By E. Poisson

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**Extra resources for A Relativist's Toolkit - The Math of Black Hole Mechanics**

**Example text**

The set of such points defines a straight line called the central axis. The previous proposition about the reduction of a collection of (force) vectors leads to an interesting condition of equilibrium. Indeed, the dynam of internal forces vanishing, we can state the following necessary condition of equilibrium: PR6 If a material system is in equilibrium, then the dynam of external forces is necessary zero. This is denoted: equilibrium where f ( e ) = ∑ f h(e ) , h ⇒ [F ] (e) f (e) = = 0.

1 supp ( f h ) is the support or line of action of fh . 36 Chapter 1 It is denoted by fS ′→S M . S ′→S a To sum up, we say: PR1 Every mechanical action is well-defined by the dynam of the mechanical action. In conclusion, the theory of dynams can be used in this context. D The dynam of mechanical actions is called the dynam of forces and is simply denoted: f F= M a (if the context is sufficiently clear). 2 CLASSIFICATION OF FORCES It is necessary to specify the various types of forces which may be applied to a material system, so we are going to give a review of these forces acting on every particle of the system.

Remark. In the hypostatic case, where the body is called partially constrained, the equilibrium is instable since the unknown forces cannot satisfy all the equilibrium equations. More explicitly, for each unsatisfied scalar equation of equilibrium, there is a corresponding possible motion, either of translation (for an equation of force), or of rotation (for an equation of moment). In engineering mechanics, it is essential to know the forces acting at different points of a solid structure. These forces can deform and break the structures; but often and in first approximation, these structures can be considered as rigid bodies.

### A Relativist's Toolkit - The Math of Black Hole Mechanics by E. Poisson

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