I know that the Hamilton's equations results from the Legendre transformation of the Lagrange equation.  It's also well-known that the Hamiltonian is equal to the energy of a system if the system is time independent but that is caused by the symmetry of $H$, not because of  the fact that the Lagrange function is given by the kinetic energy plus the potential energy. So is there a concret physical relationship between $H$ and $L$?
I would like to know the physical meaning of the , if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
Is it possible to recursively instance object particles? what do it mean by "A real nxn matrix always has a real eigenvalue when n is odd?" Shouldnt it be that regardless of odd or even there will always have a real root?  hope to hear from you guys soon:) Help ID this plant, quite possibly a weed I found this plant near my South Indian home in , India. Most probably it could be some sort of weed. Anyhoo, is this plant of any medical or commercial value? Is it a remedy for something?   It has a white tap root with little veins off the main root It has a really small yellow flower It's leaves are similar to that of a clover, except for that it has three to five leaves        Merci d'avance!  (P.S. Forgive me if this is some very common plant that I should know about, I'm in 7th grade.) For a 2x2 matrix A satisfying $A^k=I$, compute $e^A$  Oh, the exponential of a matrix is: $e^A=\sum_{i=0}^\infty\frac{1}{i!}A^i$    I thought I'd solved the $e^A$ form but I actually did something really silly, now I'm a little stuck.  If $A^k=I$ then $A^{k+1}=A$ and we have a cycle forming, so we will get:  $$\sum^{k-1}_{i=0}\sum^{\infty}_{j=1}\frac{1}{(kj+i)!}A^i$$  (Or something like this form, I don't have paper to hand and just spotted that now)  Is this what the question wants? It doesn't use the 2x2 property.  I believe the answer lies in finding an expression for the inner summation, I can do this for two terms (think of $e+e^{-1}$ all the odd power leave) The Maximum possible order for an element $S_n$ Question :   Determine all natural numbers n such that 3   divides $n\cdot2^{n}+1$  Actually I don't have any ideas to approach but  my efforts :   I see $n=1,2,7,8,13,14$ so I think :   $n=6k+1$ and $n=6k+2$ $k\in \Bbb{N} $  If I'm not wrong but I don't know how ? I prove it ? Fermat primes relation to $2^n+1$ If $2^n+1$ is prime how to show that  $n$ must be a power of $2$  Being at elementary level I am at a loss what to begin with? At  I've read that being $\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$ by using  we get $\sin'(x)=\sum_{n=0}^{\infty}(2n+1) \frac{(-1)^n}{(2n+1)!}x^{2n}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}=\cos(x)$ but the page  says that if $f(x)=\sum_{n=0}^{\infty}c_n(x-a)^n$ then inside the radius of convergence of the power series we have $f'(x)=\sum_{n=1}^{\infty}nc_n(x-a)^{n-1}$ so shouldn't it be $\sin'(x)=\sum_{n=1}^{\infty}(2n+1)\frac{(-1)^n}{(2n+1)!}x^{2n+1-1}=\sum_{n=1}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$?  Now I understand that this last result has a missing term due to the index of the series being $1$ and not $0$ but being the $1$ mandatory because of the above theorem and since trying to shift index doesn't help (we get $\sin'(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{(2n+2)!}x^{2n+2}$) how can the result that ProofWiki shows be rigorously justified?   (I mean, I think we can't just say "oh, there's a missing term!" after having applied the theorem and magically shift the index of the series to obtain the desired result) How closely can we estimate $\sum_{i=0}^n \sqrt{i}$ By looking at an integral and bounding the error? How can you construct the bijection between real numbers and functions over naturals? Calling a function from a string in C# Let's start with a definition:     Definition: Let $F$ be a field, $I$ a nonzero ideal in $F[x]$, and $g(x)$ an element of $F[x]$. Then, $I= \langle g(x)\rangle$ if and only if $g(x)$ is a nonzero polynomial of minimum degree in $I$.   Now it gets completely foggy what is meant with "minimum degree".I constantly come across the "minimal polynomial". This seems to be linked, but it seems strange that there is a division here.   Could someone give a pointer to what the second line actually means. Maybe some example or diagram (if possible) would help a lot here. How will people upgrade from 12.10 to later releases after 13.04 is EOL? What exactly is the point of memoryview in Python Aspect (simple, perfect and progressive): What are the differences? Could you please explain to me the differences between the simple, progressive and perfect aspects.  "Simple aspect" means completed action (action starts and finishes) but I don't really understand the fine details.  For example, with these sentences:     It usually rains in the fall.   He smokes 15 cigarettes a day.   I slept for a long time yesterday.     Here's what I understand: if I didn't complete sleeping I should use "I was sleeping". If it doesn't stop raining we can't say "it is finished". Is that correct?  For continuous aspect, duration is an important factor – but how?  For example:     She is wearing a jacket.   Normally "to wear" action is completed but we can use here present continuous aspect.  Another example:     She has been wearing the same jacket for 3 days.   and why can't I use continuous aspect here?     I have cleaned my room.   It means that my room is clean now. Action is finished but why can't I say "I have been cleaning room" like "They have been playing in the mud"? Action is completed but we don't care here about completed, we are caring about duration, why? Omitting 'that' in this sentence Is it better in C++ to pass by value or pass by constant reference? Is it better in C++ to pass by value or pass by constant reference?  I am wondering which is better practice.  I realize that pass by constant reference should provide for better performance in the program because you are not making a copy of the variable. Who made the decision for Gandalf's promotion from 'grey' to 'white'? In the first book of the LoTR, after his fight with the Balrog of Moria, Gandalf the Grey changed to .   My question is, who made the decision to 'promote' him from grey to white? Was it ? Or was it the ? Or was it the ? Finite Series - reciprocals of sines Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be $\dfrac{\cos1^{\circ}}{\sin^21^{\circ}}$ but I do not know how. I have tried reducing it using sum to product formulae and found out the actual value and it agrees well. Haven't been successful in telescoping it. In a sequence $a_1, a_2,\dots$ of real numbers it is observed that $a_p=\sqrt{2}$, $a_q=\sqrt{3}$, and $a_r=\sqrt{5}$, where $1\leq p&lt;q&lt;r$ are positive integers. Then $a_p$, $a_q$, $a_r$ can be terms of (A) an arithmetic progression (B) a harmonic progression (C) an arithmetic progression if and only if $p$, $q$, and $r$ are perfect squares (D) neither an arithmetic progression nor a harmonic progression  I have tried using the definition of AP and argued that if the first option is true then $\frac{\sqrt{3}-\sqrt{2}}{\sqrt{5}-\sqrt{3}}=\frac{q-p}{r-q}$  is true which implies that the left hand side is an rational number. I don't know whether the left hand side is a rational number or not. I am stuck here. Also, I don't know how correct this approach is. How do I approach and solve this problem?