The rule is that the total momentum of an isolated system is constant.   I am interested in one question. If you imagine a closed system, there will be two planets, for example: Earth and Mars. They will be attracting equally F(earth)=F(mars). At any time between two objects will be the same exchange of momentum, and the total momentum will be constant. This example is okay for me.  Then i will add a third planet to the system, such as Venus. I will line up the planets in a straight line. 1-Mars, 2-Earth, 3-Venus. And i will say that that Mars attracts the Earth with a force of 3F, and the Earth attracts the Mars force 2F, Venus attracts the Earth force 3F, And the Earth attracts Venus force 2F. (considered in a small period of time)   In that system, the momentum will be conserved. I read that the planets must attract each other equally, but what prevents them from attracting exactly as I drew? Perhaps they are hindered by Newton's third law? That is, if you combine the law of conservation of momentum and Newton's third law, than must be between two interacting objects the same exchange of momentum at any period of time (gravitational or electro-magnetic, however any forces) ? Do I understand everything correctly?   This question arose when I imagined how atoms interact with each other by electromagnetic forces, the principle with planets is a bit like atoms.
I am following the first volume of the course of theoretical physics by Landau. So, whatever I say below mainly talks regarding the first 2 chapters of Landau and the approach of deriving Newton's laws from Lagrangian principle supposing Hamilton's principle of extremum action. Please keep this view in mind while reading and answering my queries and kindly neglect the systems to which Action Principle is not applicable:  If we use homogeneity of space in Euler-Lagrange equations, we obtain a remarkable result i.e. the conservation of momentum for a closed system.  Now, this result, using the form of Lagrange for a closed system of particles, transforms into $ \Sigma F = 0 $ . Now, how from this can we conclude that the internal forces that particles exert come in equal and opposite pairs?  Is it because for 2 particles this comes out as $ F_{1} + F_{2} = 0 $ and we take the forces exerted by particles on one other to be independent of other particles (i.e. Superposition Principle) as an experimental fact?  I doubt it as whole of Newtonian Mechanics is derivable from Lagrangian Mechanics and supposed Symmetries. So, according to me, a fact like Newton's Third Law should be derivable from it without using an additional experimental fact.  I have an idea to prove it rigorously. Consider two particles $i$ and $j$. Let the force on $i$ by $j$ be $F_{ij}$ and on $j$ by $i$ be $k_{ij}F_{ij}$. Now the condition becomes $\Sigma (1+k_{ij})F_{ij}=0$ where the terms to be included and rejected in summation understood. As this must be true for any value of $F_{ij}$, we get $k_{ij}=-1$. I don't know if this argument or refinement of such an argument holds or not. I can see many questions arising in this argument and it's not very convincing to me.  I would like to hear from you people as to if it is an experimental result used or not? If not, then is the method given above right or wrong? If wrong, how can we prove it?  Addendum  My method of proof uses the fact of superposition of forces itself, so it is flawed. I have assumed that the coefficients $k_{ij}$ are constants and don't change in the influence of all other particles which is exactly what superposition principle says.  As the superposition of Forces can be derived by superposition of potential energies at a point in space and potential energy being more fundamental in Lagrangian Mechanics, I restate my question as follows:  Is the principle of superposition of potential energies by different sources at a point in space derivable from the inside of Lagrangian Mechanics or is it an experimental fact used in Lagrangian Mechanics?  I, now, doubt this to be derivable as the fundamental assumption about potential energy is only that it is a function of coordinates of particles and this function may or may not respect superposition.
What happens if I activate a product key on Steam for a game I already own? I have a Steam product key for Just Cause 2 I got for free with the purchase of my video card. However, I already have a copy of this game attached to my Steam account. What will happen if I try to activate this other key on my account? Will Steam give me a copy of Just Cause 2 to gift to somebody else? Will it simply refuse to add it? Or will it eat the key rendering it worthless?  I don't want to try this only to have the latter happen, but it would be useful to be able to add it and gift it to somebody else. How do i prove invertibility of this linear transformation Normal subgroup of prime index Generalizing the case $p=2$ we would like to know if the statement below is true.  Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal. How do you disable an application specific hotkey, for example, "Command-P"? In many applications, Command-P is used to print.  My fingers are accustomed to emacs settings and reflexively press Command-P not for printing but for browsing history. It's annoying to see the print pop-up window every time when I press Command-P. I don't need a shortcut to print. How do you block a shortcut globally? Is a sub-shell the same thing as a child-shell There are this two names: a subshell and a child-shell.  Yes, a child process will be started by any of this:  sh -c 'echo "Hello"' ( echo "hello" ) echo "$(echo "hello") echo "hello" | cat   Are all equivalent and share the same name? Do all share the same properties?    POSIX :     A shell execution environment consists of ....   But the last paragraph of above link has this:     A subshell environment shall be created as a duplicate of the shell environment, except that signal traps that are not being ignored shall be set to the default action.    And specially:     Command substitution, commands that are grouped with parentheses, and asynchronous lists shall be executed in a subshell environment. Additionally, each command of a multi-command pipeline is in a subshell environment; ....   The sh -c 'echo "Hello"' is not included there, should that be called a subshell also? How should references in non-Latin script language be treated in English paper? Let's say I want to cite a Russian text in my own paper which is written in English. Should I:   reference it as is, in the original Cyrillic alphabet, although the readers won't be able even to pronounce it? transliterate the authors' names, while keeping the rest as is? transliterate the names and translate the title of the work, journal name, etc. into English? Some one who is not good at keeping secrets. In my native language it is called "chugalkhor" but it's a slang. So I can't translate it.  What do you call such a person who can't keep secrets because his instincts don't let him. What issues should be considered when overriding equals and hashCode in Java? What issues / pitfalls must be considered when overriding equals and hashCode? How does SQL Server handle logins when the user is a part of multiple groups with various privileges? I'm trying to wrap my head around how SQL Server decides to grant certain rights based on a users login privileges.  If a user is part of a group "READ ONLY" which has access to DB1, but they are also a part of a group "SysAdmin" which has full db owner rights to the same DB1, which rights are they given when they log in?  On a similar note. What if they are part of a group but also have a standalone account with other middle of the road privileges? How can I apply an effect when a player left-clicks/right-clicks with an item in hand? How can I make content disappear when a user logs in? I am working in Wordpress, and I would like to make certain elements disappear once a user logs in. How can I do this?  I have social icons on my page that I positioned to the center of my menu bar at the top of my website.  those icons are generated by a plugin  that is what i want to disappear Unexpected output of xargs The statement below should print afc abc but it prints abc abc instead. I tried many times but every time same result Don't know why this happens could anyone explain and help me?  echo abc|xargs -i{} echo $(echo {}|sed 's/b/f/g') {} How to create a meta-package that automatically installs other packages? I'd like to be able to create a package that doesnt contain any code or programs itself, but instead installs other packages that exist in the repositories onto a computer, sort of how the ubuntu-restricted-extras package does? Cleaning unwanted overlapping duplicate meshes I have a model with hundreds of mesh objects. Unfortunately I clicked shift+D to duplicate many and did not notice. Later after working a while, I was rendering, and noticed there are double meshes overlapping each other.   How can I find all duplicate meshes and delete them. I could do it manually,  just clicking,  but it would take a long time and if I miss something i will not notice.   Is there any option,  or can anybody suggest to me,  how can I find all duplicate mesh and delete them.  Model with all double (duplicate) mesh  Final Result opengl storing multiple indices into indices buffer After parsing collada files I found out that we have to load two(or more) indices each pointing to say a vertex or normal etc. like this (&lt;p>3 0 2 0 0 0&lt;/p>)  Is there a way to load (and use)these indices to indices buffer(ELEMENT_ARRAY_BUFFER) like this (vnvnvn...) ; where v is vertex pointer, n is normal pointer from the above. with vertices loaded to VBO1 like (vvvv...) and in VBO2 like (nnnn...). So that we dont have to repeat the normals in VBO2.  VBO like (vnvnvn...) is also ok. (and not something like this ), I'm not trying to use multiple indices buffer. I'm trying to put all the given indices into one index buffer and use them alternately.  (If I wasnt clear i will try for more clarification.) How can I remove all unused materials from a .blend file? mkdir -p will create a directory; it will also make parent directories as needed.  Does a similar command exist for files, that will create a file and parent directories as needed? Jensen's Inequality implies convexity? Suppose $\phi$ is a real function on $\mathbb{R}$ such that $$\phi \Bigl(\int_0^1f(x)\,dx\Bigr)\leq \int_0^1 \phi(f)\,dx$$ for every real bounded measurable function $f$. Prove that $\phi$ is then convex.    This one is in Rudin's Real and Complex Analysis, page 74, and, to be quite honest, I am unsure what my first step to solve this problem should even be. To help think of a general methodology, I attempted this similar question, from page 71 of the same book:      Assume $\phi$ is a continuous real valued function on $(a,b)$ such that $$\phi\Bigl(\frac{x+y}{2}\Bigr)\leq\frac{\phi(x)+\phi(y)}{2}$$ for all $x, y\in (a,b).$ Prove that $\phi$ is then convex (the conclusion does not follow if continuity is omitted from the hypothesis).   I take it that the latter question is just a special case of the former (assuming $\phi$ is bounded on $(a,b)$), so I attempted to solve the latter problem first. My attempt was to note that \begin{align*} \phi\big(\lambda x + (1-\lambda)y\big) &amp;\leq \phi\Bigl(\frac{2\lambda x + 2(1-\lambda)y}{2}\Bigr) \\ &amp;\leq \frac{1}{2}\Bigl(\phi\bigl(2\lambda x\bigr)+\phi \bigl(2(1-\lambda)y\bigr)\Bigr) \\ &amp; \leq \phi\bigl(\lambda x\bigr)+\phi\bigl((1-\lambda)y\bigr) \end{align*} with the latter inequality justified simply by applying the hypothesis again twice to the previous expression.   From here, however, I am stuck; I cannot justify "popping out" the $\lambda$ and the $(1-\lambda)$ in the last inequality via some bound afforded by the hypothesis and/or the continuity of $\phi$. I am confused with the grammar here. Which is grammatically correct?       At which time will you call me?     At what time will you call me?    And why? My friend wants to consolidate his various retirement account accumulated over the past 20-30 years to make it easier to manage/track his retirement savings. He doesn't recall immediately all the accounts he currently has (one of the reasons to rollover), but we've determined he has at least the following types of accounts: 401(a), 401(k), 403(b), 457(b), and "something from the city". He's trying to track down all the relevant paperwork for all of the accounts, but we'd like to know what kind of limits he might have to watch out for.   Are there any restrictions on the types of the accounts that can be rolled over? Is there a limit to the number of accounts that can be rolled into a single IRA (whether all at once or over the lifetime of the IRA)? Is there a limit to the number of times accounts can be rolled over (can/should he do 1 or 2 at a time as he gets the paperwork in order, or wait until he has everything sorted out)? Is this a yearly limit or lifetime? Is there a dollar limit on the amount that can be rolled over? Is there a limit to the age of accounts that can be rolled over?   As far as we know, all of the accounts are pre-tax/non-Roth accounts, and that Roth vs. traditional accounts should be rolled over to separate (Roth vs. traditional) IRAs.