All Questions
Tagged with lagrangian-formalism homework-and-exercises
979
questions
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Schrödinger picture formulation of a velocity-dependent potential of the form $V(x,\dot{x}) = a + bx + cx^{2} + d\dot{x} + ex\dot{x}$
In Shankar Chapter 8, there is a section at the end of the chapter on the path integral formulation for a potential of the form $$V(x,\dot{x}) = a + bx + cx^{2} + d\dot{x} + ex\dot{x}.$$ I follow the ...
- 31
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Reparametrization invariance of Einbein action [closed]
I'm going through David Tong's online lecture notes on String theory. At the end of section 1.1.2, where he introduces the einbein action
$$S=\frac{1}{2} \int d\tau (e^{-1}\dot{X}^2-em^2),\tag{1.8}$$
...
- 11
2
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Equation of motion in conformal gravity theory?
In conformal gravity theory, the action is given by
$$L=\int \sqrt{-g}C^{abcd} C_{abcd} d^4x=\int \sqrt{-g}(R^{ab}R_{ab}- \frac{1}{3}R^2)d^4 x.$$
However, the variation of the first term $\int \sqrt{-...
- 41
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Motion around stable circular orbit
Hello I am to solve whether it is possible for body of mass $m$ to move around stable circular orbit in potentials: ${V_{1} = \large\frac{-|\kappa|}{r^5}}$ and ${V_{2} = \large\frac{-|\kappa|}{r^{\...
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Does the Lagrangian being invariant under substitution of variables imply a conserved quantity?
Consider the following Lagrangian:
$$
\mathcal{L} = \frac{Ma^2\dot\theta^2}{6} +\frac{1}{2}ma^2\left(4\dot\theta^2 + \dot\phi^2 + 4\dot\theta\dot\phi\cos(\theta - \phi) \right) - \frac{a^2k}{2}\left( ...
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2
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144
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Derivation of propagator for Proca action in QFT book by A.Zee
Without considering gauge invariance, A.Zee derives Green function of electromagnetic field in his famous book, Quantum Field Theory in Nutshell. In chapter I.5, the Proca action would be,
$$S(A) = \...
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Where does the $\eta^{\mu\nu}$ come from? (Maxwell Lagrangian, QFT) [duplicate]
From the Lagrangian in Maxwell theory
$$L = -\frac{1}{2}(\partial_{\mu} A_{\nu})(\partial^{\mu} A^{\nu}) + \frac{1}{2}(\partial_{\mu}A^{\mu})^2 - A_{\mu}J^{\mu}$$
I have to calculate $\frac{\partial L}...
- 361
2
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158
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Srednicki 36.5 symmetry question
This is from the intro to a problem 36.5 in Srednicki and not part of the problem itself. I am having trouble proving that $$\mathcal{L}=i\psi_j^\dagger\sigma^\mu\partial_\mu\psi_j$$
Has $U(N)$ ...
- 1,713
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Help with calculating Lagrangian with scalar potential
I was reading Schwartz's QFT, I came across a lagrangian density,
$$ \mathcal{L} = -\frac12 h \Box h + \frac13 \lambda h^3 + Jh ,\tag{3.69} $$
Calculating the Euler-Lagrange equation,
$$ \partial_{\mu}...
- 101
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47
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Equation of motion from lagrangian for an holonomic system with fixed constraints
We know that the lagrangian function of a holonomic system subject to fixed constraints has the form
$$\mathcal{L}(\mathbf{q,\dot{q}})=\frac{1}{2} \langle \mathbf{\dot{q},A(q)\dot{q}} \rangle - U(\...
0
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1
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139
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Dummy index question
The Maxwell's Lagrangian density is given by the equation, $$\mathcal L = -\frac{1}{4} \space F_{\mu\nu} \space F^{\mu\nu},$$ where $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$.
Hence, one ...
- 171
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Action of a Scalar Field in Path Integral Formulation Peskin & Schroeder (Pag. 285)
I'm really confused on the discretization stuff on this chapter of P&S. My question is related to the computation of the Action in scalar field theory done in page 285. When they compute the ...
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Q1.1(a) Sakurai Advanced Quantum Mechanics For energy-momentum tensor [closed]
I need to prove that the energy-momentum tensor density is defined as:
\begin{equation}
\mathcal{T}_{\mu\nu}=-\frac{\partial \phi}{\partial x_\nu}\frac{\partial\mathcal{L}}{\partial(\frac{\partial \...
- 111
3
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Problem in deriving Friedman equations from Einstein-Hilbert Lagrangian
The Einstein-Hilbert Lagrangian (along with a scalar field) in FRW spacetime reads:
\begin{equation}
\mathcal{L} = - \frac{1}{8 \pi G} (3 a \dot{a}^2 - 3 k a + \Lambda a^3) + \frac{1}{2} \dot{\phi}^2 ...
- 680
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107
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Euler-Lagrange equations with constraints
Show that if there are $M$ independent constraints $\phi_m(x_\mu,p_\mu)$ there are $M$ of the $\ddot{x}_i$'s that the Euler-Lagrange equations cannot be solved for.
Attempt of solution:
Assume that ...
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