From: mikko.levanto@iki.fi   
      
   On 2024-08-07 06:54:34 +0000, Stefan Ram said:   
      
   >   In mathematical classical mechanics, the momentum is a cotangent   
   >   vector, while the velocity is a tangent vector. I don't get this!   
      
   In the usual formalism a vector is simply a vector. What do you mean   
   with "tangent" and "cotangent"? Usually they are trigonometric   
   functions, where cotangent of x is the same as thangent of the   
   complement of x and also the inverse of the tangent of x. But   
   those definitions don't apply to vectors.   
      
   --   
   Mikko   
      
   [[Mod. note -- I think Stefan is using "tangent vector" and   
   "cotangent vector" in the sense of differential geometry and tensor   
   calculus.  In this usage, these phrases describe how a vector (a.k.a   
   a rank-1 tensor) transforms under a change of coordintes: a tangent   
   vector (a.k.a a "contravariant vector") is a vector which transforms   
   the same way a coordinate position $x^i$ does, while a cotangent vector   
   (a.k.a a "covariant vector") is a vector which transforms the same way   
   a partial derivative operator $\partial / \partial x^i$ does.   
      
   See   
     https://en.wikipedia.org/wiki/Tensor_calculus   
   for more information.   
   -- jt]]   
      
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