From: ram@zedat.fu-berlin.de   
      
   Hendrik van Hees  wrote or quoted:   
   >In the Lagrange formalism you deal with curves x^j(t) and   
   >d_t x^j(t)=\dot{x}^j   
   >obviously transform like vector components, and the Lagrangian should be   
   >a scalar. since the \dot{x}^j are vector components, and thus   
   >p_j = \partial L/\partial \dot{x}^j   
   >are the components of a co-vector.   
      
     Now I see that this explanation is also given in [1], where   
     the author writes in "2.3c. The Phase Space in Mechanics",   
      
   |For the present we wish merely to draw attention to certain   
   |basic aspects that seem mysterious when treated in most   
   |physics texts, largely because they draw no distinction there   
   |between vectors and covectors.   
      
     and   
      
   |Hamilton was led to define the functions   
   |p_i(q,q.):=dL/dq.^i               (2.26)   
   . . .   
   |Equation (2.26) is then to be considered   
   |not as a change of coordinates in TM but   
   |rather as the local description of a map   
   |p: TM --> T*M   
   |/from the tangent bundle to the cotangent   
   |bundle/.   
      
     after he has shown that the components of "p" transform   
     like the components of a covector.   
      
     [1] "The Geometry of Physics" (2001) - Theodore Frankel.   
      
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