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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.physics.research
Subject: Re: The momentum - a cotangent vector?
Date: Wed, 07 Aug 2024 11:37:02 PDT
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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]]