The question I have is, what conditions must satisfy an external force dependent on velocity so it can be a part of the lagrangian and Euler-Lagrange equations are still true.
I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized Lorentz force via a velocity-dependent potential.   I understand the generalized force   $$F_i = -\frac{\partial V}{\partial x_i} + \frac{d}{dt}\left(\frac{\partial V}{\partial v_i}\right)$$  that they derive from a Lagrangian of the form $L = \frac{1}{2}m|\vec v|^2 - V(\vec r,\vec v,t)$.  However, in the next (critical) step of the derivation, the author cites a theorem from Helmholtz saying      ...according to Helmholtz, for the existence of the Lagrangian, such a generalized force can be at most a linear function of acceleration, and it must satisfy the Helmholtz identities.   The three Helmholtz identities are then listed as:  $$\frac{\partial F_i}{\partial \dot{v_j}} = \frac{\partial F_j}{\partial \dot{v_i}},$$  $$\frac{\partial F_i}{\partial v_j} +  \frac{\partial F_j}{\partial v_i} = \frac{d}{dt}\left(\frac{\partial F_i}{\partial \dot{v_j}} +  \frac{\partial F_j}{\partial \dot{v_i}}\right),$$  $$\frac{\partial F_i}{\partial x_j} -  \frac{\partial F_j}{\partial x_i} = \frac{1}{2}\frac{d}{dt}\left(\frac{\partial F_i}{\partial v_j} - \frac{\partial F_j}{\partial v_i}\right).$$  I'm trying to understand where this theorem comes from. Razavy cited a 1887 paper by Helmholtz. I was able to find , but it is in German, so I could not verify whether or not it proved the theorem. Additionally, I could not find it in any recent literature. I searched online and in Goldstein's Classical Mechanics.   The only similar concept that I can find is in the  where we have three equations known as Helmholtz conditions. Are these two concepts one in the same? If so, how should I interpret the function $\Phi$ and the matrix $g_{ij}$ that appear in the Helmholtz conditions I found online?   If the cited theorem from Razavy does not relate from the inverse Lagrangian problem, could I have some help finding the right direction?
I have meticulously read through similar questions, answers and their comments but I still do not understand how the deletion is done. From what I understand I email the customer service and then they do what exactly?     1) They disable my account so that I can no longer use it?      2) They reassign all of my questions and answers to some other   account? To which one?      3) They delete all my questions and answers so that they go away from   public view?   What is the flexibility (if it exists) of the deletion procedure? Is it possible to erase the traces of my former association with the site. Can I request any of the following be done?     4) All of my questions and answers be deleted from public view?      5) Some of my questions and answers be deleted? If yes what criteria?      6) Can I have all of my contributions be assigned to some other   account which runs as sort of a "collective account" and already has   tons of content so that it doesn't happen like my contributions just   move to some other url but still stay grouped under that location and   therefore that location can be discovered as representing my account?      7) Can I have my contributions dispersed through several "collective   accounts?      8) Can my current public name be removed from any comments to my posts   so that my posts cannot be identified later as once belonging to me?   Please be polite with me. I wouldn't have asked if I have found the answer in the other questions. The other questions are also old and it is not clear what the situation is as per 05/2012.  UPDATE (for those interested):  I asked the Stackoverflow team about the procedure. Apparently none of the points 2-8 I asked above are possible, only 1 will take place. The link to my user page from my contributions is going to be removed and replaced with non-clickable text of an anonymized user name like "user1234". Whether this number is the same as my current user_id is not clear. That's all the information I got.  I'm going now to rename myself to "Anonymous" then request the account deletion. Those interested in the procedure will witness the exact result on this page.  Farewell. How do I know if food left at room temperature is still safe to eat? permission denied BASH All of my Firefox add-ons have been disabled suddenly, how can I re-enable them? URL to create a new Google document What is the current URL to create a new Google Document please? Since Google last updated their Docs the old URL does not work. JavaScript hard refresh of current page Which loss function is correct for logistic regression? I read about two versions of the loss function for logistic regression, which of them is correct and why?  From  (in Chinese), with $\beta = (w, b)\text{ and }\beta^Tx=w^Tx +b$: $$l(\beta) = \sum\limits_{i=1}^{m}\Big(-y_i\beta^Tx_i+\ln(1+e^{\beta^Tx_i})\Big)  \tag 1$$  From my college course, with $z_i = y_if(x_i)=y_i(w^Tx_i + b)$: $$L(z_i)=\log(1+e^{-z_i}) \tag 2$$    I know that the first one is an accumulation of all samples and the second one is for a single sample, but I am more curious about the difference in the form of two loss functions. Somehow I have a feeling that they are equivalent. Should I not answer my own questions? Given $n$ independent geometric random variables $X_n$, each with probability parameter $p$ (and thus expectation $E\left(X_n\right) = \frac{1}{p}$), what is $$E_n = E\left(\max_{i \in 1 .. n}X_n\right)$$    If we instead look at a continuous-time analogue, e.g. exponential random variables $Y_n$ with rate parameter $\lambda$, this is simple: $$E\left(\max_{i \in 1 .. n}Y_n\right) = \sum_{i=1}^n\frac{1}{i\lambda}$$  (I think this is right... that's the time for the first plus the time for the second plus ... plus the time for the last.)  However, I can't find something similarly nice for the discrete-time case.    What I have done is to construct a Markov chain modelling the number of the $X_n$ that haven't yet "hit". (i.e. at each time interval, perform a binomial trial on the number of $X_n$ remaining to see which "hit", and then move to the number that didn't "hit".) This gives $$E_n = 1 + \sum_{i=0}^n \left(\begin{matrix}n\\i\end{matrix}\right)p^{n-i}(1-p)^iE_i$$ which gives the correct answer, but is a nightmare of recursion to calculate. I'm hoping for something in a shorter form. Count days between two dates, excluding weekends (MySQL only) When to use unsigned values over signed ones? Windows 10: Pin an item to left side of Start Menu? How to trim whitespace from a Bash variable? I have a shell script with this code:  var=`hg st -R "$path"` if [ -n "$var" ]; then     echo $var fi   But the conditional code always executes, because hg st always prints at least one newline character.   Is there a simple way to strip whitespace from $var (like trim() in )?   or   Is there a standard way of dealing with this issue?   I could use  or , but I'd like to think there is a more elegant solution to this problem. How can I delete everything from my MacBook Pro before selling it? I have a Mac that runs Snow Leopard and I intend to upgrade it to Mountain Lion and sell it on ebay. However, I only have one hard-drive, and I'm sure I can't use something in my OS to delete my OS and all user data.  Is there something for this that's built into the firmware that I can do on boot, or otherwise?   I have source code on my Mac for multiple corporations' proprietary software and I can't risk somebody gaining access through the use of any sort of disk recovery tool. I plan to use a 7 pass erase rather than simply deleting the directory. VirtualBox - how to increase video memory? Usually I use beamer  with the Warsaw theme which puts theorems (and other things) in boxes. Now I switched my theme to Marburg but i still want to keep the boxes. How can one do that? 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I could also simply manually insert &lt;link&gt;s to the stylesheets that I wish to depend on rtl.css right after the call to wp_head() in my theme's header.php file, but I believe it's bad practice to do so, and I would like my theme to be commercial quality.  How can I load stylesheets after WordPress loads rtl.css? Taking notes, I was going to write this:     therefore p′ is a shortest path.   But the topic is APSP which stands for "all pairs shortest paths" so I decided to just write SP instead and...  Is it written with a like this     therefore p′ is a SP   or with an like this (because in my head I would "read" the S)     therefore p′ is an SP Switching window manager/desktop environments? In general what is the approach to switch the desktop environment or window manager on Ubuntu? The main difference between stock Ubuntu, Kubuntu and Lubuntu is supposedly the use of Unity vs. KDE vs. LXDE.  However, in order to try various desktop environments and window managers I don't want to have to reinstall the respective flavor from scratch just to get a particular desktop environments or window manager. There probably is an easier method than that.  The question: How can I easily test different desktop environments and respective compatible window managers (say KDE, LXDE, Unity, GNOME, XFCE and StumpWM)? Prove $\sum\limits_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k}$ (a.k.a. Hockey-Stick Identity) Let $n$ be a nonnegative integer, and $k$ a positive integer. Could someone explain to me why the identity $$ \sum_{i=0}^n\binom{i+k-1}{k-1}=\binom{n+k}{k} $$ holds?