For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.
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5answers
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Hamilton's Principle
Hamilton's principle states that a dynamic system always follows a path such that its action integral is stationary (that is, maximum or minimum).
Why should the action integral be stationary? On ...
20
votes
4answers
748 views
Is there a Lagrangian formulation of statistical mechanics?
In statistical mechanics, we usually think in terms of the Hamiltonian formalism. At a particular time $t$, the system is in a particular state, where "state" means the generalised coordinates and ...
17
votes
8answers
986 views
Why are L4 and L5 lagrangian points stable?
This diagram from wikipedia shows the gravitational potential energy of the sun-earth two body system, and demonstrates clearly the semi-stability of the L1, L2, and L3 lagrangian points. The blue ...
16
votes
1answer
211 views
Why does charge conservation due to gauge symmetry only hold on-shell?
While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge ...
15
votes
5answers
2k views
Why does calculus of variations work?
How does it make sense to vary the position and the velocity independently?
Edit:
Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
15
votes
3answers
838 views
How general is the Lagrangian quantization approach to field theory?
It is an usual practice that any quantum field theory starts with a suitable Lagrangian density. It has been proved enormously successful. I understand, it automatically ensures valuable symmetries of ...
14
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4answers
457 views
Is the principle of least action a boundary value or initial condition problem?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:
In analytic (Lagrangian) mechanics, the derivation of the ...
13
votes
6answers
5k views
What is the difference between Newtonian and Lagrangian mechanics in a nutshell?
What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the ...
13
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6answers
2k views
Can Noether's theorem be understood intuitively?
Noether's theorem is one of those surprisingly clear results of mathematical calculations, for which I am inclined to think that some kind of intuitive understanding should or must be possible. ...
12
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6answers
2k views
Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
12
votes
5answers
641 views
Making symmetry between E and B fields manifest in Lagrangian
Maxwell's equations are nearly symmetric between $E$ and $B$. If we add magnetic monopoles, or of course if we restrict ourselves to the sourceless case, then this symmetry is exact. This is not ...
11
votes
6answers
1k views
What is the symmetry which is responsible for conservation of mass?
According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For example conservation of energy stems from invariance to time translation.
...
11
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3answers
1k views
11
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2answers
1k views
Deriving the Lagrangian for a free particle
I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away.
Proving that a free particle ...
11
votes
1answer
205 views
Lagrangian for Euler Equations in general relativity
The stress energy tensor for relativistic dust
$$
T_{\mu\nu} = \rho v_\mu v_\nu
$$
follows from the action
$$
S_M = -\int \rho c \sqrt{v_\mu v^\mu} \sqrt{ -g } d^4 x
= -\int c \sqrt{p_\mu ...
10
votes
6answers
456 views
Lagrangian NOT of the form $ T-U$
My Physics teacher was reluctant to define Lagrangian as Kinetic Energy minus Potential Energy because he said that there were cases where a system's Lagrangian did not take this form. Are you are ...
10
votes
3answers
1k views
Galilean invariance of classical lagrangian
In QFT, the lagrangian density is explicitely constructed to be Lorenz-invariant from the beginning. However the classical free lagrangian L = 1/2 mv² is not invariant under galilean transformation. ...
10
votes
1answer
476 views
What is the historical origin of the term action
In Physics ordinary terms often acquire a strange meaning, action is one of them. Most people I talk to about the term action just respond with "its dimension is energy*time". But what is its ...
9
votes
5answers
1k views
Why not using Lagrangian, instead of Hamiltonian, in non relativistic QM?
When we studied classical mechanics on the undergraduate level, on the level of Taylor, we covered Hamiltonian as well as Lagrangian mechanics.
Now when we studied QM, on the level of Griffiths, we ...
9
votes
4answers
262 views
Is the Lagrangian of a quantum field really a 'functional'?
Weinberg says, page 299, The quantum theory of fields, Vol 1, that
The Lagrangian is, in general, a functional $L[\Psi(t),\dot{\Psi}(t)$], of a set of generic fields $\Psi[x,t]$ and their time ...
9
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2answers
216 views
Motivation for Potentials
This is a hypothetical question about "pedagogy". Let's say I am trying to take someone who has just a very small amount of knowledge about Newtonian mechanics and convince them that the Lagrangian ...
9
votes
1answer
167 views
Lagrangian for Goldstone mode + topological excitation
The XY-model Hamiltonian is the following,
$${\cal H}~=~-J\sum_{\langle i,j\rangle} \cos (\theta_i -\theta_j).$$
The Goldstone mode corresponds to term $(\nabla \theta)^2$ in the effective ...
8
votes
4answers
231 views
What makes an equation an 'equation of motion'?
Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint.
For example, in the ...
8
votes
4answers
471 views
Quantum mechanics as classical field theory
Can we view the normal, non-relativistic quantum mechanics as a classical fields?
I know, that one can derive the Schrödinger equation from the Lagrangian density
$${\cal L} ~=~ \frac{i\hbar}{2} ...
8
votes
4answers
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The meaning of action
The action
$$S=\int L \;\mathrm{d}t$$
is an important physical quantity. But can it be understood more intuitively? The Hamiltonian corresponds to the energy, whereas the action has dimension of ...
8
votes
4answers
1k views
Deriving Newton's Third Law from homogeneity of Space
I am following the first volume of the course of theoretical physics by Landau. So, whatever I say below mainly talks regarding the first 2 chapters of Landau and the approach of deriving Newton's ...
8
votes
2answers
76 views
More general invariance of the action functional
I will formulate my question in the classical case, where things are simplest.
Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the ...
7
votes
2answers
204 views
Why so many arguments for the transformation equations of generalized coordinates?
For a system of $N$ particles with $k$ holonomic constraints, their Cartesian coordinates are expressed in terms of generalized coordinates as $$\mathbf{r}_1 = \mathbf{r}_1(q_1, q_2,..., q_{3N-k}, ...
7
votes
3answers
405 views
Noether theorem with semigroup of symmetry instead of group
Suppose You have semigroup instead of typical group construction in Noether theorem. Is this interesting? In fact there is no time-reversal symmetry in the nature, right? At least not in the same ...
7
votes
1answer
306 views
Do an action and its Euler-Lagrange equations have the same symmetries?
Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations.
Can ...
7
votes
2answers
93 views
Group of symmetries of Lagrange's equations
Consider the following statements, for a classical system whose configuration space has dimension $d$:
Lagrange equations admit a smaller group of "symmetries" (coordinate change under which ...
7
votes
1answer
341 views
What corresponds to this Lagrangian density?
Is there a physical example of a field that would have the following Lagrangian density
$$
L= \sqrt{1+\phi_x^2 +\phi_y^2+\phi_z^2}
$$
where the subscripts denote partial derivatives and $\phi$ is a ...
7
votes
1answer
237 views
Lagrangian of 2D square lattice of point masses connected by springs
Zee's QFT book mentions the Lagrangian of a square 2D horizontal lattice of point masses, connected by springs, and considering only vertical displacements $q_{i}$, as
$ L = \frac{1}{2} ...
7
votes
1answer
176 views
To construct an action from a given two-point function
This is really a basic question whose answer I guess may have to do with the way we construct Feynman rules and diagrams. The question is: Suppose I have been given a two-point function (found in some ...
7
votes
0answers
121 views
Orbit through L4 and L5
I was reading the Wikipedia article on Lagrangian points and doing the requisite wiki walk through the various quasi-satellites of Earth when a question occurred to me:
Could there be a stable or ...
6
votes
9answers
2k views
Book about classical mechanics
I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...
6
votes
2answers
830 views
Why is lagrangian density correct?
The textbooks I have available explain that due to the infinite degrees of freedom of a field, the relevant object in QFT is the lagrangian density. A lagrangian is then obtained for the field by ...
6
votes
5answers
261 views
Form of the Classical EM Lagrangian
So I know that for an electromagnetic field in a vacuum the Lagrangian is $\mathcal L=-\frac 1 4 F^{\mu\nu} F_{\mu\nu}$, the standard model tells me this. What I want to know is if there is an ...
6
votes
1answer
2k views
The Euler-Lagrange equation in special relativity
How can I derive the Euler-Lagrange equations valid in the field of special relativity? Specifically, consider a scalar field.
6
votes
2answers
277 views
Why does no physical energy-momentum tensor exist for the gravitational field?
Starting with the Einstein-Hilbert Lagrangian
$$ L_{EH} = -\frac{1}{2}(R + 2\Lambda)$$
one can formally calculate a gravitational energy-momentum tensor
$$ T_{EH}^{\mu\nu} = -2 \frac{\delta ...
6
votes
1answer
35 views
Are there Trojan family or Hilda family satellites locked in Earth's orbit?
Jupiter has many Trojan asteroids located at Lagrangian points L4 and L5 and Hilda asteroids dispersed between points L3, L4, and L5.
Does the Earth have similar satellites? If so, how many?
6
votes
2answers
214 views
What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?
I have a problem with one of my study questions for an oral exam:
The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
6
votes
2answers
586 views
Can a force in an explicitly time dependent classical system be conservative?
If I consider equations of motion derived from the pinciple of least action for an explicilty time dependend Lagrangian
$$\delta S[L[q(\text{t}),q'(\text{t}),{\bf t}]]=0,$$
under what ...
6
votes
1answer
297 views
About Turbulence modeling
There is a paper titled "Lagrangian/Hamiltonian formalism for description of Navier-Stokes fluids" in PRL. After reading the paper, the question arises how far can we investigate turbulence with this ...
5
votes
2answers
620 views
Hamilton-Jacobi Equation
In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and ...
5
votes
2answers
199 views
Can auxiliary fields be thought of as Lagrange multipliers?
In the BRST formalism of gauge theories, the Lautrup-Nakanishi field $B^a(x)$ appears as an auxiliary variable
$$\mathcal{L}_\text{BRST}=-\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}+\frac{1}{2}\xi B^a B^a + ...
5
votes
1answer
86 views
Does anybody know of any good sources that explain (generically) how we form Lagrangians/Actions/Superpotentials for different field content?
I regularly find that I'll understand where the field content in a particular physics paper comes from, but then a Lagrangian or action or superpotential is stated and I don't know how it's derived. ...
5
votes
1answer
114 views
Elementary derivation of the motion equations for an inverted pendulum on a cart
Consider a cart of mass $M$ constrained to move on the horizontal axis. A massless rod is attached to the midpoint of the cart, having a mass $m$ on its endpoint. See wikipedia for a picture and for a ...
5
votes
4answers
727 views
Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
5
votes
5answers
555 views
What is the Lagrangian for a relativistic charge that includes the self-force?
The usual Lagrangian for a relativistically moving charge, as found in most text books, doesn't take into account the self force from it radiating EM energy. So what is the Lagrangian for a ...
