That is, define a function  such that  holds for all cotangent vectors in formula_12. Here, formula_13 is a vector in formula_14, the tangent space to the manifold at point . The function formula_15 is called the "momentum function" corresponding to . In local coordinates, the vector field at point may be written as  where the formula_17 are the coordinate frame on . The conjugate momentum then has the expression   where the formula_2 are defined as the momentum functions corresponding to the vectors formula_17:  The formula_1 together with the formula_23 together form a coordinate system on the cotangent bundle formula_24; these coordinates are called the "canonical coordinates". In Lagrangian mechanics, a different set of coordinates are used, called the generalized coordinates. These are commonly denoted as formula_25 with formula_1 called the generalized position and formula_27 the generalized velocity. When a Hamiltonian is defined on the cotangent bundle, then the generalized coordinates are related to the canonical coordinates by means of the Hamilton–Jacobi equations.