Principle of Alembert

The principle of Alembert is a principle stated in 1743 by Jean the Round of Alembert before Joseph-Louis Lagrange does not develop the Analytical mechanics by using it.

Statement

• Definition of a virtual displacement $\backslash \; delta\; \backslash \; vec\; R$ of a body C: it is an instantaneous, infinitesimal displacement and respecting the physical constraints of the body (rigidity, field in which it evolves/moves,…).

• the forces of constraints are the forces applying to the body, making that it respects the physical constraints (force of reaction of the table on which the body is posed, resistance of rigidity to the external forces,…).

• This principle says that the whole of the forces of constraints applied to a system does not work (does not produce nor does not consume a energy) at the time of a virtual displacement :

If the system is a body C, and that the forces of constraints are $\backslash \; vec\; F\; (X)$ for each x∈C, then for all virtual displacement $\backslash \; delta\; \backslash \; vec\; R\; (X)$ of the body, one a: $\backslash \; int\_C\; \backslash \; vec\; F\; (X).\; \backslash \; delta\; \backslash \; vec\; R\; (X)\; dx\; =\; 0$

what one can write $\backslash \; int\_C\; \backslash \; left\; (\backslash \; vec\; F\; (X)\; -\; \backslash \; dowry\; \backslash \; vec\; p\; (X)\; \backslash \; right).\; \backslash \; delta\; \backslash \; vec\; R\; (X)\; dx\; =\; 0$ with $\backslash \; vec\; p\; (X)$ the impulse and $\backslash \; vec\; F\; (X)$ the sum of the forces (which are not constraint) being exerted in $\backslash \; X\; \backslash \; in\; C$.

In fact, this principle postulates that, for example, the table on which is posed the object is passive (opposes only forces of reaction to the body) and any acceleration will not provide him.

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