F_{\mu\nu}) \propto (\mathbf{E}^2 - \mathbf{B}^2)
it appears to me that the action is not bounded from above, nor from below.
Does that mean the equations of motion we obtain by...
Homework Statement
Eulerian velocity: V_{1}=-z_{1}^{2}
V_{1}=\frac{dz_{1}}{dt}
z_{1}(t=0)=x_{1}
This is supposed to become the Lagrangian velocity of:
z_{1}=\frac{x_{1}}{1+tx_{1}}
I don't understand how to take the Eulerian velocity and transform it to Lagrangian.
Homework EquationsThe...
Homework Statement
http://img85.imageshack.us/gal.php?g=hw1y.jpg
Its an imageshack gallery
Homework Equations
Book gives completely irrelevant equations.
The Attempt at a Solution
I couldn't even solve A. I have no clue how to start this. The instructor isn't providing any...
Homework Statement
[PLAIN]http://mityaka.com/users/kolodny/img/lagprob.png
Homework Equations
L = T - V
T = \frac{1}{2}*m*U2
Vs = \frac{1}{2}*k*x2The Attempt at a Solution
I worked out the equations of motion as:
FL = m*\ddot{y}+k*y-k*b*\theta
FL*e =...
Find the Euler – Lagrange Equation when
L = -1/2 (D_p a_u)(D^p a^u) \sqrt{-g} dx^4
Use g_u_v to raise/lower indices
D_p is the covariant derivative
I am very new at this notation and am having a lot of trouble getting anywhere with this.
I know I have to take the action:
S = \int Ldt...
Homework Statement
When writing down the Lagrangian and the writing down Euler-Lagrange equation I'm having some difficulties with reasoning something.
Homework Equations
Lagrangian is:
\mathcal{L}=\frac{1}{2}mv^2-q\phi+\frac{q}{c}\vec{v}\cdot\vec{A}.
Euler-Lagrange eq...
Homework Statement
A spring of rest length L_0 (no tension) is connected to a support at one end and has a mass M attached at the other. Neglect the mass of the spring, the dimension of the mass M, and assume that the motion is confined to a vertical plane. Also, assume that the spring only...
I remember when I learned some basic continuum mechanics, Lagrangian is just a integral of lagrangian density over space, which is quite easy to accept because it's just a continuous version of L=T-U. Now I'm trying to start a bit QFT and notice that Lagrangian is an integral over space-time...
Hi, hopefully someone can check my logic.
I have a lever with a mass on top which is rising towards the vertical on a frictionless pivot. The length of the lever can change. (In reality this is a robot rocking onto a foot and straightening its leg). The intention is to bring the lever to a...
In Ryder's text, he defines the dual tensor as the anti-symmetric \tilde F^{\nu \mu} = \epsilon^{\nu \mu \alpha \beta} F_{\alpha \beta}. Later he plops down the complex scalar field Lagrangian as
L = (D_\mu \phi)(D^\mu \phi *) - m^2 \phi * \phi - \frac{1}{4} F^{\nu \mu}F_{\nu \mu}
where...
The Lagrangian finite strain tensor is defined as:
E_{i,j}=\frac{1}{2}\left(\frac{\partial x_k}{\partial X_i}\frac{\partial x_k}{\partial X_j}-\delta _{i,j}\right)
Is it in Einstein Notation so that there is a summation symbol missing, i.e. would it be the same thing if one wrote it as...
Hello
I am using Landau's mechanics Vol I for classical mechanics. On page 4 he mentions for Lagrangian of a system composed of two systems A and B which are so far away so that their interactions can be neglected.
then for the combined system we have L = LA + LB
I'm trying to...
Basic exercise for finding a Lagrangian from the Landau's "Mechanics"
Hello everyone!
Homework Statement
I've just started preparing for the classical mechanics course using only Landau & Lifgarbagez, so I'm doing everything according to their formulation.
And so I solved an exercise...
Does anyone know where I can find the lagrangian for this?
From memory I believe it looks something like
S = \frac{1}{2} \int \frac{d\tau}{e}[\dot{X}^2 +i \dot{\psi}{\psi}-2ie\nu \dot{X} \psi]
where e is the graviton and nu is the gravitino. Does anyone know of a reference that...
Starting on the topic of the Lagrangian, I have been told not to try to make intuitive sense, but just accept the nice differential equations which it goes into. Fine, but it should at least make basic sense. That L= T-V should be stationary means then d(T-V) = dT-dV = 0, i.e., dT=dV, which...
Hey, I'm doing some examples in QFT and I don't want to go too far with this one:
Doing gauge symmetries, we first introduce the Unitary spacetime-dependent gauge transformation that gives us a gauge potential. With the new gauge added Lagrangian, I want to take its variation to confirm the...
Is the Lagrangian of the neutral Proca field
\mathcal{L}=-\frac{1}{16\pi}\left(F^{\mu\nu}F_{\mu\nu}-2m^2 A_{\mu} A^{\mu}\right)
symmetric?
And How to make sure whether it's symmetric.
"invariant" Lagrangian or action
Hello everyone,
I tried to describe my question but it seems getting too complicated and confusing to write down my thoughts in detail, so I am trying to start with the following question...
Are invariance of the Lagrangian under a transformation and...
L=\frac{1}{2}m(\dot{q}_1-\dot{q}_2)^2-V(q_1,q_2)
Because if we put
p_1=\frac{\partial L}{\partial \dot{q}_1}
p_2=\frac{\partial L}{\partial \dot{q}_2}
we get
p_1=-p_2=m(\dot{q}_1-\dot{q}_2)
We can't invert to get \dot{q_1} in terms of the two momenta. We can still write down a...
I am self trying to understand Lagrangian mechanics and I have come across with Degree of freedom and constraints which I think I understood in bits. So please try to explain these terms to me. I use Goldstein's Classical Mechanics.
Anyone know of a Lagrangian given in terms of E and B (or equivalently the tensor F) that yields Maxwell equations? A link or reference would be appreciated.
I can write down such a Lagrangian which yields the two second-order Maxwell equations, but not the usual four 1st order equations...
I heard that in classical field theory, terms in the Lagrangian cannot have more than two derivatives acting on them. Why is this?
In quantum field theory, I read somewhere that having more than two derivatives on a term in the Lagrangian leads to a violation of Poincare invariance. Is this...
Homework Statement
Hi all, I need to derive differantial equations of system with lagrange multiplier method, a disk is rolling and a bar is fixed onto the point of a disk
http://img130.imageshack.us/img130/1669/adsziss.jpg
By deniz120 at 2010-05-31
Homework Equations
The...
Homework Statement
A particle moves so that \vec{x} \equiv [x_0 exp(2t^2), y_0 exp(-t^2), z_0 exp(-t^2)] . Find the velocity of the particle in terms of x_0 and t (the LAgrangian description) and show it can be written as \vec{u} \equiv (4xt, -2yt, -2zt), the Eulerian description...
Homework Statement
If I consider a 3 dimensional problem related to a spring pendulum (a mass attached to the end of a spring of constant k and the other end of the spring remains fixed) and I want to find the equations of motion of the mass (for several initial conditions), what system of...
Hi,
so today in maths we defined the Lagrangian as L = T-V and stated Hamilton's Principle, which says that the actual path of a conservative system is the one which minimises the action
A(q)=\int^{t_{2}}_{t_{1}}L dt.
I'm a bit confused about this. What does the action represent in physical...
Hi,
I am going through my notes trying to get things straight in my head and something is confusing me. I also have the feeling it will turn out not to be very complicated, which is why I am a bit frustrated with this.
I know that there is something used in general relativity called the...
Homework Statement
A block of mass m moves on a horizontal, frictionless table. It is connected to the centre of the table by a massless spring, which exerts a restoring force F obeying a nonlinear version of Hooke's law,
F = -kr + ar^3
where r is the length of the spring. Show that the...
Homework Statement
Consider the spherical pendulum. In other words a particle with mass m constrained to move over the surface of a sphere of radius R, under the gravitational acceleration \vec g.
1)Write the Lagrangian in spherical coordinates (r, \phi, \theta) and write the cyclical...
Hello everyone,
I have a problem with 4 degrees of freedom, 2 of which are superfluous. My goal is to derive the equations of motion.
I have derived the equations connecting the superfluous DoFs with the independent ones, however they are nonlinear and interlaced, which means that I cannot...
Hi guys, I have a question about finding a lagrangian formulation of a theory.
If I have a system for which I know the equations of motion but not the form of the lagrangian, is it possible to find the lagrangian that will give me those equations of motion? Is there a systematic way of doing...