Variation Definition and 540 Threads

in order to extend hamilton's principle to include holonomic constraints, out lecturer did the following : when we are under constraints, we cannot consider the variations of the coordinates as independent of each other. we know that the constraint equations are fa = 0. we can multiply each...

On another forum someone has been reporting on their efforts to characterise the background radiation in their lab prior to running some experiments. They don't have much data but what they have appears to have a daily cycle. They weren't looking for one and they haven't done enough work to...

Dear all, I have a headache and I need you help, with the following problem. I need to create excel sheet for the calculation (estimate) of the size of the orifice on the orifice plate that will produce a pressure reduction rate in pipeline at 0,5 bar per minute. It is not important to...

Hello; If a system receives a thermal energy Q, can it keep its entropy constant (that is, with equal value before it receives the energy) without wasting the energy received?

Sorry for all the questions. Reviewing for my midterm next week. Fun fun. If someone could take a look at my proof for (a) and help me out with (b) that'd be awesome! (a) Let $\Delta$ be a partition of $[a, b]$ that is a refinement of partition $\Delta'$. For a real-value function $f$ on $[a...

Define $f(x)=sinx$ on $[0, 2\pi]$. Find two increasing functions h and g for which f = h−g on $[0, 2\pi]$. I know that if f is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions. However, we didn't do any examples of this in class. Is there a...

Homework Statement Show that the variation of gravity with height can be accounted for approximately by the following potential function V = mgz(1+z/re) in which re is the radius of the Earth. find the force given by the above potential function. Homework Equations V = GM/r The Attempt at a...

I've read the twin paradox and if I am correct the resolution is that one twin accelerates and decelerates so he comes back younger. But I have a different scenario that I would like to ask: What if you have two twins equally distant from a point in space and completely at rest relative to each...

<Moderator's note: Moved from a technical forum and thus no template.> Problem: One dimensional quartic oscillator, V(x) = cx^4 (c is a constant) Use the trial function e^(-aplha(x^2)/2) to determine the value of the appropriate variational integral W. I've attached a picture of my work. I...

Homework Statement I'm perpetually confused keeping track of the energetics of a nuclear reaction and I broke down my conceptual questions into the following parts statement a-In a fission reaction, the two medium sized daughter nuclei each have more binding energy per nucleon than the original...

Homework Statement The question I am working on is the one in the file attached. Homework Equations y = u1y1 + u2y2 : u1'y1 + u2'y2 = 0 u1'y1' + u2'y2' = g(t) The Attempt at a Solution I think I have got part (i) completed, with y1 = e3it and y2 = e-3it. This gives a general solution to the...

When a classical field is varied so that ##\phi ^{'}=\phi +\delta \phi## the spatial partial derivatives of the field is often written $$\partial _{\mu }\phi ^{'}=\partial _{\mu }(\phi +\delta \phi )=\partial _{\mu }\phi +\partial _{\mu }\delta \phi $$. Often times the next step is to switch...

A cat in a box with a poison that may or may not be released, that hinges on a random event. Until the box is opened, the cat is both dead and alive, until a single state is forced by an observation. Would it be just as valid to pose that the cat and the poison existing in the box is also...

I need to calculate the energy of the ground state of a helium athom with the variational method using the wave function: $$\psi_{Z_e}(r_1,r_2)=u_{1s,Z_e}(r1)u_{1s, Z_e}(r2)=\frac{1}{\pi}\biggr(\frac{Z_e}{a_0}\biggr)^3e^{-\frac{Z_e(r_1+r_2)}{a_0}}$$ with ##Z_e## the effective charge considered...

Homework Statement How to determine variation of mean momentum of a nucleon with the mass number A of nucleus? Homework Equations R=R_0A^(1/3) The Attempt at a Solution Can't find a solution with elementary approach.

Hi there. Everyone knows about Torricelli's theorem that says about , in a too big container (opened) the speed of the liquid is given by: v=√(2gh) This result is just for containers that have a hole in the side and the fluid goes out perpendicular to the gravity. And also this result is just...

Hello everyone, I'm sure a lot of you know that the Christoffel symbols are not tensors by themselves but, their variation is a tensor. I want to revive a post that was made in 2016 about this: The Variation of Christoffel Symbol and ask again "How is that you can calculate ∇ρδgμν if δ{gμν} is...

I work as part of a team of fourteen. No big challenge to work out the probability of two of us sharing a birthday. It's a well-documented puzzle. In my team, we've gone one better: we have two dates where two people have birthdays on that day! Trying to work out the probability has us...

I'd like to ask an specific question. If c changes (for whatever reason*), does the rest mass of a given particle changes, asuming E is conserved? Let's say, for a given particle, the following initial condition: Placed in a vacuum**. Rest mass m0. Particle's energy E. Propagation speed of...

Homework Statement Solve the DE by variation of parameters: y'' - y = cosh(x) Homework EquationsThe Attempt at a Solution I got m = 1 and m = -1 so y = c_1e^x + c_2e^{-x} + y_p y_p = u_1e^x + u_2e^{-x} The wonksian gave me -2 so u_1' = \frac{\begin{vmatrix} 0 & e^{-x} \\ cosh(x)...

Homework Statement Distance Earth-Sun at perihelion = 1.471×108 km Distance Earth-Sun at aphelion = 1.521×108 km Sun mass = 2.0×1030 kg Earth mass = 5.972×1024 kg G = 6.67×10-11 m3/kg⋅s What is the change in Newton of the attraction force between the Sun and the Earth from the perihelion to the...

Homework Statement Homework EquationsThe Attempt at a Solution this is in the cats frame let the radial length be r then ## v cos\theta = v _{\theta}\\ v - vsin \theta = v _r \\ \frac{dr}{dt} = \frac{dr}{d \theta} \frac{d \theta}{dt} = v - vsin \theta = \frac{dr}{d \theta} \frac{v cos...

I am reading Polchinski's review on AdS/CFT https://arxiv.org/abs/1010.6134. I have a very simple question, and please help me out. Thanks in advanced. The question abou formula (3.19) The scalar effective bulk action is given by $$ S_0=\frac{\eta}{2}\epsilon^{1-D}\int d^Dx \phi_{\rm cl}...

Homework Statement Find the ∆S per mol between liquid water at -5 ºC and ice at -5ºC at 1020hPa Data: ∆CP,m fusion = 37,3 J K-1 mol-1 ∆fusH = 6,01 kJ mol-1 The answer is 21.3 J/K mol Homework Equations Usually I solve these problems by steps when they are at P=1 atm but since its at P=1020...

Homework Statement Variation during reproduction is beneficial to the species but not necessarily for the individual? Homework Equations Not any The Attempt at a Solution I only know about variation during reproduction is beneficial to the species but I don't know anything about how it is not...

Homework Statement In the last picture , we can see that at time between 0 and infinity , the variation of pore pressure across depth is parabolic curve , why is it so ? Homework EquationsThe Attempt at a Solution I think it's incorrect . I think it should be linearly decrease with depth ...

a microchannel of length 2L and width h in the thermal cycling region. the temperature profile ...(1) the cyclic temperature profile leads to a time dependent density ...(2) using the mass conservation equation i.e. ...(3) and momentum balance equation i.e. ...(4) we have to find the exact...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding another aspect of the proof of Proposition 1.3...

I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding aspects of Proposition 1.3 ...Proposition 1.3 and its...

I am looking for the proof of delta variation of determinant of metric but still I find difficulty ? Can I get the full proof here

Homework Statement Determine the y_particular solution Homework Equations The Attempt at a Solution I've tried this for hours but still don't get the correct value. This is what I get: The question is the same as the one I found from...

Homework Statement In a article I have found this transformation (exp to bessel function) . I have two questions. Homework EquationsThe Attempt at a Solution a)where did the Cos go after setting n=1 and n=-1 ? in the third equations ( it is equal to -wmt-pi/2)? why?) b)how did the writer...

Consider a hypothetical pipe of length L and diameter d with both end closed. The pipe is filled with a some fluid. Let there be a ultrasound source of frequency f=v/2L at one end of the pipe and a sensor to measure pressure at the other end of the pipe, where v is the speed of ultrasound wave...

Homework Statement H=I^2RT and H=(V^2)T/R. In one case H is directly proportional to R while in the other it is not where H is the energy consumed. Explain the apparent difference between the two cases Homework Equations None The Attempt at a Solution In the first case electricity is being...

what is the difference between δ- variation and ∆-variation in variational principle, used in classical mechanics?

This link shows us how to derive Hamilton's generalised principle starting from D'Alembert's principle. While I had no trouble understanding the derivation I am stuck on this particular step. I can't justify why ## \frac{d}{dt} \delta r_i = \delta [\frac{d}{dt}r_i] ##. This is because if I...

so df/dy' is yy'/ √(1+y'^2) then we are supposed to do y' . [ yy'/ √(1+y'^2) ] - y√(1+y'^2) how does this bring equation 2 in the image ?

In Hawking-Ellis Book(1973) "The large scale structure of space-time" p69-p70, they derive the energy-momentum tensor for perfect fluid by lagrangian formulation. They imply if ##D## is a sufficiently small compact region, one can represent a congruence by a diffeomorphism ##\gamma: [a,b]\times...

$\tiny{242.17.8}$ 2000 $\textrm{Solve the given equation by variation of parameters.}$ \begin{align*}\displaystyle y''-10y'+25y&=2e^{5x}\\ \end{align*} $\textrm{the homogeneous equation:}$ \begin{align*}\displaystyle x^2-10x+25&=0\\ (x-5)^2&=0\\ x&=5\\...

Homework Statement X-ray pulses, visible-light pulses, and radio pulses (the latter corrected for dispersion in the interstellar plasma) emitted by an astronomical object called a “pulsar” are all observed to arrive simultaneously at the Earth — with an uncertainty of only 200 microseconds. The...

Is the Hubble constant decreasing over cosmological timescales?

Starting with the action for a free scalar field $$S[\phi]=\frac{1}{2}\int\;d^{4}x\left(\partial_{\mu}\phi(x)\partial^{\mu}\phi(x)-m^{2}\phi^{2}(x)\right)=\int\;d^{4}x\mathcal{L}$$ Naively, if I expand this to second-order, I get $$S[\phi+\delta\phi]=S[\phi]+\int\;d^{4}x\frac{\delta...

I have 2 rubidium clocks which both are generating 10MHz sine wave. Some reason my system is giving bad data. So, I am deciding to inspect every single part of the system. I am trying to check the rubidium clock and make sure it have to be in nanosecond time variation, because: t=1/f = 1/10e6 =...

The cost of running a machine is partly constant and partly varies as the number of parts machined. Write an expression to show the cost? My Reasoning: Let C = cost Let x = partly constant Let y = partly varies Let k = constant of proportionality C = x + yk Right?

What is the formula to solving a problem like this? Thanks in advance!

Is it possible to tell with certainty whether μ and ε for vacuum change over time within a very long period of time? I know that we are measuring a space expansion, and we can tell that not objects are moving away from each other, but the space itself expands. However, this seems to be similar...

Homework Statement P(average) for a speaker is 10 W. Gamma is 1.4 (ratio of specific heats), molar mass is 28.8 g/mol, air temperature is 50F, and pressure is 1atm. Find Pmax at 100 I have this equation that gives Intensity = (Pmax^2)/(2*Rho*v) where rho is density, and v is speed of sound...

Homework Statement "By choosing the lower limit of integration in Eq. (28) in the text as the initial point ##t_0##, show that ##Y(t)## becomes ##Y(t)=\int_{t_0}^t(\frac{y_1(s)y_2(t)-y_t(t)y_2(s)}{y_1(s)y_2'(s)-y_1'(s)y_2(s)})g(s)ds## Show that ##Y(t)## is a solution of the initial value...

In lagrangian variation we are trying to minimize the action S = ∫t2t1 L dt. Consider a simple case of free particle. Imagine In a world that everyone one only knows how to solve ODE, Using euler lagrange equation, one has d2x/dt2 = 0 , give that we know the initial position of particle in the...

I have to find functions that maximise certain criterea. The problem can however not be put "under a single integral", for example I've to find ##f(t)##, ##g(t)## that maximise: ## \int_0^{t_e}f(t)^2dt\int_0^{t_e}g(t)^2dt - (\int_0^{t_e}f(t)g(t)dt)^2 ## With ## -1 \leq f(t)\leq1## and ## -1...