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3
votes
1answer
196 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
418 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
322 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
906 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
114 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
202 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
221 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
257 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
422 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
138 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
825 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
207 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
299 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
318 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
10k 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
283 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 ...