What is Lagrangians: Definition and 58 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.
Basically, the stress energy tensor is given by $$T_{uv}=-2\frac{\partial (L\sqrt{-g})}{\partial g^{uv}}\frac{1}{\sqrt{-g}}.$$ It makes easy to calculate stress energy tensor if the variation of Lagrangian with the metric tensor is known. But it is possible to retrieve matter Lagrangian if the...
I had used the same constraint as the solution manual says.
So my two Lagrangian would be
##L_1=\frac{1}{2}m_A\dot{x_A}^2+\frac{1}{2}m_B\dot{x_B}^2+\frac{1}{2}m_C\dot{x_C}^2+m_Cgx_C+T(x_A+x_B+2x_C-c)##
whereas c is just a constant.
Of course, I have to write my Lagrangian using constraints...
Hi Pfs
When instead of the variables x,x',t the lagrangiean depends on the trandformed variables q,q',t , time may be explicit in this lagrangian and q' (the velocity of q) may appear outside. I am looking for a toy model in which tine is not explicit in L but where the velocities appear somhere...
In this article [1] we can read an explanation about Wilson's approach to renormalization
I have read that Kenneth G Wilson favoured the path integral/many histories interpretation of Feynman in quantum mechanics to explain it. I was wondering if he did also consider that multiple worlds...
I am currently studying QFT from this book.
I have progressed to the chapter of QED. In the course, the authors have been writing the Lagrangian for different fields as and when necessary. For example, the Lagrangian for the complex scalar field is $$\mathcal{L} \ = \ (\partial ^\mu...
Some or all of the planets are thought to have migrated long ago under the gravitational influence of Jupiter. Would the trojan matter at their L4 and L5 points have followed during their migration to new orbits? In other words, while the L4 and L5 points are approaching or receding from the...
I have some questions about scalar field Lagrangians, using the box notation defined as \Box \equiv \frac{\partial^2}{\partial t^2} - \nabla^2 . It's a basic, perhaps silly issue, but somehow I've managed to sweep it under the rug for a long time.
So, usually, the Lagrangian of a free scalar...
Suppose one starts with the standard Klein-Gordon (KG) Lagrangian for a free scalar field: $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}$$ Integrating by parts one can obtain an equivalent (i.e. gives the same equations of motion) Lagrangian...
What is the intuitive reasoning for requiring that a Lagrangian describing a free-field contains terms that are at most quadratic in the field?
Is it simply because this ensures that the EOM for the field are linear and hence the solutions satisfy the superposition principle implying (at least...
In field theory a typical Lagrangian (density) for a "free (scalar) field" ##\phi(x)## is of the form $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi -\frac{1}{2}m^{2}\phi^{2}$$ where ##m## is a parameter that we identify with the mass of the field ##\phi(x)##.
My question is...
Consider the following Lagrangian:
##YHLN_{1}^{c} + Y^{c}H^{\dagger}L^{c}N_{1} + \text {h.c.},##
where ##L=(N_{0}, E')## and ##L^{c} = (E^{'c}, N_{0}^{c})## are a pair of ##SU (2)## doublets and ##N_{1}## and ##N_{1}^{c}## are a pair of neutral Majorana fermions...
Homework Statement
Two blocks of equal mass, m, are connected by a light string that passes over a massless pulley. One block hangs below the pulley, while the other sits on a frictionless horizontal table and is attached to a spring of constant k. Let x=0 be the equilibrium position of the...
Hi
On page 176 of Physics from Symmetry it says (note 9)
If we assume Ψ describes our particle directly in some way what would U(1) the transformed solution Ψ' = e^iθ Ψ be which is equally allowed describe.
He is speaking of allowed solutions of Lagrangian's. Its true for all Lagrangian's I...
For relativistic particle dynamics, there are two different approaches to choosing a Lagrangian that give the same equations of motion:
The quadratic form is:
\mathcal{L} = \frac{m}{2} g_{\mu \nu} U^\mu U^\nu
where U^\mu = \frac{d x^\mu}{d \tau}
This is for the action that involves...
The title sort of says it all, but I'll clarify a bit. Is there any intuition for what Lagrangians are and what action is. I'm asking in all generality, not just for classical mechanics.
Hello everyone,
my teacher asked in last day class as a curiosity to be discussed: As a function of the space-time dimension "d", which Lagrangians containing an scalar and a fermion field are renormalisable?. Then he encouraged us to think in the interaction vertex of the form...
Consider a theory with the Lagrangian \mathcal L=\mathcal L_{free} + \mathcal L_{int} . I think if we say \mathcal L_{free} \gg \mathcal L_{int} , this means that the equations of motion will be much near to the free equations. But I'm not sure that we can prove that if in an equation of...
Hey gang,
I'm re-working my way through gauge theory, and I've what may be a silly question.
Promotion of global to local symmetries in order to 'reveal' gauge fields (i.e. local phase invariance + Dirac equation -> EM gauge field) is, as far as i can tell, always done on the Lagrangian...
Lagrangian mechanics, as you know, is very useful in today's physics. But there is a point that I can't understand.
In cases where we can write L=T-V , Lagrangian mechanics is very useful because for some problems, it gives us a easier way than Newtonian mechanics to derive the equations of...
I'm in my junior year and recently took classical mechanics. We did not cover Lagrangian or Hamiltonian mechanics which was very shocking to me. My instructor said that Lagrangian mechanics involve very little physical intuition and therefore, time would be better spent with Newtonian mechanics...
Homework Statement
Write down the Lagrangian of a simple pendulum in terms of it's angle θ to the vertical suspended from a pivot attached to a moving carriage at constant velocity ##v##. Suppose that the carriage is now moving at a velocity ##v(t)=at## so it is accelerating uniformly. Show...
I’m not very good with english, it isn’t my native language..., but I’m going to explain my question...
I’m reading the first book of Landau's series ,it’s about clasical mechanics.
In the second chapter you can find a problem about the conservation's theorem
Homework Statement
the problem...
I’m not very good with english, it isn’t my native language..., but I’m going to explain my question...
I’m reading the first book of Landau's series ,it’s about clasical mechanics.
In the second chapter you can find a problem about the conservation's theorem
the problem says The...
Hi all,
If I take an action involving two point particles coupled together by a delta function contact interaction is it possible to carry out the variation with respect to the fields? For e.g.
S = \int dt \frac{1}{2} \dot{x}^{2} + \int \int dt \dot{x}(t) \delta^{D}\left(x(t) - y(t')\right)...
Hi guys,
The title pretty much says it. Say you have a very simple 3D Lagrangian:
L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - V
So How do you tell what is conserved from a generic potential?
I know for example that if V = V(x,y,z) then the total linear momentum is not...
For a Lagrangian L(x^k,\dot{x}^k) which is homogeneous in the \dot{x}^k in the first degree, the usual Hamiltonian vanishes identically. Instead an alternative conjugate momenta is defined as
y_j=L\frac{\partial L}{\partial \dot{x}^j}
which can then be inverted to give the velocities as a...
Homework Statement
Let be the lagrangian given by
L(x,y,\dot{x},\dot{y})=\frac{m}{2}(\dot{x^2} +\dot{y^2})-V(x^{2}+y^{2})
and
L(x,y,\dot{x},\dot{y})=\frac{m}{2}(\dot{x^2} + \dot{y^2})-V(x^{2}+y^{2}) - \frac{k}{2}x^{2}
and the transformation
x'=\cos\alpha x - \sin\alpha y
y'=\sin\alpha x +...
Homework Statement
I have a lagrangian written as:
\mathcal{L}_H = \text{Tr}\left[\,(D_\mu \Phi)^\dagger D^\mu \Phi\right] - \mu^2 \text{Tr}\left[\,\Phi^\dagger \Phi\right] - \lambda (\text{Tr}\left[\,\Phi^\dagger \Phi\right])^2
Where the field is:
\Phi \equiv \frac{1}{\sqrt{2}}(i...
So let's say we have a mechanical system described by some Lagrangian L=L(q_i,\dot{q}_i), where the qi's are the generalized coordinates of the system. Does the condition
\frac{\partial L}{\partial q_i}=0
give the equilibrium configurations of the system? Intuitively it seems so, but I can't...
I am steadily working my way through D'Inverno and have reached Chapter 20. On page 272 there is a problem which goes along the lines of ... here is a Lagrangian ... show that the Einstein tensor can be derived from it. The Lagrangian in question is a 'quadratic' Lagrangian and has four terms...
Hi,
I'm trying to clear up a confusing point in the book by José and Saletan, concerning equivalent Lagrangians (in the sense that they give you the same dynamics). It is clear that if
L_1 - L_2 = \frac{d\phi ( q,t )}{dt},
then L_1 and L_2 will have the same equations of motion. However...
I've heard that using Lagrangians to solve mechanics problems is much more efficient and easier than using Newton's laws. In your opinion, is it too early for a student to learn lagrangians for a first year due to a lack of exposure of the mathematics required?
Hi,
In SUSY we introduce chiral superfield, vector superfield, then build some invariants and get the SUSY lagrangians which after decomposition into normal fields (F and D terms) gives us for example ordinary QED plus some other terms. And we call this SUSY-QED.
I have following question...
I'm looking for a good reference work on time-dependent Lagrangians. For example, the Lagrangian and resultant Euler-Lagrange equations for a forced harmonic oscillator. All the classical textbooks just skip over this subject area. Obviously the system is non-energy conserving. In deriving...
Okay, so two equal masses are connected by spring with spring constant k. The kinetic energy is obviously 1/2*m*x1dot^2 +1/2*m*x2dot^2. Please excuse my notation. x1 and x2 are the positions, x1dot and x2dot are the velocities. L is the length of the spring when not stretched.
So anyway...
Are More Complicated Lagrangians "Wrong"?
When deriving physics from a postulated Lagrangian (like in Landau's books) we demand the simplest (i.e. the one with the lowest order terms) that obeys some symmetries. Are more complicated Lagrangians "wrong"? Or are they actually better...
Hi!
Our TA told us, that it may be not always possible to change lagrangian into hamiltonian using Legendre transformation. As far as I'm concerned the only such possibility is that we can not substitute velocity (dx/dt) with momenta and location(s). And so, we've been tryging to come up with an...
Meaning of "h.c." in Lagrangians (& elsewhere?)
I am fairly new to particle physics and am puzzled by an abbreviation I often see in Lagrangians here (though it may not be particular to that application): " + h.c." is tacked on after other terms. What does this denote? Apologies if I've missed...
(1) How can a generalized velocity function, \dot{q}, be "independent" of the corresponding generalized position function, q. One is the derivative of the other.
(2) How can any Lagrangian function be "time-independent", given that its component functions are defined as functions that depend...
I find the Lagrangian associated with the Dirac equation given in texts as
\mathcal{L}=\bar{\psi}\left(i\gamma^\mu \partial_\mu - m\right)\psi
or
\mathcal{L}=i\bar{\psi}\gamma^\mu \partial_\mu \psi- m\bar{\psi}\psi
\mathcal{L}=i \psi^{\dagger}\gamma^0\gamma^\mu \partial_\mu \psi-...
Is it possible to do statics using lagrangians? (specifically building up a compound shape like a bridge from constrained point masses). Where could I see an example of this?
This is a continuation of an original thread first posted by me on May 11th 2010. Altabeh has been very kindly trying to guide me towards a resolution.
I started the thread when I realized that in producing an answer to (i) of Problem 11.7 in D'inverno, I had ignored the term...
There is nothing particular quantum about this question but I'm posting it here because I think the quantum folks are likely more familiar with the topic. Hope that's ok.
There are two ways of looking at field Lagrangian densities in relation to particle Lagrangians.
(1) A particle (one...
Oh dear! I thought I had cracked chapter 11 and had done all the problems. However, when I came to write up the answers I realized my answer to Problem 11.7 didn't work. I thought I had a simple answer to (i) but then used the same process for (ii) and ended up with +Rab. My approach to (i) was...
Hi,
I have a relatively straight forward question. If we have a Lagrangian that only depends on time and the position coordinate (and its derivative), how can I decide whether angular momentum is conserved?
That is, if the Lagrangian specifically does not have theta or phi dependence, does...
I know this is getting really ridiculous but I have yet another question on Lagrangians...
This is our Lagrangian:
L=\frac{1}{2}m\dot{\vec{x}}^{2}+e\vec{A}.\dot{\vec {x}}
Using the fact that:
\vec P= \frac{\partial L}{\partial \dot{\vec{x}}}=m \dot{\vec{x}} + e\vec A...