I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete answer to this problem for general partial differential equations ? PDEs arise, for example, in Classical Field Theories of modern Physics and almost all that know originate from a variational principle. Please provide a summary of the answer (for the PDE case) and suggest books where I can learn about this subject of inverse problems in calculus of variations for ODE's and PDE's .
2026-04-05 01:15:18.1775351718
Inverse problem in calculus of variations
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This problem is also referred to as the Helmholtz equation. The answer can be found using the variational (bi)-complex as discussed in
http://math.uni.lu/~michel/data/VARIATIONNAL%20BICOMPLEX.pdf
by Ian Anderson.