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In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by L.\u00a0Schwartz and P.-A.\u00a0Meyer, to construct second-order counterparts of those classical structures. These will allow us to study symmetries of stochastic differential equations (SDEs), to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton\u2013Jacobi\u2013Bellman (HJB) equations. Indeed, stochastic prolongation formulae will be derived to study symmetries of SDEs and mixed-order Cartan symmetries. Stochastic Hamilton\u2019s equations will follow from a second-order symplectic structure and canonical transformations will lead to the HJB equation. A stochastic variational problem on Riemannian manifolds will provide a stochastic Euler\u2013Lagrange equation compatible with HJB one and equivalent to the Riemannian version of stochastic Hamilton\u2019s equations. A stochastic Noether\u2019s theorem will also follow. The inspirational example, along the paper, will be the rich dynamical structure of Schr\u00f6dinger\u2019s problem in optimal transport, where the latter is also regarded as a Euclidean version of hydrodynamical interpretation of quantum mechanics.<\/jats:p>","DOI":"10.1007\/s00332-023-09917-x","type":"journal-article","created":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T14:02:28Z","timestamp":1686146548000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["From Second-Order Differential Geometry to Stochastic Geometric Mechanics"],"prefix":"10.1007","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1798-8313","authenticated-orcid":false,"given":"Qiao","family":"Huang","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5390-0753","authenticated-orcid":false,"given":"Jean-Claude","family":"Zambrini","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,7]]},"reference":[{"key":"9917_CR1","volume-title":"Foundations of Mechanics","author":"R Abraham","year":"1978","unstructured":"Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. 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