# Equilibrium equations in mechanics: why there are only two kinds?

Off late, I am not seing any random thoughts on iMechanica.  This is my humble attempt at reviving such discussions.  Please contribute with your comments...

In Engineering Mechanics (Statics) we have two kinds of equilibrium equations: one set based on forces and another set based on moments.  Why can't we have some other kinds?  One perfectly reasonable argument is that we have only two kinds of rigid body motions: translations and rotations.  Therefore, we have two associated sets of equilibrium equations.  That brings up a question: is it somehow related to three-dimensional nature of our space?  Can we have more kinds of equations if we have a four-dimensional space?  Recently I read that it should be possible to perform inversions (say, from a left-handed helix to a right-handed one), if we have four dimensions.  Then, can we have some equilibrium equations associated with such inversions if we have a four-dimensional space?

### Re.: Equilibrium equations in 4D

Ummm... If you can explain your point via a suitable example (even if only out of "random thoughts"), complete with some graphical illustration, then it would be easier to analyze it.

Here, please think: After using appropriate dimensional analysis to take the relevant terms of the inertia of an object to unity, the "simple" force is quantitatively the same as the linear acceleration that it undergoes. The "angular force" i.e. the moment (of a force) is, the angular acceleration that it undergoes. What, if any, would be the nature of any additional kind of force(s) in a 4D space?

Hint: In general, before you can introduce forces, i.e., before you can begin working out the kinetics, you first have to state the kind of kinematics there is. What kind of a new kinematics do you see as necessitated by the introduction of the fourth dimension? If you can state this part (say with some illustrative graphics to go with it), then answering your question would be as simple as defining a new kind of acceleration corresponding to it.

Have fun i.e. random thoughts!

--Ajit

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