Editing Newtonian dynamics (section)

==Relation to Lagrange equations==
Mechanical systems with constraints are usually described by [[Lagrangian mechanics#Lagrange equations of the second kind|Lagrange equations]]:
{{NumBlk|:|<math>
\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</math>,|{{EquationRef|16}}}}
where <math>T=T(q^1,\ldots,q^n,w^1,\ldots,w^n)</math> is the kinetic energy the constrained dynamical system given by the formula ({{EquationNote|12}}). The quantities <math>Q_1,\,\ldots,\,Q_n</math> in 
({{EquationNote|16}}) are the inner [[tensor#Tensor valence|covariant components]] of the tangent force vector <math>\mathbf F_\parallel</math> (see ({{EquationNote|13}}) and ({{EquationNote|14}})). They are produced from the inner [[tensor#Tensor valence|contravariant components]] <math>F^1,\,\ldots,\,F^n</math> of the vector <math>\mathbf F_\parallel</math> by means of the standard [[raising and lowering indices|index lowering procedure]] using the metric ({{EquationNote|11}}):
{{NumBlk|:|<math>
Q_s=\sum^n_{r=1}g_{sr}\,F^r,\qquad s=1,\,\ldots,\,n</math>,|{{EquationRef|17}}}}
The equations ({{EquationNote|16}}) are equivalent to the equations ({{EquationNote|15}}). However, the metric ({{EquationNote|11}}) and
other geometric features of the configuration manifold <math>\displaystyle M</math> are not explicit in ({{EquationNote|16}}). The metric ({{EquationNote|11}}) can be recovered from the kinetic energy <math>\displaystyle T</math> by means of the formula
{{NumBlk|:|<math>
g_{ij}=\frac{\partial^2T}{\partial w^i\,\partial w^j}</math>.|{{EquationRef|18}}}}