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.
4
votes
1answer
226 views
Differentiating inside an integral sign
I'm reading John Taylor's Classical Mechanics book and I'm at the part where he's deriving the Euler-Lagrange equation.
Here is the part of the derivation that I didn't follow:
I don't get how ...
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 ...
8
votes
4answers
466 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
$${\cal L} ~=~ \frac{i\hbar}{2} ...
4
votes
2answers
183 views
Is the form of the Lagrangian relevant before the renormalization procedure?
In the renormalization procedure, is writing things like
$$\varphi=\sqrt{Z_{\varphi}}\ \varphi_R\ ,\ \ m_0^2=Z_m\ m_R^2\ ,\ \ g_0=Z_g \mu^{\epsilon}\ g_R$$
and
$$Z_i=1+\sum_{\nu=1}^\infty ...
3
votes
1answer
383 views
When is the principle of virtual work valid?
The principle of virtual work says that forces of constraint don't do net work under virtual displacements that are consistent with constraints.
Goldstein says something I don't understand. He says ...
3
votes
1answer
124 views
Showing constraint is nonholonomic
One example of a nonholonomic constraint is a disk rolling around in the cartesian plane that is constrained to not be slipping.
These leads to the constraint $dx - a \sin\theta d\phi = 0$ and $dy - ...
2
votes
3answers
204 views
Why can't we ascribe a (possibly velocity dependent) potential to a dissipative force?
Sorry if this is a silly question but I cant get my head around it.
1
vote
1answer
123 views
Question on 1st order Lagrangian Derivation in Faddeev-Jackiw Formalism
I'm looking at this reference (sorry it's a postscript file, but I can't find a pdf version on the web. This paper describes a similar procedure).
The topic is the Faddeev-Jackiw treatment of ...
3
votes
3answers
285 views
Calculating lagrangian density from first principle
In most of the field theory text they will start with lagrangian density for spin 1 and spin 1/2 particles. But i could find any text where this lagrangian density is derived from first principle.
2
votes
2answers
197 views
How do I show that there exists variational/action principle for a given classical system?
We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
3
votes
3answers
544 views
Is there a valid Lagrangian formulation for all classical systems?
Can one use the Lagrangian formalism for all classical systems, i.e. systems with a set of trajectories $\vec{x}_i(t)$ describing paths?
On the wikipedia page of Lagrangian mechanics, there is an ...
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votes
3answers
282 views
How can we represent the motion of a particle in 2D space using Lagrange's equations?
Can we represent the motion of a particle in 2D space using Lagrange's equations? This is what I tried. Please tell me what is wrong?
Consider a particle on a plane have the co-ordinates $(x,y)$ with ...
2
votes
1answer
318 views
Origins of the principle of least time in classical mechanics
Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least ...
2
votes
1answer
148 views
Differentiation of the action functional
In the QFT book by Itzykson and Zuber, the variation of the action functional $I=\int_{t_1}^{t_2}dtL$ is written as:
$$\delta I=\int_{t_1}^{t_2}dt\frac{\delta I}{\delta q(t)}\delta q(t)$$
How is ...
11
votes
1answer
204 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
$$
S_M = -\int \rho c \sqrt{v_\mu v^\mu} \sqrt{ -g } d^4 x
= -\int c \sqrt{p_\mu ...
8
votes
2answers
75 views
More general invariance of the action functional
I will formulate my question in the classical case, where things are simplest.
Usually when one discusses a continuous symmetry of a theory, one means a one-parameter group of diffeomorphisms of the ...
7
votes
2answers
92 views
Group of symmetries of Lagrange's equations
Consider the following statements, for a classical system whose configuration space has dimension $d$:
Lagrange equations admit a smaller group of "symmetries" (coordinate change under which ...
1
vote
1answer
120 views
Is the number of independent constants of a system equal to the number of degree of freedom of it?
Maybe the question is not very clear myself since I am not a physics major.But can you help me make this question clearer and then give me some comments on it?
I got that this holds in gravitional ...
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vote
0answers
310 views
Find equations of motion from given Lagrangian density [closed]
Could someone help me solve this probably not very hard problem?
Given Lagrangian Density:
$\mathcal ...
2
votes
1answer
420 views
Principle of Least Action; Newton's 2nd Law of Motion
This question is based on the description of Longair in his book "Theoretical Concepts in Physics".
He starts by giving some provisions:
Conservative force field
Fixed times $t_1$ and $t_2$
Object ...
3
votes
2answers
312 views
what this Lagrangian stands for?
i saw this Lagrangian in notes i have printed:
$$
L(x,dx/dt) = (m^2(dx/dt)^4)/12 + m(dx/dt)^2*V(x) -V^2(x)
$$
what is it? is it physical? it seems like it doesn't have a right units of energy,
...
3
votes
2answers
362 views
Conversion of the Nambo-Goto action into the Polyakov action?
I`ve read that the Nambo-Goto action containing the induced metric $\gamma_{\alpha\beta}$
$$\tag{1} S_{NG} ~=~ -T\int_{\tau_i}^{\tau_f} d\tau \int_0^{\ell} d\sigma \sqrt{-\gamma}$$
can be converted ...
1
vote
1answer
426 views
Problems that Lagranges equations of the 1st kind can solve whereas the 2nd kind can't?
Can anyone give examples of mechanics problems which can be solved by Lagrange equations of the first kind, but not the second kind?
16
votes
1answer
207 views
Why does charge conservation due to gauge symmetry only hold on-shell?
While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge ...
12
votes
6answers
2k views
Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
9
votes
2answers
215 views
Motivation for Potentials
This is a hypothetical question about "pedagogy". Let's say I am trying to take someone who has just a very small amount of knowledge about Newtonian mechanics and convince them that the Lagrangian ...
2
votes
1answer
203 views
What variables does the action $S$ depend on?
Action is defined as,
$$S ~=~ \int L(q, q', t) dt,$$
but my question is what variables does $S$ depend on?
Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$?
In ...
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. ...
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?
11
votes
3answers
1k views
7
votes
1answer
175 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 ...
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 ...
-1
votes
1answer
141 views
Quantum tunneling in Field theory with Time dependent potential
What should be the limits of integration for euclidean action $S(\phi)$ in 3d and 4d? This action is negatively exponentiated to calculate the decay rate. I suspect that it is variable limit problem.
...
5
votes
5answers
554 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 ...
4
votes
1answer
732 views
Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time
My question is in reference to Landau's Vol. 1 Classical Mechanics. On Page 6, the starting paragraph of Article no. 4, these lines are given:
If an inertial frame $К$ is moving with an ...
7
votes
1answer
340 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}
$$
where the subscripts denote partial derivatives and $\phi$ is a ...
7
votes
2answers
204 views
Why so many arguments for the transformation equations of generalized coordinates?
For a system of $N$ particles with $k$ holonomic constraints, their Cartesian coordinates are expressed in terms of generalized coordinates as $$\mathbf{r}_1 = \mathbf{r}_1(q_1, q_2,..., q_{3N-k}, ...
4
votes
1answer
511 views
invariance of lagrangian in Noether's theorem
Noether's theorem needs the lagrangian to be invariant.
However, given a lagrangian $L$, we know that the lagrangians $\alpha L$ (where $\alpha$ is any constant) and $L + \frac{df}{dt}$ (where $f$ is ...
1
vote
0answers
369 views
What are the forces of constraint if there are multiple equivalent constraints?
Suppose a large (rigid) block is sitting on top of two smaller blocks of equal height $1$, both of which rest on the ground. We wish to find the position of the block (easy) and the forces of ...
8
votes
4answers
2k views
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 ...
2
votes
1answer
364 views
What gauge is used in the Lagrangian for a non-relativistic point particle in an electromagnetic potential
For the Lagrangian $$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$ of a non-relativistic point particle in an electromagnetic potential, what gauge is used for the electromagnetic ...
12
votes
5answers
639 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 ...
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 ...
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 ...
4
votes
3answers
602 views
Hanging chain in a planet's gravitational field
The curve for a chain hanging between two poles in a uniform gravitational field is known as the catenary.
Is there known an expression for the curve of a hanging chain on a planet of mass $M$ which ...
5
votes
1answer
269 views
formal framework for talking about 'minimal couplings'
usually on physical theories one would have Lagrangians or Hamiltonians with multiple fields; say, a vector $A_{\mu}$ and a scalar $\phi$ and one would postulate ad hoc a coupling between the fields ...
3
votes
3answers
150 views
Are there measurable effects to scaling the action by a constant?
Classically, we obtain the equations of motion by finding a path which has an action that is stationary with respect to small changes in the path. That is the path for which:
$\delta S =0$
Scaling ...
10
votes
1answer
473 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 ...
5
votes
2answers
665 views
How the Lagrangian of classical system can be derived from basic assumptions?
It is well known that the Lagrangian of a classical free particle equal to kinetic energy. This statement can be derived from some basic assumptions about the symmetries of the space-time. Is there ...
1
vote
2answers
406 views
Geodesics and trajectories
I'm a mathematician studying Arnold's Mathematical Methods of Classical Mechanics.
On p. 83 the following definition is given.
Let $M$ be a differentiable manifold, $TM$ its tangent bundle, and ...
