Bounded Variation in Time

Abstract : Describing a motion consists in defining the state or position $q$ of the investigated system as a function of the real variable $t$, the time. Commonly, $q$ takes its values in some set $Q$, suitably structured for the velocity $u$ to be introduced as the derivative of $t\to q$, when it exists. This, in fact, makes sense if $Q$ is a topological linear space or, more generally, a differential manifold modelled on such a space. For smooth situations, classical dynamics rests, in turn, on the consideration of the acceleration. This is the derivative of $t \to u$, if it exists in the sense of the topological linear structure of $ Q$, or, when $Q$ is a manifold, in the sense of some connection. But, from its early stages, classical dynamics has also had to face shocks, i.e. velocity jumps. For isolated shocks, one traditionally resorts to the equations of the dynamics of percussions. Even in the absence of impact, it has been known for a long time that systems submitted to such nonsmooth effects as dry friction may exhibit time discontinuity of the velocity. Furthermore, nonsmooth mechanical constraints may also prevent $t\to u$ from admitting a derivative. In all these cases, the laws governing the motion can no longer be formulated in terms of acceleration.
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Jean Jacques Moreau. Bounded Variation in Time. Topics in Nonsmooth Mechanics, 1987, 9783764319076. ⟨hal-01363799⟩

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