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
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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
 
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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.
 

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