Let ( X , L ) be a real dynamical system with n degrees of freedom. Here X is the configuration space and L = L ( t , q ( t ) , v ( t ) ) the Lagrangian, i.e. a smooth real-valued function such that q ( t ) ∈ ∈ X , and v ( t ) is an n -dimensional "vector of speed". (For those familiar with differential geometry, X is a smooth manifold, and L : R t × × T X → → R , where T X is the tangent bundle of X ) .