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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0answers
21 views
Derivation of Dirac equation using the Lagrangian density for Dirac field
How can I find Dirac equation using the Lagrangian density for Dirac field?
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0answers
32 views
Lagrangian with a general constraint [closed]
Can any body help me out to solve this problem?
I am familiar with mechanism of Lagrangian and I can solve some problems with constraints but this one is really hard to solve.
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vote
1answer
79 views
Retrieving Maxwell's equations from the minimum action principle
I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps.
Starting with the action:
$$S = \int dt \int ...
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0answers
32 views
Lagrangian of electromagnetic tensor in light cone coordinates? [closed]
I have Lagrangian Density of Electromagnetic field Tensor in light cone coordinates using D'Alembertian operator and Lagrangian density in Cartesian coordinates. I couldn't figure out the way to ...
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1answer
56 views
Why does Lagrangian of free particle depend on the square of the velocity ?
Why does Lagrangian of free particle depend on the square of the velocity ?
For example, $L(v^4)$ also doesn't depend on direction of $v$.
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0answers
73 views
Optimal tunnel shape for travelling inside the earth [duplicate]
Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other ...
4
votes
2answers
84 views
Independent systems and Lagrangians
Definition 1:
The notion of independent systems has a precise meaning in probabilities. It states that the (joint) probability or finding the system ($S_1S_2$) in the configuration ($C_1C_2$) is ...
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0answers
28 views
2 Masses attached to the roof by strings [closed]
There are two masses $M>>m$. $M$ is attached from a fixed surface by a string and ;m'mass is attached to $M$ by a string. Lengths L1, L2, c/s area A1, A2 of cords are given.I have to find ...
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1answer
82 views
A Type of Pendulum
Is there any chance that $$rtl(\ddot\omega+\ddot\phi)\cdot\sin{(\phi+\omega t)}- gl\dot\phi \cdot \sin{\phi} + ltr\dot\omega(\dot\phi^2-\dot\omega)\cdot \cos{(\phi+\omega t)}-gtr\cos{(\omega ...
2
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1answer
66 views
Determinant for a coupled fluctuation Lagrangian
Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the ...
1
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1answer
45 views
Total energy is extremal for the static solutions of equation of motions
In physics total energy is extremal for the static solutions of equation of motions.
Can anyone explain this sentence to me?
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3answers
140 views
What is the mathematical justification for the quadratic approximation to the energy of a spring in a one-dimensional lattice?
It follows easily from this draw, the length $l$ of this spring as a function of the vertical distance $x$, as $l(x)=\sqrt{1+x^{2}}$
Now, $l$ can be expressed as a MacLaurin expansion:
$$l(x) = ...
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0answers
60 views
Small oscillations [closed]
I am asked to consider a fixed homogeneous rod of length $2L$ and mass density $\rho$ It is centered around $O$. A particle with mass M is moving in the same plane. The attractive force between the ...
2
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3answers
130 views
Lagrangian mechanics and time derivative on general coordinates
I am reading a book on analytical mechanics on Lagrangian. I get a bit idea on the method: we can use any coordinates and write down the kinetic energy $T$ and potential $V$ in terms of the general ...
1
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1answer
103 views
Potential in Relativistic Scalar Field Theory
My intention is to establish a Soliton equation. I have cropped a page from Mark Srednicki page no 576.
I have understand the equation (92.1) but don't understand that how they guessed the ...
2
votes
2answers
105 views
Different approaches to calculating the Christoffel symbols
I would be very grateful to whoever can debug the following calculations...
We have the metric for static spacetime: $$ds^2 = -\exp(2U(\vec x))dt^2+h_{ij}(\vec x) d x^i d x^j$$
I want to find the ...
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1answer
75 views
Higher order covariant Lagrangian
I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at ...
1
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0answers
40 views
Lagrangians for non-local equations of motion
Say I have a multicomponent field $X_a(x,t)$ such that I know it Fourier modes satisfy the following equation of motion,
$(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(k,t) = e^t \int \frac{d^3p ...
2
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2answers
95 views
Lorentz invariance of the action for free relativistic particle
I tried to check the Lorentz invariance of the standard special relativity action for free particle directly: ($c=1$)
$$
S=\int L dt=-m\int\sqrt{1-v^{2}}dt
$$
Lorentz boost:
$$ ...
2
votes
2answers
59 views
Where is the magnetic self energy term in $L$ for a charged particle in an electromagnetic field?
In the Lagrangian for a charged particle in an electromagnetic field
$$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$
the energy of the particle is contained in the kinetic term, ...
1
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0answers
104 views
Deriving torque from Euler-Lagrange equation
How could you derive an equation for the torque on a rotating (but not translating) rigid body from the Euler-Lagrange equation? As far as I know from my first class in Classical Mechanics, there is ...
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1answer
82 views
A small oscillations of a rod on the cylinder
Let's have the next case.
A rod (with mass $m$, length $L$ and a momentum of inertia $I$) at the initial time is located on a cylinder (with radius $R$) surface so that it's (rod's) center of mass ...
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1answer
107 views
Lagrangian formulation for relativistic case
Lagrangian for a real scalar field:
$$\mathcal{L}=\frac{1}{2}\eta^{\mu \nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^2\phi^2 $$
Can someone simply drive me how can I write it from ...
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2answers
119 views
How the boundary term in the variation of the action vanishes
Can someone explain a little more that why the last term in equation (1.5) vanishes?
Reference:
David Tong, Quantum Field Theory: University of Cambridge Part III Mathematical Tripos, Lecture ...
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3answers
153 views
Must the Lagrangian always be known for the Euler-Lagrange equations to be of any use?
When studying classical mechanics using the Euler-Lagrange equations for the first time, my initial impression was that the Lagrangian was something that needed to be determined through integration of ...
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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. ...
2
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1answer
73 views
Varying an action (cosmological perturbation theory)
I am stuck varying an action, trying to get an equation of motion. (Going from eq. 91 to eq. 92 in the image.)
This is the action
$$S~=~\int d^{4}x \frac{a^{2}(t)}{2}(\dot{h}^{2}-(\nabla h)^2).$$
...
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0answers
93 views
Small oscillations: diagonal matrix [closed]
I'm solving an exercise about small oscillations.
I name $T$ the kinetic matrix and $H$ the hessian matrix of potential.
The matrix $\omega^2 T- H$ is diagonal and so find the auto-frequencies is ...
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1answer
45 views
if i want action to be positive number then it require that $\tau_i$ be bigger than $\tau_f$, isn't it true? [closed]
the action is the length of the geodesic
$S=-E_o\int_i^f d\tau$
we get an action that is minimised for the correct path.
if i want action to be positive number then it require that $\tau_i$ be ...
4
votes
2answers
176 views
Why is the Lagrangian quadratic in $\dot{q}$? [duplicate]
My teacher said we only consider Lagrangians which are quadratic in $\dot{q}$, and we don't take other Lagrangians. I couldn't understand why. Can anyone please explain this?
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5answers
260 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 ...
3
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3answers
123 views
Virtual differentials approach to Euler-Lagrange equation - necessary?
I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for the Euler-Lagrange equation. The whole notion of, and ...
1
vote
2answers
150 views
Null geodesic given metric
I (desperately) need help with the following:
What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$
I don't know how to transform a metric into a geodesic...! There is no need to ...
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2answers
137 views
Lagrange-Euler equations for a bead moving on a ring
A bead with mass $m$ is free to glide on a ring that rotates about an axis with constant angular velocity. Form the Lagrange-Euler equations for the movement of the bead.
Solution: Let us ...
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1answer
54 views
Strong interaction and the Lagrangian for electromagnetic interaction
The Lagrangian for electromagnetic field has the following expression:
$$
L = -\frac{1}{c^{2}}A_{\alpha}j^{\alpha} - \frac{1}{8 \pi c}(\partial_{\alpha} A_{\beta})(\partial^{\alpha}A^{\beta})
$$
(I ...
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1answer
74 views
Linear/ non linear Scalar field theory
How do I understand that the action for the free relativistic scalar field theory is non linear? What will be the associated interaction potential of that equation?
1
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1answer
103 views
Euler-Lagrange for constrained system
Suppose we have Euler-Lagrange system with generalized coordinate $r_1$ and $r_2$, and input $u_1$ and $u_2$. I know how to prove this system is indeed Euler-Lagrange system.
Suppose now if we have a ...
-1
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1answer
142 views
Scalar field lagrangian and potential
This question is a continuation of this Phys.SE post.
Scalar field theory does not have gauge symmetry, and in particular, $\phi\to\phi−1$ is not a gauge transformation. but why?
and
I want see the ...
2
votes
2answers
136 views
Does a constant factor matter in the definition of the Noether current?
This is a very basic Lagrangian Field Theory question, it is about a definition convention. It takes much more time to typeset it than answering, but here it is:
Consider a field Lagrangian with only ...
1
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2answers
110 views
In Noether's theorem, what is a “classical solution of the equations of motion”?
I'm reading a book which states that:
for each generator of a global symmetry transformation, there is a
current $j^{\mu}_{a}$ which, when evaluated on a classical solution
of the equations of ...
2
votes
1answer
84 views
Calculating the (on-shell) action of a free particle
I am having difficulty with the first problem from Feynman and Hibbs' book.
For a free particle $L = (m/2)\dot{x}^2$. Show that the (on-shell) action $S_{cl}$ corresponding to the classical ...
1
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1answer
187 views
What is the Lagrangian from which the Klein-Gordon equation is derived in QFT?
Is there a well-known Lagrangian that, writing the corresponding eq of motion, gives the Klein-Gordon Equation in QFT? If so, what is it?
What is the canonical conjugate momentum? I derive the same ...
3
votes
2answers
88 views
How is the physical Lagrangian related to the constrained minimization Lagrangian?
If we're minimizing an energy $V(q)$ subject to constraints $C(q) = 0$, the Lagrangian is
$$L = V(q) + \lambda C(q).$$
I have fairly solid intuition for this Lagrangian, namely that the energy ...
0
votes
1answer
94 views
Fast question about Lagrangian
I've seen some problems solved in a weird way, I just want to be sure: the whole kinetic energy has to be in the lagrangian, right? For example, if we have a particle fixed in a plane with spherical ...
1
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1answer
143 views
How would you use the Euler-Lagrange equation to predict the motion of projectiles with linear (Stokes) drag (but no wind)?
My first instinct would be to use the force
$$\vec{F} =- \alpha \vec{v}$$
and therefore
$$V(\vec{r}) = \alpha \int_C \vec{v}\cdot d\vec{s} = \alpha \int_C \vec{v}\cdot \vec{v} dt = \alpha \int_C ...
1
vote
2answers
190 views
Why lagrangian is negative number?
In the special relativistic action for a massive point particle,
$$\int_{t_i}^{t_f}\mathcal {L}dt,$$
why is the Lagrangian
$$\mathcal {L}=-E_o\gamma^{-1}$$
a negative number?
2
votes
2answers
152 views
Where does the mass term come from in the Proca Lagrangian?
There are many good books describing how to construct the Lagrangian for an electromagnetic field in a medium.
$$
\mathcal{L}~=~-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}-\frac{1}{c}j^{\nu}A_{\nu}
$$
...
1
vote
0answers
52 views
relevant 4-dimensional theory with interacting vector field
A simple langragian that gives the simplest interaction is $\mathcal{L}=(\partial\phi)^2+(m\phi)^2$ where $m$ is some constant. Does anyone know of theory in four dimensions which is physically ...
0
votes
1answer
290 views
Hamilton's equations for a simple pendulum
I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with
$$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$
Now Hamilton's equations will be:
$$\dot ...
20
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4answers
745 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 ...
