The talk will be roughly in three parts, with the last two strongly related. In the first part of the talk I shall discuss the use of affine connections for writing the equations of motion for ``simple'' mechanical systems with constraints (i.e., those whose Lagrangians are kinetic energy minus potential energy). The second part of the talk deals with using the Gibbs-Appell equations to arrive at equations of motion for general Lagrangian systems (here general means formulated on a bundle over the R and with nonlinear constraints). In the last part of the talk, I remain in the general framework and talk about variational principles, first Gauss's Principle of Least Constraint, then a generalisation of the Lagrange-d'Alembert Principle.
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