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1
vote
1answer
383 views

Questions regarding solving the Brachistochrone problem using Lagrangian

brachistochrone problem: Suppose that there is a rollercoaster. There is point 1 ($0,0$) and point 2 ($x_2, y_2)$. Point 1 is at the higher place when compared to the point 2, so the rollercoaster ...
3
votes
0answers
252 views

Integrals given by Landau [closed]

Discussion about Landau's "Theoretical Minimum" has already been posted here. Unfortunately I couldn't find much about some examples of questions he gave to students. There are three questions in the ...
16
votes
3answers
950 views

When is Lebesgue integration useful over Riemann integration in physics?

Riemann integration is fine for physics in general because the functions dealt with tend to be differentiable and well behaved. Despite this, it's possible that Lebesque integration can be more ...
2
votes
2answers
7k views

Determining the center of mass of a cone

I'm having some trouble with a simple classical mechanics problem, where I need to calculate the center of mass of a cone whose base radius is $a$ and height $h$..! I know the required equation. But, ...
6
votes
2answers
397 views

What the circled integral?

What the circled integral $$ \oint $$ means? I saw this symbol in a lot of books about advanced physics. How is his definition? What kind of integral it is? It is used only in physics or also in ...
6
votes
4answers
3k views

How do you do an integral involving the derivative of a delta function?

I got an integral in solving Schrodinger equation with delta function potential. It looks like $$\int \frac{y(x)}{x} \frac{\mathrm{d}\delta(x-x_0)}{\mathrm{d}x}$$ I'm trying to solve this by ...
4
votes
1answer
378 views

Number of unique 2-electron integrals

Consider 2-electron integrals over real basis functions of the form $$(\mu\nu|\lambda\sigma) = \int d\vec{r}_{1}d\vec{r}_{2} \phi_{\mu}(\vec{r}_{1}) \phi_{\nu}(\vec{r}_{1}) r_{12}^{-1} ...
1
vote
1answer
4k views

Calculation of Distance from measured Acceleration vs Time

I have an Accelerometer connected to a device that feeds the instant values of the acceleration in the 3 directions. I've tried to calculate the distance for a vertical movement using these values ...
0
votes
1answer
85 views

Vehicle acceleration

What I'm essentially doing is Kalman Filter. If anyone is familiar with (but it doesn't really matter in this case). Consider the following formulas: $$x_k=x_{k-1}+v_{k-1}dt+a_{k-1}\frac{dt^2}{2}$$ ...
2
votes
1answer
359 views

Meaning of $d\Omega$ in basic scattering theory?

In basic scattering theory, $d\Omega$ is supposed to be an element of solid angle in the direction $\Omega$. Therefore, I assume that $\Omega$ is an angle, but what is this angle measured with respect ...
3
votes
1answer
229 views

Symplectic integrators of the pendulum equation?

In particular, a symplectic integrator to solve: $$\ddot{\theta} + \dfrac{g}{l} \sin(\theta) = 0.$$ I'm currently using velocity verlet - by realizing that $$\ddot{\theta} = -\nabla (-cos(\theta)) ...
4
votes
2answers
436 views

About an electrostatics integral and a delta-function kernel

I'm having trouble with an integral and I would like some pointers on how to "take" it: $$ \int \limits_{-\infty}^{\infty}\frac{3\gamma a^{2}d^{3}\mathbf r}{4 \pi \left( r^{2} + ...
4
votes
1answer
395 views

Integration of partition-function over many momentum variables

My integral looks like $$Z(\beta) = \frac{1}{h^3}\int d^3p\ \exp{\left(-\frac{\beta}{2m}\sum^{3N}_{i=1}p_i^2\right)}.$$ I'm confused about how to integrate over seemingly 3N variables in only a ...
7
votes
3answers
1k views

Principal value integral

I am reading A. Zee, QFT in a nutshell, and in appendix 1 he has: Meanwhile the principal value integral is defined by: $$\int dx\,{\cal P}{1\over x}f(x)~=~ \lim_{\epsilon \rightarrow 0} \int ...
1
vote
0answers
122 views

How to integrate twice of this viscous term?

I am reading a paper, and I do not understand why the author said the following term when integrated twice will become, $\int\limits_\Omega {{\rm{d}}\Omega {{\bf{\psi }}^{\bf{u}}}\cdot\nabla ...
0
votes
2answers
219 views

How was transformed an integral below?

I know how transform an integral below, $$ \iint f(\mathbf v_{1})f(\mathbf v_{2})d^3\mathbf v_{1}d^3\mathbf v_{2}, $$ using relative speed coordinates: we just use $$ m_{1} \mathbf v_{1} + ...
2
votes
0answers
250 views

Why does this integral come out imaginary?

Im working through Zee and I'm having a little trouble with some integrals. I'm trying to reproduce the analogue of the inverse square law for a 2+1 D universe and I figured I could start with the ...
3
votes
1answer
304 views

Spinor integration

I am learning on-shell methods for one loop integrals from this paper: Loop amplitudes in gauge theory: modern analytic approaches by Britto. Starting with formula (18) spinor integration is ...
1
vote
1answer
464 views

Equations of motion in 2D [closed]

I'm struggling with a seemingly simple problem in 2D motion. Basically, the question is, given accelerations in $x$ and $y$ ($a_x$ and $a_y$) as well as the angular velocity ($\omega$), how can we ...
13
votes
1answer
156 views

Monte Carlo integration over space of quantum states

I am currently facing the problem of calculating integrals that take the general form $\int_{R} P(\sigma)d\sigma$ where $P(\sigma)$ is a probability density over the space of mixed quantum states, ...
0
votes
1answer
910 views

How to find acceleration given position and velocity? [closed]

Sorry for this very simple question but I am still very new to the laws of motion. I am dealing with 2-dimensional vectors in my programming environment and I'm following these slides to learn about ...
0
votes
1answer
248 views

Expansion of Helmholtz energy

To get an expansion of Helmholtz energy of a) an ideal gas b) a Van der waals gas we must integrate $\left ( \frac{\delta A }{\delta V} \right )_{T}=-P$ I saw the solution is : Can you ...
1
vote
0answers
303 views

How to find the electric field at a point based on a uniformly charged surface

What is the general solution to finding the electric field at a point based on some (or multiple) charged surfaces. I know that we can perform a line/surface integral if a charge is close to a wire or ...
3
votes
1answer
367 views

Basic Grassmann/Berezin Integral Question

Is there a reason why $\int\! d\theta~\theta = 1$ for a Grassmann integral? Books give arguments for $\int\! d\theta = 0$ which I can follow, but not for the former one.
7
votes
5answers
15k views

How to get distance when acceleration is not constant?

I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question. The equation for distance of an accelerating object with constant ...
14
votes
1answer
363 views

Why isn't the Gear predictor-corrector algorithm for integration of the equations of motion symplectic?

Okumura et al., J. Chem. Phys. 2007 states that the Gear predictor-corrector integration scheme, used in particular in some molecular dynamics packages for the dynamics of rigid bodies using ...