What are some interesting calculus of variation problems? [closed]

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Reference for mathematics of statistical mechanics

Before answering, please see our policy on resource recommendation questions. Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Explain the nature of the resource so that readers can decide which one is best suited for them rather than relying on the opinions of others. Answers … Read more

What’s wrong with the following way to calculate mean lifetime from half-life?

So I understand that half-life T1/2 is the time for the amount N(0) to reduce by half. Basically, N(t)=N(0)2−t/T1/2 My question is that why can’t I use this to directly pull out the the mean lifetime τ, and why τ=ln(2)T1/2 is correct. Assume T1/2=1 for easiness, I make the argument: Since a particle has a … Read more

Physics interpretation of Sobolev space

What is the physics interpretation of Sobolev space? Hs,p:={u∈Lp(Rn):F−1((1+|⋅|2)s/2F(u))∈Lp(Rn)}, s≥0,1<p<∞ (Sobolev space) For example, if n=3,s=2,p=2. Is there an interpretation? Ask this, to find out if the equation F−1((1+|⋅|2)s/2F(u))=g with g∈L2(Rn) has any physical interpretation or application. Answer The (non-homogeneous) Sobolev spaces Hp,s are the subspaces of Lp of functions that admit s weak derivatives … Read more

Is using traditional continuous calculus appropriate to study discrete Nature events such as quantum physics? [duplicate]

This question already has answers here: Does the Planck scale imply that spacetime is discrete? (2 answers) Is time continuous or discrete? (12 answers) Closed 9 months ago. I was wondering why traditional calculus is used for studying quantum physics considering that there seems to be no continuum at quantum scale but discrete. For example, … Read more

Calculation of spherical Bessel functions – meaning of (1xddx)l\left(\frac{1}{x}\frac{d}{dx}\right)^{l}

I’m trying to understand the calculation of spherical Bessel functions in chapter four of Griffiths’ Introduction to Quantum Mechanics (2nd ed, p142). He gives j2(x)=(−x)2(1xddx)2sinxx=x2(1xddx)xcosx−sinxx3 =3sinx−3xcosx−x2sinxx3. I can’t see how he arrives at this answer. I think my problem is the (1xddx)2 bit (the general term for jl(x) is (1xddx)l ). I’m assuming this means … Read more

Square root of a matrix appears in massive gravity. How to solve √A+B\sqrt{A+B} perturbatively?

A=diag{λ1,…,λn}, where λi can be any number and not necessarily a small number, λi>0, B is a positive definite symmetric matrix, and max{Bij}≪min{λi}. Note that the perturbative calculation of square root of I+B is very easy, where B is a small matrix. √(I+B)=I+12B−18B2⋯ In general how to calculate the square root of A+B perturbatively? This … Read more

How to do the integrals over the multivariate delta function?

How to do this integration? ∫∞−∞dq∫∞−∞dpδ(E−p22m−k2q2)=2π√mk I obtained the result using Mathematica, I am not even sure it is correct. Anyways, I’d love to know how one can evaluate this by hand. I am familiar with the Delta function and the identities on how to evaluate those integrals over infinities. Answer The simplest way to … Read more

Are certain fields of physics axiomatized?

Everything from Quantum mechanics can be derived from six (?) postulates. Similarly classical electrodynamics can be reduced to Maxwell’s equations and Lorentz force law, and special relativity is based on two postulates. Are there similar sets of postulates for Newtonian mechanics/gravity or other fields of physics, like quantum field theory or string theory? Answer Many … Read more

Why is the angle of a pendulum as a function of time a sine wave?

OK so I’m trying to understand why the angle of a pendulum as a function of time is a sine wave. I can’t really find an explanation online and when I do find something partial there are certain symbols I don’t understand. d2θdt2+glsinθ=0 This is the equation I found on Wikipedia. What I don’t understand … Read more