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This article attempts to provide a rigorous definition of the looser, simpler idea presented in the article Canonical Conjugate Variables .


DEFINITION

Given a manifold ''Q'', a Vector Field ''X'' on the Tangent Bundle ''TQ'' can be thought of as a function acting on the Cotangent Bundle , by the duality between the tangent and cotangent spaces. That is, define a function
  • Q o \mathbb{R}

    • such that

:P_X(q,p)=p(X_q)
  • Q. Here, X_q is a vector in T_qQ, the tangent space to the manifold ''Q'' at point ''q''. The function P_X is called the momentum function corresponding to ''X''.


In Local Coordinates , the vector field ''X'' at point ''q'' may be written as
:X_q=\sum_i X^i(q) rac{\partial}{\partial q^i}
where the \partial /\partial q^i are the coordinate frame on TQ. The conjugate momentum then has the expression
:P_X(q,p)=\sum_i X^i(q) \;p_i
where the p_i are defined as the momentum functions corresponding to the vectors \partial /\partial q^i:
:p_i = P_{\partial /\partial q^i}
  • Q; these coordinates are called the canonical coordinates.



GENERALIZED COORDINATES

In Lagrangian Mechanics , a different set of coordinates are used, called the Generalized Coordinates . These are commonly denoted as (q^i,\dot{q}^i) with q^i called the generalized position and \dot{q}^j 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 .


SEE ALSO