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Explore advanced concepts in geometric mechanics through this lecture from the ESI's Thematic Programme on "Infinite-dimensional Geometry." Delve into Lagrange's fundamental principle that connects motion laws to equilibrium, bridging dynamics and statics through differential geometry. Learn how variational calculus integrates differential geometry into infinite-dimensional trajectory spaces of physical systems, and understand the evolution to symplectic geometry's application in mechanics and multisymplectic geometry in classical field theory. Master geometric structures and procedures essential to Geometric Mechanics, with particular focus on challenging mechanical systems involving singular Lagrangians and nonholonomic constraints. Examine mechanics on algebroids and discover how Dirac algebroids serve as tools for deriving phase equations in both Hamiltonian and Lagrangian frameworks for systems with nonholonomic constraints.
