Using the Einstein-Hilbert action for a Universe with just the cosmological constant ##\Lambda##:
$$S=\int\Big[\frac{R}{2}-\Lambda\Big]\sqrt{-g}\ d^4x$$
I would like to derive the equations of motion:
$$\Big(\frac{\dot a}{a}\Big)^2+\frac{k}{a^2}=\frac{\Lambda}{3}\tag{1}$$
$$2\frac{\ddot...
Hello, I have been working on the three-dimensional topological massive gravity (I'm new to this field) and I already faced the first problem concerning the mathematics, after deriving the lagrangian from the action I had a problem in variating it
Here is the Lagrangian
The first variation...
Hi All,
Anyone willing to help out in explaining what eigenfreuqncy for this oscilatory system, would be? Also if anybody knows the equation to calulate this stuff please, if you're willing to share I'd be greatful!
Thanks, regards.
This image shows the equations.
I managed to almost get equation 5, but my partial derivative is not squared but instead multiplied by mu, and also I don't have a factor of 1/2.
Here is an image of the work I have. I'm sorry for any sloppiness. I tried to be as concise as possible when writing...
In the Classical Dynamics of Particles and Systems book, 5th Edition, by Stephen T. Thornton and Jerry B. Marion, page 220, the author derived Equation (6.67) from Equation (6.66) which is the following:
Equation (6.67):
$$\left(\frac{\partial f}{\partial y} − \ \frac{d}{dx}\frac{\partial...
First of all, disclaimer: This isn't an official assignment or anything, so I'm not even sure if there is a resonably simple solution.
Consider the following sketch.
(Forgive me if it isn't completely clear, I didn't want to fiddle around for too long with tikz...)
Let us assume that we can...
Homework Statement
Let ##U## be a plane given by ##\frac{x^2}{2}-z=0##
Find the curve with the shortest path on ##U## between the points ##A(-1,0,\frac{1}{2})## and ##B(1,1,\frac{1}{2})##
I have a question regarding the answer we got in class.
Homework Equations
Euler-Lagrange
##L(y)=\int...
Hi all,
I was working on a problem using Euler-Lagrange equations, and I started wondering about the total and partial derivatives. After some fiddling around in equations, I feel like I have confused myself a bit.
I'm not a mathematician by training, so there must exist some terminology which...
Homework Statement
Prove that snell's law ## {n_1}*{sin(\theta_1)} ={n_2}*{sin(\theta_2)} ## is derived from using euler-lagrange equations for the time functionals that describe the light's propagation, As described in the picture below.
Given data:
the light travels in two mediums , one is...
Hi,
I'm trying to solve the following problem
##\max_{f(x)} \int_{f^{-1}(0)}^0 (kx- \int_0^x f(u)du) f'(x) dx##.
I have only little experience with calculus of variations - the problem resembles something like
## I(x) = \int_0^1 F(t, x(t), x'(t),x''(t))dt##
but I don't know about the...
In my quest to understand the Euler-Lagrange equation, I've realized I have to understand the chain rule first. So, here's the issue:
We have g(\epsilon) = f(t) + \epsilon h(t). We have to compute \frac{\partial F(g(\epsilon))}{\partial \epsilon}. This is supposed to be equal to \frac{\partial...
Hello everyone,
Reading Landau and Lifshitz Course of Theoretical Physics Volume 1: Mechanics (page 3) I got suck in the following step (and I cite in italics):
The change in S when q is replaced by q+δq is
\int_{t_1}^{t_2} L(q+δq, \dot q +δ\dot q, t)dt - \int_{t_1}^{t_2} L(q, \dot q, t)dt...
We have a car accelerating at a uniform rate ## a ## and a pendulum of length ## l ## hanging from the ceiling ,inclined at an angle ## \phi ## to the vertical . I need to find ##\omega## for small oscillations.
From the Lagrangian and Euler-Lagrange equations, the equation of motion is...
Suppose one starts with the standard Klein-Gordon (KG) Lagrangian for a free scalar field: $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}$$ Integrating by parts one can obtain an equivalent (i.e. gives the same equations of motion) Lagrangian...
I'm currently studying Quantum Field Theory and I have a confusion about some mathematics in page 30 of Mandl's Quantum Field Theory (Wiley 2010).
Here is a screenshot of the relevant part: https://www.dropbox.com/s/fsjnb3kmvmgc9p2/Screenshot%202017-01-24%2018.10.10.png?dl=0
My issue is in...
Hello I am little bit confused about lagrange approximation to geodesic equation:
So we have lagrange equal to L=gμνd/dxμd/dxν
And we have Euler-Lagrange equation:∂L/∂xμ-d/dt ∂/∂x(dot)μ=0
And x(dot)μ=dxμ/dτ. How do I find the value of x(dot)μ?