I recently posted another thread on the General Physics sub forum, but didn't get as much feedback as I was hoping for, regarding this issue. Let's say I have two Lagrangians:
$$ \mathcal{L}_1 = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\mu A^\mu)^2 $$
$$...
Ok, thank you for the feedback. BTW, the following Lagrangian gives rise to the same equations of the motion as the above one: $$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\nu A_\mu)(\partial^\mu A^\nu) $$
So they should differ only by a total...
@ Okefenokee: The E field is the gradient of the electric potential field, not the charge distribution.
2. Yes, but only if the curl has a singularity at a point. For instance the field (-y/r^2, x/r^2) (the so-called irrotational vortex) has zero curl everywhere, except at the center, where...
Yes, I generally agree with you that statics can only get you so far. You'll eventually have to include horizontal and angular acceleration to figure out how it actually moves. However, the problem is first easier to visualize in terms of equilibrium of forces and torques, and I find the picture...
Hmm, you need to know to some rigid body statics to understand this. The tipping over isn't just a matter of how much force you apply, but a conjugation of that and the point of application of the force. Let's say you have a large rectangular box, upright, and you apply some horizontal force F...
I'm assuming you're talking about a body whose mass is much smaller than the Earth's, dropped with zero velocity from an arbitrary distance from the Earth. If so, please check out this article or this one. Basically the body follows a degenerate ellipse trajectory, similar to the trajectory of...
Hey everyone,
I wasn't really sure where to post this, since it's kind of classical, kind of relativistic and kind of quantum field theoretical, but essentially mathematical. I'm reading and watching the lectures on Quantum Field Theory by Cambridge's David Tong (which you can find here), and...
You can figure out the differentials dx and dy from the general formula for a multivariable differential: $$ x = x(r, \theta) = r \cos \theta \Leftrightarrow dx = \frac{\partial x}{\partial r}dr+\frac{\partial x}{\partial \theta}d\theta = \cos \theta\,dr - r \sin \theta\,d\theta $$
$$ y = y(r...
So, I understand that the derivative operator, D=\frac{d}{dx} has translational invariance, that is: x \rightarrow x - x_0, and its eigenfunctions are e^{\lambda t}. Analogously, the theta operator \theta=x\frac{d}{dx} is invariant under scalings, that is x \rightarrow \alpha x, and its...
Ok, so I was playing around with some Z transforms. I'm sorry about the long derivation, but I'm a bit unsure of the mathematical rigor, and want to make sure every step is clear. I started with the recurrence relation defining the factorial:
$$n!:
u_{n+1}=(n+1)u_n=u_n+nu_n $$ $$
u_0 = 1
$$...
I had thought about how the fact that the norm of the basis elements isn't really defined since the integral doesn't converge, but didn't really reach any conclusion... How can you justify integrating over a finite interval?
You mention the Hilbert basis won't guarantee pointwise convergence...
Hey everyone,
I've been reading up a bit on control systems theory, and needed to brush up a bit on my Laplace transforms. I know how to transform and invert the transform for pretty much every reasonable function, I don't have any technical issue with that. My only problem is that some...
Hey everyone,
I started reading up on GR a couple of days ago, and I'm somewhat stuck on the concept of a free-falling IRF. I understand that an observer on a free-falling small spaceship would experience the laws of physics in a rather simple form, eliminating the need for a force of gravity...
Exactly, that's what I was trying to say with recursively differentiating the original equation to obtain an equation for the nth derivative. But thank you, still ;)
Cheers
I don't know anything about that particular book, but any introductory ODE textbook should suit you fine. There's not really that much to know about separable equations, just a couple of tricks to recognize if a certain equation can be made separable under a certain substitution, or rearranging...
It's not an example of bad mathematics... There is a formal theory of differential forms, and bringing the dx to the other side is just simply used to express a relation between the differentials... Of course, without resorting to differentials, you can just interpret it as a change of variable...
Thank you for your answer. My teacher's suggestion was to turn the equation into an equivalent system of linear differential equations, and then use Picard's theorem for systems to prove uniqueness. I guess I wasn't supposed to directly invoque the existence and uniqueness theorem for linear...
Homework Statement
\frac{d^2y}{dt^2} + t\frac{dy}{dt} + t^3y = e^t;\ \ \ y(0) = 0, \ \ y'(0) = 0
Show that the solution is unique and has derivatives of all orders. Determine the fourth derivative of the solution at t = 0.
2. The attempt at a solution
I'm somewhat lost here... Trying to...
I don't know if you have learned the concept of determinants, but a matrix A is invertible if and only if its determinant is nonzero. The determinant for 2x2 matrices of the form
(a b)
(c d)
is given by det A = ad - bc
Computing the determinant and factoring the quadratic polynomial you'll get...
Oh, that makes a lot of sense! So one gets:
\int_{\gamma}\ f^*(z)\ f'(z)\ dz = \int_a^b\ f(\gamma(t))^*\cdot f'(\gamma(t))\cdot \gamma'(t)\ dt = \int_a^b\ ((f\circ\gamma)(t))^*\cdot \frac{d}{dt}(f\circ\gamma)(t)\ dt
= [\ |\ f(\gamma(t))\ |^2\ ]_a^b - \int_a^b\ \frac{d}{dt}((f\circ\gamma)^*)(t)...