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From this:

Normal mode oscillations. If the Hamiltonian turns out to be a quadratic function of coordinates and momenta for a system of $N$ objects, e.g. $$H=\sum_{ij} M_{ij} q_i q_j + \sum_{ij} M_{ij} p_i p_j$$ then you can simply do a canonical transformation along the eigenvectors of $M_{ij}$ to diagonalize $M_{ij}$, and your system separates into independent harmonic oscillators.

Can someone please elaborate a little bit on this? I don't know how to "simply do a canonical transformation along the eigenvectors of $M_{ij}$ to diagonalize $M_{ij}$".

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1 Answers1

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More generally, given a semipositive definite quadratic real Hamiltonian $$H=\frac{1}{2}z^I H_{IJ} z^J~\geq~ 0 \tag{1}$$ in $2n$ canonical coordinates $$(z^1, \ldots z^{2n})~=~(q^1,\ldots, q^n,p_1, \ldots, p_n),\tag{2}$$ one may show that there always exists a real & linear symplectic transformation $$Z~=~ S z, \qquad S\in Sp(2n,\mathbb{R}),\tag{3}$$ that brings the Hamiltonian on diagonal form, cf. Ref. 1.

Note that the coordinate transformation must be symplectic in order to preserve the canonical commutation relations (CCR).

References:

  1. V.I. Arnold, Mathematical methods of Classical Mechanics, 2nd eds., 1989; Appendix 6.
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