Path: ...!3.eu.feeder.erje.net!feeder.erje.net!fu-berlin.de!uni-berlin.de!individual.net!not-for-mail From: Mikko Newsgroups: sci.physics.research Subject: Re: The momentum - a cotangent vector? Date: Wed, 07 Aug 2024 11:37:02 PDT Organization: - Lines: 27 Approved: Jonathan Thornburg [remove -color to reply]" References: X-Trace: individual.net LoiTDD2SRz9vO/JU6D3Y5QjqaXYtkvYxEG37eNHqJKRDBBlcFvUM8b9eiG Cancel-Lock: sha1:7Iz6VKLeJ2XY98Im2L5VcVgsj70= sha256:tdiDUrD1qzbcF8+IuP4yqAVfvXWWD/UJBuudiy4cjPA= Bytes: 1882 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]]