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  1. DuckAmuck

    I Momentum operator acting to the left

    Right. I get that but I’m not sure where I go wrong here.
  2. DuckAmuck

    I Momentum operator acting to the left

    Is the following true if the momentum operator changes the direction in which it acts? \langle \phi | p_\mu | \psi \rangle = -\langle \phi |\overleftarrow{p}_\mu| \psi \rangle My reasoning: \langle \phi | p_\mu | \psi \rangle = -i\hbar \langle \phi | \partial_\mu | \psi \rangle \langle...
  3. DuckAmuck

    A Separation of variables possible in this problem?

    Is it possible to use separation of variables on this equation? au_{xx} + bu_{yy} + c u_{xy} = u + k Where u is a function of x and y, abck are constant. I tried the u(x,y) = X(x)Y(y) type of separation but I think something more clever is needed. Thank you.
  4. DuckAmuck

    I Elliptical Orbit and Kepler's equation

    It is easy to find that the equation for an ellipse is: $$1 = x^2/a^2 + y^2/b^2$$ Then according to Kepler's equation: $$x = a(\cos(E)-e)$$ $$y = b\sin(E)$$ where E is the eccentric anomaly and e is the eccentricity. If you plug the Kepler's equations' x and y into the equation for the ellipse...
  5. DuckAmuck

    A Elliptical orbit parameterized by time

    Yes. I get something non-analytic.
  6. DuckAmuck

    A Elliptical orbit parameterized by time

    It is fairly trivial to do this with a circular orbit: $$(x,y) = (cos(\omega t),sin(\omega t))$$ where t is time, and $$\omega = \sqrt{GM/r^3}$$ How this parametric equation look for an elliptical orbit?
  7. DuckAmuck

    A Did NASA use something more efficient than a Hohmann Transfer?

    $$t = \pi \sqrt{\frac{(r_1+r_2)^3}{8GM}}$$ $$r_1 + r_2 = 4e8 m$$ $$M = 6e24 kg$$ $$t=4.9 days $$
  8. DuckAmuck

    A Did NASA use something more efficient than a Hohmann Transfer?

    I see. Free return trajectories are faster then? Since they have enough energy to return a craft with minimal/no burns? I’m just trying to account for the discrepancy in time between the Hohmann transfer time of 5 days and the actual 3 day time.
  9. DuckAmuck

    A Did NASA use something more efficient than a Hohmann Transfer?

    Thanks. Are Bi-elliptic transfer orbits quicker than Hohmann transfers as well? Are they more akin to what the apollo missions used?
  10. DuckAmuck

    A Did NASA use something more efficient than a Hohmann Transfer?

    It's common knowledge that it takes about 3 days to get to the moon. With a Hohmann transfer, I get a transit time of 5 days, not 3. I see NASA used something called "trans-lunar injection". Is this distinct from a Hohmann transfer, and more time efficient? What makes this trajectory different...
  11. DuckAmuck

    I Inelastic Collision Question

    In collisions that are inelastic or partially elastic, how can we predict how much of the energy lost to the surroundings becomes heat, and how much becomes sound? What determines that fraction?
  12. DuckAmuck

    A Compactification of spatial extra dimensions

    Hi everyone, I am looking at a paper on compact dimensions. Equation 65 makes sense except for the term of 4*pi*n*R in the denominator. Why is it 4*pi and not 2*pi? I cannot rationalize this. Please help. Thank you. https://arxiv.org/ftp/hep-ph/papers/0609/0609260.pdf
  13. DuckAmuck

    I Universe Expansion

    So it is not yet known/understood?
  14. DuckAmuck

    I Universe Expansion

    So the universe is expanding, and galaxies are getting farther apart from one another on average. Does this motion count the same as ordinary motion, in that if a galaxy is being expanded away from us at 0.5c, that clocks in that galaxy would appear to tick slower at 0.866 the rate of clocks here?
  15. DuckAmuck

    I Invariant Mass in a Gravitational Field

    None. I took a class that covered SR. That's why I am asking here.
  16. DuckAmuck

    I Invariant Mass in a Gravitational Field

    Ok. What is it I am computing? How do I compute the velocity, momentum and energy of the photon in this scenario?
  17. DuckAmuck

    I Invariant Mass in a Gravitational Field

    Some follow-up questions. If there's a photo traveling toward the center of a gravitational body, we have: m^2 = 0 = g_{\mu\nu} p^\mu p^\nu If we simplify by saying motion is along the x-axis: g_{00} E^2 + g_{11} p^2 = 0 Plug in the Schwartzchild metric, and we get \frac{\left(1 -...
  18. DuckAmuck

    I Invariant Mass in a Gravitational Field

    Thanks for clarifying this. It was what I was thinking but did the math wrong.
  19. DuckAmuck

    I Invariant Mass in a Gravitational Field

    I see now. So it should be written: 0 = g_{\mu\nu}p^\mu p^\nu - \eta_{\mu\nu} p'^\mu p'^\nu
  20. DuckAmuck

    I Invariant Mass in a Gravitational Field

    In Special Relativity, you learn that invariant mass is computed by taking the difference between energy squared and momentum squared. (For simplicity, I'm saying c = 1). m^2 = E^2 - \vec{p}^2 This can also be written with the Minkowski metric as: m^2 = \eta_{\mu\nu} p^\mu p^\nu More...
  21. DuckAmuck

    I Time dependent Lagrangian

    That term comes from the chain rule of δL I have seen the least action principle shown as 0 = δS = ∫δL dt, which I guess is misleading. I have seen the form you have, and that makes more sense. You are explicitly minimizing with respect to epsilon.
  22. DuckAmuck

    I Time dependent Lagrangian

    Can you show why? How is the term ∂L/∂t δt handled in Least Action?
  23. DuckAmuck

    I Time dependent Lagrangian

    If a Lagrangian has explicit time dependence due to the potential changing, or thrust being applied to the object in question, how does calculus of variations handle this? It's easy to get the Lagrange equations from: δL = ∂L/∂x δx + ∂L/∂ẋ δẋ What is not clear is how this works when t is an...
  24. DuckAmuck

    I Random Unit Vector Angle Difference

    Silly me, yes you can just forget about it being angles. Uniform distribution sample - uniform distribution sample = non-uniform sample. Still not sure why this is.
  25. DuckAmuck

    I Random Unit Vector Angle Difference

    The vectors aren't *really* vectors computationally. I'm just generating angles using a uniform random number generator. Then taking the differences between them.
  26. DuckAmuck

    I Random Unit Vector Angle Difference

    I am simulating random angles from 0 to 2π with a uniform distribution. However, if I take the differences between random angles, I get a non-uniform (monotonically decreasing) distribution of angles. In math speek: Ai = uniform(0,2π) dA = Ai - Aj dA is not uniform. Here is a rough image of...
  27. DuckAmuck

    B Question about how the nabla interacts with wave functions

    Is the following true? ψ*∇^2 ψ = ∇ψ*⋅∇ψ It seems like it should be since you can change the direction of operators.
  28. DuckAmuck

    Solving for y: e^(y) = y^(2) - 2

    This cannot be solved analytically. You have to use an approximation. For example, you can take the approximation (taylor series) of e^y = 1 + y + y^2/2 Then you will have 0 = y^2 - 2y - 6, and can solve for y. You will get -1.6458 which is approaching the correct answer. To get more correct...
  29. DuckAmuck

    I Can any matrix be expressed as the product of two vectors?

    Thanks. I was trying to think of a counter example. This is very obvious.
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