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I had the intuition that, in classical mechanics, when the trajectory of a body is known, then analysis of its motion can be done in the linear space of that trajectory, if all forces are projected on the tangent of the trajectory.

My original idea was to bypass the consideration of angular momentum, and work only with linear momentum. I even wrote a question for a formalization of this idea, with no success, which surprised me.

Luckily, I found a hint on how to get started on it, and I managed to write that formalization and answer my own question, with some very elementary differential geometry. And I realized that I get a coherent problem of mechanics, but in a 1-dimensional world, which may even include several interacting masses (sharing the same trajectory), and has momentum conservation.

But my mathematical knowledge and ability stops there.

So my question is the following. If I am analysing the motion of masses, with the knowledge that they never leave a known surface, can I play the same trick, projecting all forces on tangents to that surface, so that I can analyse my problem in a 2-dimensional space, and benefit of some simplifications. For example an angular momentum is a scalar in 2D. A problem is obviously that coordinates are harder to define on a non-developable surface.

Can there be a concept of inertial frame in this projection space.

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babou
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    Are you familiar with Lagrangian mechanics with constraint equations? It's a mathematical tool very well suited to doing exactly this. – David Z Sep 06 '13 at 23:25

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Your idea is realized within the formalism of Lagrangian mechanics in terms of so-called constraints. It is based on the introduction of suitable generalized coordinates which capture the degrees of freedom of the system. You then introduce the effect of forces acting in your system (e.g. forces keeping a point mass on a certain trajectory like a circle) by constraint equations. These allow for an elegant solution of the problem in terms of the Euler-Lagrange equations. In case that the constraints are given by algebraic equations, one speaks of holonomic constraints.

Instead of giving a thorough mathematical treatment on the concept, I would rather refer to the rich literature on the subject. For a good introduction with nice examples I can recommend these lecture notes.

Frederic Brünner
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