I'd like to understand better the assertion that planets are spheres because of gravity.  The thing which I believe is true, but don't completely get, is that any arrangement of a sufficient number of molecules, all with zero velocity, in isolation from anything else, will over time form a sphere due to gravitational attraction between the molecules.  If this is true, what is the argument for why this will happen? (i.e. what is the proof that I can't construct a weird initial arrangement of molecules for which it won't happen?) Can we do some math and see it pop out? If this is false, what additional stipulations do I need?
Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical?  Let's assume that a planet is 'born' when lots of particles of uniform density are piled together, and constrained only by the Newtonian gravity of the $N$ particles with no external gravity.    My question is: Is it possible to use the variational calculus starting from this situation to prove that planets should be spherical?
Prepositional Phrase - confused - at or in Gift to a supportive professor? As you might have seen from my history I had a very bad advisor who deliberately underpaid me, stole my research work and undermined my career at multiple phases.  Now I am done with my university and leaving next month. A professor in my department had been very supportive of me during these times, giving me advice on how to handle this situation, gave me recommendation letters to apply for PhD programs at other universities and also got me a internship. I just want to thank her for her help. I usually would have got a champagne or something but I know she doesn't drink alcohol. Any advice? How do I make log4j clear a log at startup? How can I apply Newton's sums to solve this problem? Equivalent statement of AC or not? Here's a corollary of AC stating that: if there exists an onto mapping $f:A\to B$, then there exists a 1-1 mapping $g:B\to A$. What I'm confused about is that whether or not  AC can be derived from this proposition, in return? I have tried but haven't found a proof. Any help would be appreciated. move file by list in file (with leading whitespace) UnboundLocalError on local variable when reassigned after first use Inductive proof of a formula for Fibonacci numbers May someone help me? I am trying to use induction to prove that the formula for finding the $n$-th term of the Fibonacci sequence is:   $$F_n=\frac{1}{\sqrt{5}}⋅\left(\frac{1+\sqrt{5}}{2}\right)^n-\frac{1}{\sqrt{5}}⋅\left(\frac{1-\sqrt{5}}{2}\right)^n.$$  I tried to put $n=1$ into the equation and prove that if $n=1$ works then $n=2$ works and it should work for any number, but it didn't work. I need to prove that this formula gives the $n$th Fibonacci number. How do I stop sharing call history between two iPhones? My husband and I both just upgraded to iOS 8.3.  Now I notice that if I have a missed call it shows up on his notice screen and all my calls show up in his recent call history (and vice versa).  Both phones don't ring and they don't need to be near each other or connected by Wi-Fi.  We do share an iTunes account.  How do I stop sharing call history between the two phones? What type of path in shebang is more preferable? In scripts, first line should specify the path to interpreter. But on different servers Linux, Unix, or BSD this path could be different.  What is more preferable?  #!/usr/bin/env bash    or  #!/bin/bash When lecturing about conservation of energy in the 1960s, Richard Feynman remarked:     It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and when we add it all together it gives “28” —always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.    50 some years later, have we made any progress in terms of understanding the fundamental nature of what energy is? What is the difference between a HMAC and a hash of data? I'm arriving to the US by air, but departing via land. How do I end form I-94 now that it is only electronic? What is a reliable method to locate floating islands in Terraria? Inverse of $f(x)=\sin(x)+x$ What is the inverse of   $$f(x)=\sin(x)+x.$$  I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet.  What about  $$f(x)=\sin(a \cdot x)+x$$  where $a$ is a known real constant.  Thank you for taking the time to read this question! Sorry if this has been asked before... .       The  has no sign of the user sharth The  also does not have any sign of related activity   I would like to know why the user's name is associated with the question. At least why their display name shows up as having modified the question timeline activity.  Possible reason that I can think of is that, .  Is this the reason?  Is there a way (for normal users), to look for such details, other than looking at the user activity and the question's revision history. My MWE:  \documentclass{report}  \usepackage{amsmath}  \begin{document}  \begin{align} y &amp;= a \\ =&amp; -2c_1\bar{z}_1^2 + 2\bar{z}_1\bar{z}_2 + 2\bar{z}_2 \left ( -\bar{z}_1 -c_2z_2 - \frac{3\epsilon}{2}\bar{z}_2 -\sigma_{2,1}\cos(x_1)\Delta x_1 \notag \right. \\ \left. &amp;- \sigma_{2,2}\Delta x_2 -\sigma_{2,3} \Delta u +c_2 \chi_2 \vphantom{\frac{1}{1}} \right ) - 2\gamma \sigma_{2,1}^2\cos^2(x_1) \Delta x_1^2 - \gamma \sigma_{2,2}^2 \Delta x_2^2   - 2\gamma \sigma_{2,3}^2 \Delta u^2 \end{align}  \end{document}   The error I get:  Extra }, or forgotten \right.   The expected result: How do I show that $x$ is a root of $p(x)$ in $F[x]/\langle p(x)\rangle$? Let $F$ be a field, and $p(x)$ an irreducible polynomial in $F[X]$. Define $K=F[X]/\langle p(x)\rangle$. For every $a\in F$, denote $\bar{a}=\left&lt;p(x)\right&gt;+a$. How do I show that $\bar{x}\in K$ is a root of $p(x)$?. How can you sync Blender configs between multiple machines? I want to use some constants, height and width, when creating an ellipse but it won't work with \newcommand. Here is a minimal example:  \documentclass[12pt, a4paper]{article}  \usepackage{amsthm, tikz} \usetikzlibrary{arrows, shapes, trees, positioning} \begin{document}  \begin{figure} \centering \begin{tikzpicture}       \newcommand*{\width}{2}%     \newcommand*{\height}{0.5*\width}%      \draw[fill=gray!40] (0, 0) ellipse (\width and \height); \node (Y) at (0,0) {\(Y\)};      \end{tikzpicture} \end{figure}  \end{document}    I get the following error:  error: line 14:  Package PGF Math Error: Unknown operator `a' or `an' (in '2and 0.5*2'). ...ay!40] (0,0) ellipse (\width and \height)