Re: The momentum - a cotangent vector?

From	Mikko <mikko.levanto@iki.fi>
Newsgroups	sci.physics.research
Subject	Re: The momentum - a cotangent vector?
Date	2024-08-07 11:37 -0700
Organization	-
Message-ID	<v8vbib$2fuke$1@dont-email.me> (permalink)
References	<cotangent-20240806233433@ram.dialup.fu-berlin.de>

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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]]

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Thread

The momentum - a cotangent vector? ram@zedat.fu-berlin.de (Stefan Ram) - 2024-08-07 06:54 +0000
  Re: The momentum - a cotangent vector? Mikko <mikko.levanto@iki.fi> - 2024-08-07 11:37 -0700
    Re: The momentum - a cotangent vector? ram@zedat.fu-berlin.de (Stefan Ram) - 2024-08-08 07:02 +0000
      Re: The momentum - a cotangent vector? Hendrik van Hees <hees@itp.uni-frankfurt.de> - 2024-08-08 07:49 +0000
        Re: The momentum - a cotangent vector? ram@zedat.fu-berlin.de (Stefan Ram) - 2025-12-05 13:41 -0800
      Re: The momentum - a cotangent vector? Mikko <mikko.levanto@iki.fi> - 2024-08-08 11:00 +0000
    Re: The momentum - a cotangent vector? Mikko <mikko.levanto@iki.fi> - 2024-08-08 21:15 -0700
  Re: The momentum - a cotangent vector? Hendrik van Hees <hees@itp.uni-frankfurt.de> - 2024-08-09 13:53 -0700
  Re: The momentum - a cotangent vector? ram@zedat.fu-berlin.de (Stefan Ram) - 2024-08-10 06:16 +0000

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