Celestial mechanics with symplectic integrators

AI Thread Summary
The discussion revolves around understanding the algorithm in a paper on celestial mechanics using symplectic integrators. Key points of confusion include the meaning of the sum from 1 to n in equation 2, the interpretation of n, and the application of exponential operators in the equations. There is uncertainty about the general solution q(t) = exp(tau * F) * q(t-tau) and its relation to dq/dt = Fq. Additionally, the relevance of equation 7 to previous equations is unclear, and there is a need for clarification on the concept of degrees of freedom in this context. Overall, the participant seeks specific guidance to grasp these concepts before their presentation.
Masklin
Messages
2
Reaction score
0

Homework Statement



The problem is that I just don't understand how the algorithm described here in section 2 hangs together... I have to present this on Thursday morning and that sensation of 'I'll never understand this soon enough' is growing ominously.

Homework Equations



Equations 2,6,7 and 8 are a mystery to me. I could write them out here but they're already in the paper and without their context it wouldn't help much I think.

Why is there a sum from 1 to n in equation 2? What is n? It doesn't say...

And, how is q(t) = exp(tau * F) * q(t-tau) a general solution to dq/dt = Fq ?

Shouldn't it be q(t) = q(tau) * exp(Fq) ?

How does one knows in which order to apply the exponential operators in equation 6?

Where does equation 7 fit in with anything introduced previously?

The Attempt at a Solution



My attempt at a solution is asking for help here... I've googled but my questions are far too specific for that to help. =(

Please please help!Masklin
 
Physics news on Phys.org
Equ. 2 is an application of the chain rule and then Equ. 1. n is the number of the degrees of freedom.
 
Degrees of freedom - what does that mean in this context? The dimensionality of x and p, or the number of bodies? Or something else?
 
The dimension of x.
 
Thread 'Variable mass system : water sprayed into a moving container'
Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
Back
Top