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We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an application, we obtain a representation formula for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation\n  \\begin{align*}\n  D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0\n  \\end{align*}\n  and study the related Lax-Oleinik evolution."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.05769","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-07-12T14:37:20Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"4307b2a6de5b4a6ddea26cadec331eaca23850c1e42e16c39fd99137345ea190","abstract_canon_sha256":"1458c0ef19c665360c9242884ff96b01259bf985dde1932518372eab75fbdbe7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:44.473951Z","signature_b64":"g4I364eO3CG7hQHABDId7nkBFDMUKXyiYW0ciZ/UOz/NImq/ULvI1lMWzHTimNu/qnGJ3/VM2AZHUeohg4v7DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2ffc214babb2e8528e25f672b5d2358618762f1636c5f09e338c24e3c32165c","last_reissued_at":"2026-05-17T23:40:44.473355Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:44.473355Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Herglotz' variational principle and Lax-Oleinik evolution","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Jun Yan, Kaizhi Wang, Liang Jin, Piermarco Cannarsa, Wei Cheng","submitted_at":"2019-07-12T14:37:20Z","abstract_excerpt":"We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in \\cite{CCWY2018} in the time-dependent case. 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