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)

Read more about this topic: Newtonian Dynamics

Famous quotes containing the words relation to and/or relation:

“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)

“There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
—Adrienne Rich (b. 1929)