Recent content by phi1123

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    How to Optimize a Functional with Nested Integrals in Calculus of Variations?

    Ah, I see your point. Okay suppose we interchange the variables s and t in the second integral, and then exchange the order of integration (not sure if this is justified). Then we can combine the integrals and factor out the ##\eta(s)## such that $$ 0=\int\int\left( \frac{\partial}{\partial...
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    How Can One Solve This Complex Trigonometric Integral Analytically?

    Well if you're really interested in getting an analytical solution, you could try expanding the ##(7.625+.275 \cos(4x))^{1.5}## as a taylor series. At that point you'd have a sum of integrals that are just powers of trig functions, which should be integrable (though ugly). You could then...
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    How to Optimize a Functional with Nested Integrals in Calculus of Variations?

    So I've been thinking about this problem some more (sorry I didn't reply earlier). Following your suggestion, let's assume only ##\phi## is varied for simplicity. In that case we want to minimize a functional of the form...
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    How to Optimize a Functional with Nested Integrals in Calculus of Variations?

    Okay, so I've run into a rather weird functional that I am trying to optimize using calculus of variations. It is a functional of three functions of a single variable, with a constraint, but I can't figure out how to set up the Euler-Lagrange equation. The functional in question is (sorry it's...
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    Analysis of the physics in Interstellar

    So I recently watched the new movie Interstellar, and I've been inspired to do some more general relativity. At one point in the movie they mention that 1 hour on a planet orbiting a black hole is 7 years back on Earth, and so I decided my first project would be to figure out exactly how close...
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    Calculus of variations: multiple variables, functions of one variable

    Simply put, can you find the function which extremizes the integral J[f]=\iint L\left(x,y,f(x),f(y),f'(x),f'(y)\right) \,dx \,dy Where ##f## is the function to be extremized, and ##x## and ##y## are independent variables? A result seems possible by using the usual calculus of variation...
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    Problems with a Charged Particle in a Magnetic Field

    Okay, so I guess I'm still confused by the whole choice of Gauge thing. You say: Does this mean that there is a degenerate energy level for each choice of Gauge? If not, how does the same state correspond to several different wavefunctions?
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    Problems with a Charged Particle in a Magnetic Field

    I was wondering about what the wavefunction of a particle in a magnetic field would look like, so after some quick work and a little research, I found the the Hamiltonian is \hat{H} = \frac{(\hat{p}+qA)^{2}}{2m} where A is the vector potential such that B=∇×A. I thought that the vector...
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    General question about wavefunctions

    Homework Statement Is it possible given a wavefunction ψ(x,t) to find the probability that the particle is at a particular location over an interval of time? Homework Equations The Attempt at a Solution Intuitively, given that the probability of finding the particle in a region...
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