Newtonian Dynamics - Relation To Lagrange Equations

Relation To Lagrange Equations

Mechanical systems with constraints are usually described by Lagrange equations:

$\frac{dq^s}{dt}=w^s,\qquad\frac{d}{dt}\left(\frac{\partial T}{\partial w^s}\right)-\frac{\partial T}{\partial q^s}=Q_s,\qquad s=1,\,\ldots,\,n$ ,

(16)

where is the kinetic energy the constrained dynamical system given by the formula (12). The quantities in (16) are the inner covariant components of the tangent force vector (see (13) and (14)). They are produced from the inner contravariant components of the vector by means of the standard index lowering procedure using the metric (11):

$Q_s=\sum^n_{r=1}g_{sr}\,F^r,\qquad s=1,\,\ldots,\,n$ ,

(17)

The equations (16) are equivalent to the equations (15). However, the metric (11) and other geometric features of the configuration manifold are not explicit in (16). The metric (11) can be recovered from the kinetic energy by means of the formula

$g_{ij}=\frac{\partial^2T}{\partial w^i\,\partial w^j}$ .

(18)

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