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.
The Lagrangian, $$\mathcal L(x)= \frac 1 2 \partial^{\mu} \phi (x) \partial_{\mu} \phi (x) - \frac 1 2 m^2 \phi (x)^2$$ for a scalar field ##\phi (x)## is said to be Lorentz invariant and to transform covariantly under translation.
What does it mean that it transforms covariantly under translation?
I read that a Lagrangian is renormalizable if it contains only triple and quadruple vertices, or at most four powers of the fields.
Where can I read more about the precise mathematical conditions?
Hi all currently got a lagrangian function which i've found to be :
\begin{equation}\mathcal{L}=\frac{1}{2}m(\dot{x}^2+\dot{y}^2+4x^2\dot{x}^2+4y^2\dot{y}^2+8xy\dot{x}\dot{y})- mg(x^2+y^2)
\end{equation}
Let us first calculate
$$(\frac{\partial L}{\partial \dot{x}})$$ which leads us to...
I was studying a derivation of noether's theorem mathematically and something struck my eyes.
Suppose you have ##L(q, \dot q, t)## and you transform it and get ##L' = L(\sigma(q, a), \frac{d}{dt}\sigma(q,a), t)##. ##\sigma## is a transformation function for ##q##
Let's represent ##L'## by...
Hi, as we know SM is symmetric under SU(3) X SU(2) X U(1) , But my question is , how can we check the invariance of terms in Lagrangian under these symmetries - thanks
The tutor solved the problem using kinetic spinning energy though I find it very difficult and confusing to do so, therefore, I would like to know if there is a way to solve the problem using effective potential energy,
Veff = J2/(2mr2
below is a sketch of the problem
hi, I have go through many books - they derive Dirac equation from Dirac Lagrangian, KG equation from scalar Lagrangian - but my question is how do we get Dirac or scalar Lagrangian at first place as our starting point - kindly help in this regard or refer some book - which clearly elaborate...
hi, when we say in SM , we can add terms having dimension 4 or less than that- in this to what dimension we are refering ? kindly help how do you measure the dimension of terms in Lagrangian. thanks
I'm following the derivation in Lancaster and Blundell. First, the Lagrangian for the free particle is ##L=-\frac {mc^2} {\gamma}## and the action ##S=\int -\frac {mc^2} {\gamma} \, dt##. Then, EM is "turned on" with the potential energy ##-qA_{\mu}dx^{\mu}##. Then, they say, the action becomes...
I am writing a 2D hydrocode in Lagrangian co-ordinates. I have never done this before, so I am completely clueless as to what I'm doing. I have a route as to what I want to do, but I don't know if this makes sense or not. I've gone from Eulerian to Lagrangian co-ordinates using the Piola...
If we have a natural unit Lagrangian, where some fundamental quantities have been excluded to ease calculations...and aim to restore it's S.I units back, do we just have to plug back the fundamental quantities that were initially excluded Into the Lagrangian...or we use some specific scaling...
We have a Lagrangian of the form:
$$
\mathcal{L} = \overline{\psi} i \gamma_{\mu} \partial^{\mu} \psi - g \left( \overline{\psi}_L \psi_R \phi + \overline{\psi}_R \psi_L \phi^* \right) + \mathcal{L}_{\phi} - V(|\phi|^2)
$$
Essentially, what we are studying is spontaneous symmetry breaking...
Can somebody explain why the kinetic term for the fluctuations was already diagonal and why to normalize it, the sqrt(m) is added? Any why here Z_ij = delta_ij?
Quite confused about understanding this paragraph, can anybody explain it more easily?
One of the constraints is given as ##r=R##, which is very obvious. The second constraint is however given as
$$\phi - \frac {2\pi} h z=0$$
where ##h## is the increase of ##z## in one turn of the helix. Physically, I can't see where this constraint comes from and how ##\phi=\frac {2\pi}h z##.
I had an interesting thought and it just might be me, but I'm looking forward to hearing your thoughts, but not like the thoughts - "yeah, Landau is messy, complicated, don't read that, e.t.c". Just think about what you think about my thoughts.
Landau first starts to mention the variational...
Using free scalar field for simplicity.
Hi all, I have a question which is pretty simple, we have the path integral in QFT in the presence of a source term:
$$
Z[J] = \int \mathcal{D}\phi \, e^{i \int d^4x \left( \frac{1}{2} \phi(x) A \phi(x) + J(x) \phi(x) \right)}
$$
So far so good. Now...
I feel like I might be posting many questions, so I hope you're not angry with me on this.
I'm following Landau's book and try to understand why L = K - U but first we need to figure out why it contains K.
Landau discusses two inertial frames where the speed of one frame relative to another is...
When we have ##L(q, \dot q, t)##, The change in action is given by:
##\int_{t_1}^{t_2} dt (\frac{\partial L}{\partial x} f(t) + \frac{\partial L}{\partial \dot x} \dot f(t))## when we change our true path ##x(t)## by ##x(t) + \epsilon f(t)##.
Now, attaching the image, check what Landau says...
This isn't a homework problem (it's an example from David Tong's QFT notes where I didn't understand the steps he took), but I am confused as to how exactly to take the partial derivative of the Lagrangian with respect to ##\partial(\partial_\mu \mathcal{A}_\nu)##. (Note the answer is...
I have been learning emma noether’s theorem where every symmetry results in conservstion law of something, sometimes momentum, sometimes angular momentum, sometimes energy or sometimes some kind of quantity.
Every explanation that I have listened to or learned from talks about homogeneity of...
Let's say we got one particle in x direction only and we got some motion x(t) which we figured it out through Lagrangian.
In Noether's theorem for coordinate transformation symmetry, we start with the following:
x(t)' = x(t) +εf(t) (ε - some number) - I denoted new path with x(t)'
I'm...
Hello! I have a Lagrangian of the form:
$$L = \frac{mv^2}{2}+f(v)v$$
where ##f(v)## is a function of the velocity. I would like to derive the equation of motion in general, without writing down an expression for ##f(v)## yet. I have that ##\frac{\partial L}{\partial x} = 0##. However, what is...
We were introduced the lagrangian for a particle moving in an eletromagnetic field (for context, this was a brief introduction before dealing with Zeeman effect) as $$\mathcal{L}=\dfrac{m}{2}(\dot{x}^2_1+\dot{x}^2_2+\dot{x}^2_3)-q\varphi+\dfrac{q}{c}\vec{A}\cdot\dot{\vec{x}}.$$ A...
I just calculated the Lagrangian of a particle of mass ##m## in a radially symmetric potential ##V(r)##. I thought it would be a good idea to switch to spherical coordinates for that matter. What I get is
$$
L = \frac{1}{2} m \left( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \dot{\varphi}^2...
I want to use the Lagrangian approach to find the equation of motion for a mass sliding down a frictionless inclined plane. I call the length of the incline a and the angle that the incline makes with the horizontal b. Then the mass has kinetic energy 1/2m(da/dt)2 and the potential energy should...
I found a the answer in a script from a couple years ago. It says the kinetic energy is
$$
T = \frac{1}{2} m (\dot{\vec{x}}^\prime)^2 = \frac{1}{2} m \left[ \dot{\vec{x}} + \vec{\omega} \times (\vec{a} + \vec{x}) \right]^2
$$
However, it doesn't show the rotation matrix ##R##. This would imply...
Hi,
I am not quite sure whether I have solved the following problem correctly:
I have now set up Lagrangian in general, i.e.
$$L=T-V=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)-mgz$$
After that I imagined how ##x##,##y## and ##z## must look like and got the following:
$$x=\beta \cos^2(\alpha r)...
I'm having trouble understanding how to find out whether or not a stationary point is a minimum and I'm hoping for some clarification. In my class, we were shown that, using Euler's equation, the straight-line path:
with constants a and b results in a stationary point of the integral:
A...
Let E be a fixed immutable quantity. E can be freely exchanged between T and V, as long as $$T + V = E$$
1. What does the quantity $$\int_x T - V $$ signify? What is the importance of this quantity?
--------------------
Let E now be the budget of a factory. E can either be spent on T or V in...
The system is shown below. It consists of a rod of length ##L## and mass ##m_b## connecting a disk of radius ##R## and mass ##m_d## to a collar of mass ##m_c## which is in turn free to slide without friction on a vertical and rigid pole. The disk rolls without slipping on the floor. The ends...
Hello! I have the following Lagrangian:
$$L = \frac{1}{2}mv^2+fv$$
where ##v = \dot{x}##, where x is my coordinate and f is a function of v only (no explicit dependence on t or x). What I get by solving the Euler-Lagrange equations is:
$$\frac{d}{dt}(mv+f+\frac{\partial f}{\partial v} v) =...
I'm just starting to get into QFT as some self study. I've watched some lectures and videos, read some notes, and am trying to piece some things together.
Take ##U(1)_{EM}: L = \bar{\psi}[i\gamma^{\mu}(\partial_{\mu} - ieA_{\mu}) - m]\psi - 1/4 F_{\mu\nu}F^{\mu\nu}##
This allegedly governs spin...
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...
I am trying to solve this and get the equations of motion using the Lagrangian method.
I could do all the steps but the equations (especially the third one) seems..weird.
What am I doing wrong? Sorry if the equations aren't in their simplest form, they are pulled straight from Wolfram...
hi, i have seen lagrangian density for spin 0 , spin 1/2, spin 1 , but i am not getting from where these langrangian densities comes in at a first place. kindly give me the hint.
thanks
I'm having trouble following a proof of what happens when the Dirac Lagrangian is put into the Euler-Lagrange equation. This is the youtube video: and you can skip to 2:56 and pause to see all the math laid out. I understand the bird's eye results of the Dirac Lagrangian having an equation of...
I've learned that in canonical quantization you take a Lagrangian, transform to a Hamiltonian and then "put the hat on" the fields (make them an operator). Then you can derive the equations of motion of the Hamiltonian.
What is the reason that you cannot already put hats in the QFT Lagrangian...
I understand the process of the calculus of variations.
I accept that a proper Lagrangian for Dynamics is "Kinetic minus potential" energy.
I understand it is a principle, the same way F=ma is a law (something one cannot prove, but which works)
Still... what do you say to students who say "I...
Hi all experts!
I would like to read about the Lagrangian of a classical (non-quantum), real, scalar, relativistic field and how it is derived. What is the best book for that purpose?Sten Edebäck
Hi,
I am reading Robert D Klauber's book "Student Friendly Quantum Field Theory" volume 1 "Basic...". On page 48, bottom line, there is a formula for the classical Lagrangian density for a free (no forces), real, scalar, relativistic field, see the attached file.
I like to understand formulas...
In https://arxiv.org/pdf/1709.07852.pdf, it is claimed in equation (1) and (2) that when we take non-relativistic limit, the following Lagrangian (the interaction part)
$$L=g \partial_{\mu} a \bar{\psi} \gamma^{\mu}\gamma^5\psi$$
will yield the following Hamiltonian
$$H=-g\vec{\nabla} a \cdot...
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...
Setting up coordinates for the problem
##L=\frac{1}{2}M_1 \dot{x}^2+\frac{1}{2}M_2(\dot y-\dot x)^2+M_1gx+M_2g(l_a-x+y)##
After using Euler Lagrange for x component and y component separate and substitute one to another then I get that ##\ddot{x}=\frac{M_1-2M_2}{M_1}g##
whereas on the...
What are the rules for writing a good Lagrangian?
I know that it should be a function of the position and its first order derivatives, because we know that we only need 2 initial conditions (position and velocity) to uniquely determine the future of the particle.
I know that the action has to be...
If the standard model Lagrangian were generalized into what might be called "core capabilities" what would those capabilities be? For example, there are a lot of varying matrices involved in the standard model Lagrangian and we can generalize all of them as the "core capability" of matrix...
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...
Hello,
I am using the Lagrange multipliers method to find the extremums of ##f(x,y)## subjected to the constraint ##g(x,y)##, an ellipse.
So far, I have successfully identified several triplets ##(x^∗,y^∗,λ^∗)## such that each triplet is a stationary point for the Lagrangian: ##\nabla...
How would you unify the two Lagrangians you see in electrodynamics?
Namely the field Lagrangian:
Lem = -1/4 Fμν Fμν - Aμ Jμ
and the particle Lagrangian:
Lp = -m/γ - q Aμ vμ
The latter here gives you the Lorentz force equation.
fμ = q Fμν vν
It seems the terms - q Aμ vμ and - Aμ Jμ account for...
I need to use hermiticity and electromagnetic gauge invariance to determine the constraints on the constants. Through hermiticity, i found that the coefficients need to be real. However, I am not sure how gauge invariance would come into the picture to give further contraints. I think the...