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  1. J

    Adding a total derivative to the Lagrangian

    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: $$ \mathcal{L}_1 = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\mu A^\mu)^2 $$ $$...
  2. J

    Calculating derivatives of a Lagrangian density

    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...
  3. J

    Figuring symmetries of a differential operator from its eigenfunctions

    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...
  4. J

    Representing a factorial through its pseudo Z transform

    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: $$n!: u_{n+1}=(n+1)u_n=u_n+nu_n $$ $$ u_0 = 1 $$...
  5. J

    Creating intuition about Laplace & Fourier transforms

    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...
  6. J

    Confused about the Equivalence Principle and Inertial Reference Frames

    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...
  7. J

    Proving that a solution to an IVP is unique and infinitely differentiable

    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...
  8. J

    Proving that an integral is a pure imaginary

    Homework Statement Show that \int_{\gamma}\ f^*(z)\ f'(z)\ dz is a pure imaginary for any piecewise smooth closed curve \gamma and any C^1 function f whose domain contains an open set containing the image of \gamma 2. The attempt at a solution I have tried to approach it from some...
  9. J

    Probability fluid velocity and quantum entanglement question

    Hey everyone, I'm a second year undergraduate student in aerospace engineering and I've been learning a bit of quantum mechanics for the past few weeks, for recreational purposes. I've been following this textbook: http://www-thphys.physics.ox.ac.uk/people/JamesBinney/qb.pdf On pages 40-41...
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