deriving Definition and 962 Threads

Hello, using computation simulation, can the statistical behavior of many particles be derived through deterministic classical mechanics?

Using the Einstein-Hilbert action for a Universe with just the cosmological constant ##\Lambda##: $$S=\int\Big[\frac{R}{2}-\Lambda\Big]\sqrt{-g}\ d^4x$$ I would like to derive the equations of motion: $$\Big(\frac{\dot a}{a}\Big)^2+\frac{k}{a^2}=\frac{\Lambda}{3}\tag{1}$$ $$2\frac{\ddot...

I have been able to prove to myself that, based on Einstein's two postulates and the the Pythagorean theorem, that time dilates. From here how do I prove that length contracts? (All of this observing a frame that is moving relative to the proper frame at uniform velocity.)

I'm a bit confused on the derivation above. I understand what the goal of the derivation is, as it derives Gauss's Law using the solid angle, but i was wondering if someone could kind of fill in the steps the author skipped and explain the use of the solid angle.

Hello, I am learning how to use calculus to derive the formula for kinetic energy now, I understandthe majority of the steps in how to do this, however, there is one step where I get totally lost, I will post a picture of the steps and I will circle the part where I get lost. If you see the...

I'm not a newbee to C++ but I've been out of the loop for a while. I know it's discouraged, if not directly a no-go, to derive from STL classes. Something that has to do, if I remember correctly, with the STL classes having no virtual destructors. It will work in the short run but may...

With the assumptions of Inviscid flow, no pressure gradient and no body force terms in 1-D Navier Stokes becomes 1-D nonlinear convection equation; And if we assume velocity of wave propagation is constant value c, equation becomes 1-D linear convection equation; This is online derivation and...

I'm trying my hand at deriving Lorentz transformations using 3 postulates - it's a linear transformation, the frames are equivalent, so they see the same speed of each other's origins and that the speed of light is the same. Let's say frame ##S## is moving at velocity ##v## in the...

OK, so I just want to show ω = E/h = kv, but I keep running into errors, I don't know why. So, let's start with momentum: p^2 / 2m = E p^2 = 2mE p = sqrt(2mE) h/λ = sqrt(2mE) hk = sqrt(2mE) k = sqrt(2mE)/h So far so good. Now let's start with conserved Energy E= ½ mv^2 2E/m = v^2 v =...

Does anybody know of a derivation of the van der Waals equation from the molecular kinetic theory of gases, but without using the tools of statistical physics (such as partition functions)?

The formula for general oscillation is: The formula for underdamping oscillation is: where λ = -γ +- sqart(γ^2 - ω^2), whereas A+ and A- , as well as λ+ and λ-, are complex conjugates of each other. After some operations, we get: x(t) = Ae^(-γx)[e^i(θ+ωx) +e^-i(θ+ωx)], where A is the modulus...

**I realize some of my inline math delimiters '\(' and '\)' are not acting on the text for some reason, and it looks clunky. I spend 20-30 minutes trying to understand why this is, but I can't. My limited LaTeX experience is in Overleaf, and these delimiters work fine in that compiler. My...

I am trying to derive the adjoint / tangent linear model matrix for this partial differential equation, but cannot follow the book's steps as I do not know the math. This technique will be used to solve another homework question. Rather than posting the homework question, I would like to...

I assumed a uniform distribution of charge within the droplet such that ##E = \frac{q}{4\pi\epsilon_{0}r^{2}}## at the outside surface. I then said that the pressure acting at the surface would be the force on a charge element ##dq## within an area ##dA## on the surface, divided by the area...

Consider a free particle with rest mass ##m## moving along a geodesic in some curved spacetime with metric ##g_{\mu\nu}##: $$S=-m\int d\tau=-m\int\Big(\frac{d\tau}{d\lambda}\Big)d\lambda=\int L\ d\lambda$$...

This is the equation given. I attempted to use Radial Equation, obtained from separating variables, to solve for ##E_1##.

I tried finding the solution of the equation itself but it hasn't helped! Links to concepts would be greatly appreciated...thank you...

Hello everyone I was hoping someone could shed some light on the following:- I am trying to derive the equation of Momentum from Newton's 2nd Law. What I know is the following:- I don't know how to get from Force = Mass * Acceleration TO Momentum = Mass * Velocity. I have attempted to...

I have an identity $$\vec{\nabla} \times (\frac{\vec{m} \times \hat{r}}{r^2})$$ which should give us $$3(\vec{m} \cdot \hat{r}) \hat{r} - \vec{m}$$ But I have to derive it using the Einstein summation notation. How can I approach this problem to simplify things ? Should I do something like...

Li=1/2*∈ijkJjk, Ki=J0i,where J satisfy the Lorentz commutation relation. [Li,Lj]=i/4*∈iab∈jcd(gbcJad-gacJbd-gbdJac+gadJbc) How can I obtain [Li,Lj]=i∈ijkLk from it?

Note that the wave equation we want to derive was introduced by Alfven in his 1942 paper (please see bottom link to check it out), but he did not include details on how to derive it. That's what we want to do next. Alright, writing the above equations we assumed that: $$\mu = 1 \ \ \ ; \ \ \...

I am trying to derive the gravitational binding energy of the cluster. Its given as $$U = -\alpha \frac{GM^2}{r}$$ Now for the derivation I started from $$dU = -\frac{GM(r)dm}{r}$$ I I tried to write ##dm = \rho(r)4 \pi r^2dr## and do it from there but I could not do much. Any ideas how can...

Exercise statement: Given the action (note ##G_{ab}## is a symmetric matrix, i.e. ##G_{ba} = G_{ab}##): $$S = \int dt \Big( \sum_{ab} G_{ab} \dot q^a\dot q^b-V(q)\Big)$$ Show (using Euler Lagrange's equation) that the following equation holds: $$\ddot q^d +...

Hello, I just have a quick question on deriving the kinetic energy formula using calculus. I understand most of it, I just have a question about one of the steps. here are the steps. Begin with the Work-Energy Theorem.The work that is done on an object is related to the change in its kinetic...

Here's what I have done: Let t1 be the time taken for the ball to reach the ground after the first bounce, and ux be the initial speed in the horizontal direction of ball. Also, let v be the speed after the bounce. For horizontal, $$\dfrac {4}{3}H=U_{x}t_{1} $$ For vertical, $$H=\dfrac...

The metric for 2-sphere is $$ds^2 = dr^2 + R^2sin(r/R)d\theta^2$$ Is there an equation to describe the area of an triangle by using metric. Note: I know the formulation by using the angles but I am asking for an equation by using only the metric.

Homework Statement: A quantity of ideal gas initially at 20 atmospheres in a volume of 1 litre and a temperature of 300K suddenly expands and comes to equilibrium with a volume of50 litres and a pressure of 1 atmosphere. If the process is irreversible and involves the transfer of heat and work...

So I have managed to derive the magnetic field of a current carrying wire, however, I seem to have some enquiries on deriving the one for the loop. In the formula where ##\frac {ds * r} {r^2}##, I know that it will become ##ds sin \theta.## However compared to the theta in the wire, the theta...

Perhaps I should ask this question in the math section of these forums. But, I'm stumped and I don't think this equation is solvable? Find theta given all the other variables:

So I think I do understand how to do this, but the thing is my answers are always incorrect. Will need some guidance/help on this. ##B =\frac{\mu_{0}I}{4\pi}\int\frac{dysin\theta}{r^2} ## ##y = Rtan\phi## ##dy = Rsec^2\phi d\phi## ##B = \frac{\mu_{0}I}{4\pi}\int\frac{dysin\theta}{r^2} =...

I have been attempting this physics problem for a while now. I am not sure how to begin to attempt this problem with the equations of motion formulas.

PS: This is not an assignment, this is more of a brain exercise. I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition...

Hi, I am studying a paper by Yann Bugeaud: http://irma.math.unistra.fr/~bugeaud/travaux/ConfMumbaidef.pdf on page 13 there is an inequality (16) as given below- which is obtained from - , on page 12. How the inequality (16) is derived? I couldn't figure it out. However one of my...

Okay so I'm not quite sure about the normal reaction forces. The condition is that they are both equal for the sum of frictional forces to be zero but then technically when substituting into the number equations in place of [ n(1) + n(2) ] either 2n(1) or 2n(2) makes me confused as then n(1)...

Hi, I'm worried I've got a grave misunderstanding. Also, throughout this post, a prime mark (') will indicate the transformed versions of my tensor, coordinates, etc. I'm going to define a tensor. $$T^\mu_\nu \partial_\mu \otimes dx^\nu$$ Now I'd like to investigate how the tensor transforms...

Hi all, I have attached an image of a page out of the book I am using for context. The blue arrow in Figure 12-3 describes the motion of the particle. I figured the net force would need to be in the same direction, but apparently the net force opposes the motion. So, in Figure 12-3 the pressure...

I have tried to derive the lorentz transformations but there is a part of it that requires substitution into two equations when t=0. How do I do that

I'm trying to derive the lever law by myself, however, I'm stuck. Please follow the logic of my calculations. Every object in the picture has the same mass. I want to prove that, under the effect of the gravitational force, I can replace the objects in A and C with the two objects in B, and...

I'm trying to derive the electric and magnetic fields of a plane wave from the four-potential ##\mathbf{A} = (A^t , \mathbf{a}) ## in the Lorenz gauge. Given: ##\mathbf{A}(\mathbf{R}) = \Re \left( \mathbf{C} e^{i \mathbf{K} \cdot \mathbf{R}} \right)## for constant future-pointing lightlike...

Hi, it's been a while since I last posted. Anyway, so I went through the trouble of enrolling in two finite element analyses classes and yet, they still don't teach how the 2D formulation has been made. I'll list the things that 'I know' already to get some things clear. I know how to derive...

Number of states in that volume of k-space, ##n(k)dk## is: $$n(k)dk = (\frac{L^3}{4 \pi^3}) \cdot 4 \pi k^2 dk = \frac{L^3}{\pi^2}dk$$. Then the notes state that by defintion, ##n(k)dk = n(E)dE##, and hence $$n(E)d(E) = \frac{L^3}{\pi^2}dk$$. I don't quite see why this is true - isn't it the...

Homework Statement L = 20mH = 20 x 10-3 H i = 40 mA for t≤0 i = A1e-10,000t + A2e-40,000t A for t≥0 The voltage at t=0 is 28 V. I have to find the equation for the voltage for t>0. Then I have to find the time when power is zero. Homework Equations v(t) = L* di/dt p(t) = L*i* di/dt The...

## \newcommand{\ihat}{\hat{\boldsymbol{\imath}}} \newcommand{\jhat}{\hat{\boldsymbol{\jmath}}} \newcommand{\khat}{\hat{\boldsymbol{k}}} ## Several times now I've seen the following technique for deriving the component form of the dot product. It always felt clean and simple until last night when...

I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using $$dxdydz = \left (\frac{\partial x}{\partial r}dr +...

Hi, The origin of this question was contemplating how to express the impedance of an inductor as a function of frequency, for non sinusoidal voltage wave-forms such as triangle waves, but in particular rectangular pulse trains. So going back to basics, I watched this video: He derives the...

Hi, I was trying to see where the equation N = No e-λt came from and it is derived from dN/dt = -λN which is discussed very well in this thread in post #2 (https://www.physicsforums.com/threads/derivations-of-the-decay-constant-equation.213312/). I understand the steps except for the reason why...

Homework Statement The problem is taken from Morin's book on classical mechanics. I found out Lagrangian of motion. Now to solve, we need small angle and small x approximation. The small angle approximation is easy to treat. But how to solve small x approximation i.e how do I apply it...

Homework Statement I want to try to derive the rocket equation and then add additional effects like gravity, air resistance, etc. Here's the equation that I found online: Homework Equations p (momentum) = mass * velocity F (force) = Δp / Δt The Attempt at a Solution [/B] pi = mv Here is...

Homework Statement This post contains the answer to my thread of 10th August... [/B] in which I asked if anyone could point out how to derive ##\pi^{ij} = \sqrt {^{(4)}g} (^{(4)} \Gamma ^0 \,_{pq} - g_{pq} ^{(4)} \Gamma ^0\, _{rs} g^{rs}) g^{pq} g^{jq}## from ##\mathfrak {L}## = (4)R...