remark 23.19 The same conditions govern the Helmholtz equation

open in the book · parts/03-classical-mechanics/06-hamilton-jacobi.tex:595 · p. 830

remark -- no derivation owed by its kind

Rests on

depends_on theorem 23.17 Stäckel's conditions are sufficient ¶
- depends_on definition 23.13 Additive separability ¶
  - depends_on definition 23.5 Complete integral ¶
    - depends_on equation 23.2 eq:hj-equation ¶
  - depends_on equation 23.15 eq:hj-characteristic-equation ¶
- depends_on definition 23.16 Stäckel matrix; Stäckel system ¶
  - depends_on equation 23.23 eq:hj-orthogonal-hamiltonian ¶
- depends_on equation 23.15 eq:hj-characteristic-equation ¶ ↺
- proves proof ch:06-hamilton-jacobi@proof-6 ¶
depends_on theorem 10.34 Separable systems for the Helmholtz operator ¶
- depends_on equation 10.25 eq:pde-helmholtz-equation ¶
- depends_on lemma 10.28 The separation constant ¶
  - depends_on definition 10.2 Linear, semilinear, quasilinear ¶
    - depends_on definition 10.1 Partial differential equation; order ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 6.2 Open set ¶
  - proves proof ch:08-pdes@proof-9 ¶
- proves proof app:A-long-proofs@proof-82 ¶
- proves proof ch:08-pdes@prooflink-3 ¶

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type	direction	node	provenance	where
`depends_on`	→	Stäckel's conditions are sufficient	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:617
`depends_on`	→	Separable systems for the Helmholtz operator	declared	parts/03-classical-mechanics/06-hamilton-jacobi.tex:617