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.
6
votes
1answer
295 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 ...
0
votes
2answers
239 views
How to think of the harmonic oscillator equation in terms of “acceleration = gradient”
This is related to another question I just asked where I learned that the equation of motion of a harmonic oscillator is expressed as:
$$\ddot{x}+kx=0$$
What little physics I grasp centers on ...
1
vote
2answers
226 views
Can cos(x) or sin(x) be the function of stationary action?
Is there a way to express $\cos(x(t))$ (or $\sin(x(t))$) as the solution to the Euler-Lagrange equation, in other words is there a sense in which this function is the path of stationary action?
4
votes
1answer
258 views
Lagrangian dynamics with initial conditions: motion of free particle
I am very new to Lagrangian dynamics so I am trying to get my head around the practical usage. So far on here all I could find were proofs and they did not make much sense to me, especially when time ...
4
votes
2answers
264 views
Gauge fixing and equations of motion
Consider an action that is gauge invariant. Do we obtain the same information from the following:
Find the equations of motion, and then fix the gauge?
Fix the gauge in the action, and then find the ...
3
votes
3answers
200 views
Is the Lagrangian “math” or “science”?
I've seen in class that we can get from Lagrangian to derive equations of motion (I know its used elsewhere in physics, but I haven't seen it yet). It's not clear to me whether the Lagrangian itself ...
14
votes
4answers
455 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 ...
3
votes
4answers
596 views
Lagrangian to Hamiltonian in Quantum Field Theory
While deriving Hamiltonian from Lagrangian density, we use the formula
$$\mathcal{H} ~=~ \pi \dot{\phi} - \mathcal{L}.$$
But since we are considering space and time as parameters, why the formula
...
4
votes
1answer
366 views
Deriving the action and the Lagrangian for a free particle in Relativistic mechanics
My question relates to
Landau, Classical Theory of Field, Chapter 2 - Relativistic Mechanics, paragraph 8 - The principle of least action.
As stated there, To determine the action integral for a ...
3
votes
3answers
262 views
What is the difference between manifest Lorentz invariance and canonical Lorentz invariance?
I often read that the Lorentz symmetry is manifest in the path integral formulation but is not in the canonical quantization - what does this really mean?
4
votes
1answer
171 views
speed of sound and the potential energy of an ideal gas; Goldstein derivation
I am looking the derivation of the speed of sound in Goldstein's Classical Mechanics (sec. 11-3, pp. 356-358, 1st ed). In order to write down the Lagrangian, he needs the kinetic and potential ...
2
votes
1answer
149 views
Lagrangian and Equilibrium Points
I'm wondering whether you can tell quickly just from looking at a Lagrangian whether a given point $q^0$ is an equilibrium point. Obviously all you have to do is verify it satisfies the E-L equations, ...
17
votes
8answers
985 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 ...
5
votes
1answer
265 views
Lagrangian density for a Piano String
So I'm trying to do this problem where I'm given the Lagrangian density for a piano string which can vibrate both transversely and longitudinally. $\eta(x,t)$ is the transverse displacement and ...
1
vote
1answer
144 views
Improved energy-momentum tensor
While still dealing with this issue, I've stumbled upon this answer to a question asking about the conserved quantity corresponding to a scaling transformation. It mentions that in accordance with ...
2
votes
2answers
146 views
Is the gravitational constant G a minimum value in some sense?
Assume a central body of mass $M$, and call $a$ the acceleration of a test
body at a distance $r$ due to any interaction whatsoever with the central
body. Is is correct to say that the ratio $a r^2/ ...
3
votes
1answer
257 views
How do you know if a coordinate is cyclic if its generalized velocity is not present in the Lagrangian?
Goldstein's Classical Mechanics says that a cyclic coordinate is one that doesn't appear in the Lagrangian of the system, even though its generalized velocity may appear in it (emphasis mine). For ...
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 ...
2
votes
2answers
206 views
What is the significance of action?
What is the physical interpretation of
$$ \int_{t_1}^{t_2} (T -V) dt $$
where, $T$ is Kinetic Energy and $V$ is potential energy.
How does it give trajectory?
2
votes
2answers
1k views
Deriving Lagrangian density for electromagnetic field
In considering the (special) relativistic EM field, I understand that assuming a Lagrangian density of the form
$$\mathcal{L} =-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{c}j_\mu A^\mu$$
and ...
2
votes
1answer
164 views
Do Lagrangian points actually maintain a fixed distance?
I was reading on up Lagrangian points and the restricted three-body problem.
From what I was able to tell, the Lagrangian points are 5 points in a two-body system such that a third body would be ...
1
vote
1answer
88 views
Cyclic co-ordinates implying the constant velocity motion of center of mass of a system of particles
I'm reading the section on Central Force in my textbook (Goldstein's Classical Mechanics has a similar argument in the chapter titled "The Central Force Problem", first section), where we have the ...
0
votes
2answers
198 views
Advice on classes: Theoretical Mechanics vs E&M II
So I'm having a tough time deciding between courses next semester. I'm a rising 3rd year undergrad math major whose goal is to get a solid understanding of theoretical physics through advanced math ...
3
votes
1answer
170 views
Question about units in Lagrangian dynamics (inertia matrix)
I have a 3 degree of freedom system and my equation of motion is like this:
$$M(q)q_{dd} + C(q,q_d)q_d+G(q)~=~0$$
$M(q)$: inertia matrix
$C(q,q_d)$: Coriolis-centrifugal matrix
$G(q)$: potential ...
3
votes
2answers
1k views
Lagrangian of two particles connected with a spring, free to rotate
Two particles of different masses $m_1$ and $m_2$ are connected by a massless spring of spring constant $k$ and equilibrium length $d$. The system rests on a frictionless table and may both oscillate ...
5
votes
2answers
619 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 ...
1
vote
1answer
68 views
Smooth trajectory on a smooth manifold
Physicists talk about a smooth trajectory of a particle on a smooth manifold and they label it as q(t) where q_1(t)....q_n(t) are component functions coming from the homeomorphism. I don't see how we ...
3
votes
1answer
316 views
Noether current for the Yang-mills-higgs lagrangian
I am trying to calculate the Noether's current, more specifically, the energy density of the Yang-mills-Higgs Lagrangian. Please refer to the equations in the Harvey lectures on Magnetic Monopoles, ...
1
vote
2answers
283 views
Why is ${\partial^i}{\partial_i\phi}$ = ${\partial^i {\phi}}{\partial_i{\phi}}$?
This notation can be found on page 254 of Victor Stenger's Comprehensible Cosmos and in David Tong's Lectures on QFT (Equation 2.4 http://www.damtp.cam.ac.uk/user/tong/qft/two.pdf), and in
EDIT: on ...
1
vote
2answers
262 views
M-theory no lagrangian?
Is there any formulated lagrangian (density) for M-theory? If not, why is there no lagrangian?
If not, is this related to many vacua existing?
Thnx.
2
votes
1answer
209 views
Spontaneous symmetry breaking and 't Hooft and Polyakov monopoles
What is spontaneous symmetry breaking from a classical point of view. Could you give some examples, using classical systems.I am studying about the 't Hooft and Polyakov magnetic monopoles solutions, ...
2
votes
4answers
292 views
Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system?
I am a Physics undergraduate, so provide references with your responses.
Landau & Lifshitz write in page one of their mechanics textbook:
If all the co-ordinates and velocities are ...
4
votes
0answers
296 views
General equation of motion for elementary particles
Elementary particles can be grouped into spin-classes and described by specific equations, see below:
Is there a general Lagrangian density from which all these equations can be derived?
A ...
3
votes
1answer
247 views
Gauge-invariant field strength term in Yang-Mills Lagrangian
I am reading the chapter of non-abelian gauge invariance from Peskin and Schroeder. Why is the term $-\frac{1}{4}(L_{\mu\nu}^i)^{2} $ gauge invariant?
3
votes
2answers
326 views
The Faddeev-Popov Lagrangian
This is a non-abelian continuation of this QED question.
The Lagrangian for a non-abelian gauge theory with gauge group $G$, and with fermion fields and ghost fields included is given by
$$
...
1
vote
1answer
125 views
How does General Relativity Emerge from Brans-Dicke Gravity with an Infinite Omega Parameter?
The action for the Brans-Dicke-Jordan theory of gravity is
$$
\\S =\int d^4x\sqrt{-g} \;
\left(\frac{\phi R - \omega\frac{\partial_a\phi\partial^a\phi}{\phi}}{16\pi} + ...
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
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 ...
3
votes
3answers
229 views
The number of independent variables in the Lagrangian and Hamiltonian methods in Classical Mechanics
It's told in Landau - Classical Mechanics, that in the Hamiltonian method, generalized coordinates $q_j$ and generalized momenta $p_j$ are independent variables of a mechanical system. Anyway, in the ...
0
votes
0answers
153 views
Comparing Lagrangian in Special Relativity vs General Relativity for a weak gravitational field
This is a sequel to this question.
Who knows a difference between the Lagrangian in SR and GR for a weak gravitational field in non-relativistic case? What is the reason of this difference?
4
votes
3answers
348 views
Generalizing Heisenberg Uncertainty Priniciple
Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$
$$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$
...
3
votes
2answers
77 views
A charged particle moves in a plane subject to the oscillatory potential
A charged particle moves in a plane subject to the oscillatory potential:
$U(r)=\frac{m\omega^2 r^2}{2}$
There is also a constant EM-field described by:
$\vec{A}=\frac{1}{2}[\vec{B}\times\vec{r}]$
...
4
votes
2answers
877 views
Lagrangian mechanics vs Hamiltonian mechanics
First of all, what are the differences between these two: Lagrangian mechanics and Hamiltonian mechanics?
And secondly, do I need to learn both in order to study quantum mechanics and quantum field ...
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 ...
3
votes
1answer
282 views
The form of Lagrangian for a free particle
I've just registred here, and I'm very glad that finally I have found such a place for questions.
I have small question about Classical Mechanics, Lagrangian of a free particle. I just read Deriving ...
11
votes
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 ...
3
votes
1answer
304 views
A Question about Virtual Work related to Newton's Third Law
In describing D'Alembert's principle, the lecture note I was provided with states that the total force $\mathbb F_l$ acting on a particle can be taken as,
$$\mathbb F_l=F_l+\sum_mf_{ml}+C_l,$$
...
3
votes
2answers
134 views
Does locality emerge from (classical) Lagrangian mechanics?
Consider a (classical) system of several interacting particles. Can it be shown that, if the Lagrangian of such a system is Lorenz invariant, there cannot be any space-like influences between the ...
2
votes
1answer
219 views
Constructing the “most general” two-particle spin interaction with $SU(2)$ symmetry
Suppose I want to write down an interaction term for an action for spin 1/2 fermions that is $SU(2)$-symmetric.
I start from the most naive general form of such an action:
$$S_{int} ~=~ \int_{4321} ...
2
votes
3answers
2k views
Finding Lagrangian of a Spring Pendulum
I'm trying to understand Morin's example of a spring pendulum. What I don't get is his expression for $T$. I can understand the $\dot x^2$ term in the brackets. But I don't understand the $(l + ...
