Newtonian Dynamics - Internal Presentation of The Velocity Vector

Internal Presentation of The Velocity Vector

The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function (7):

\displaystyle\mathbf v=\sum^n_{i=1}\frac{\partial\mathbf r}{\partial q^i}\,\dot q^i .

(8)

The quantities are called internal components of the velocity vector. Sometimes they are denoted with the use of a separate symbol

\displaystyle\dot q^i=w^i,\qquad i=1,\,\ldots,\,n

(9)

and then treated as independent variables. The quantities

\displaystyle q^1,\,\ldots,\,q^n,\,w^1,\,\ldots,\,w^n

(10)

are used as internal coordinates of a point of the phase space of the constrained Newtonian dynamical system.

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