Talk:Multiple-scale analysis

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Mathematics Mid‑priority

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Hamiltonian bound

I made a small edit and someone reverted it, so this is worth a discussion. The Hamiltonian for the Duffing eq. is H = p^2/2 + q^2/2 + (eps/4) q^4. This is clearly not symmetric in q and p, but the stated bounds on p and q are. So this confused me. The bound on p is right: to maximize p set q = 0, then p_max = sqrt(2 H0) with H0 the initial value of H. The same logic applies for q, but we then have q_max^2/2 + (eps/4) q_max^4 = H0. This gives a more complicated, and smaller, q_max than sqrt(2 H0) as written on the page. Yes, it is true that q < sqrt(2 H0), which we can see by dropping the eps term in the q_max eq. while still keeping it in H0. So a lower q_max exists, which reflects the asymmetry between q and p.

I realize the article is about multiple-scale analysis. The main point that q and p are both bounded by the Hamiltonian nature of the system. I got tripped up by H being asymmetric in q and p, but the bounds being symmetric. So it's worth a comment. Though my original one wasn't clear. I'll try again. Dstrozzi (talk) 13:03, 27 March 2025 (UTC)[reply]