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Nonholonomic Behavior in Robotic Systems
Published in Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, 2017
Richard M. Murray, Zexiang Li, S. Shankar Sastry
The tools we develop are based on a variety of results from differential geometry and nonlinear control theory, more specifically Frobenius’ theorem and nonlinear controllability. To keep the mathematical prerequisites to a minimum, we do all the calculations in ℝn and restrict ourselves to drift-free control systems (i.e., control systems whose state remains fixed when the input is turned off). Many of the proofs in this section rely on some properties of manifolds which we have omitted from the discussion; they can be skipped without loss of continuity. A good introduction to nonlinear control theory which includes many of the necessary differential geometric concepts can be found in Isidori [44] or Nijmeijer and van der Schaft [84].
On the Symmetry of Blast Waves
Published in Nuclear Technology, 2021
The Frobenius theorem of differential geometry gives integrability conditions for a system of differential one-forms on a surface. This theorem extends the concept of integrating factors from a single equation to a system of equations. The Frobenius theorem has at least two natural forms: one which gives a functional representation of differential one-forms, and one which expresses an equivalent concept in terms of the closure properties of a differential ideal defined by differential one-forms. Following Ivey and Landsberg,28 the Frobenius theorem in terms of differential one-forms is Theorem Let be a manifold of dimension , and let be pointwise linearly independent. If there exist one-forms such that for all , then through each point there exists a maximal connected -dimensional manifold such that for . This manifold is unique, in the sense that any other such connected submanifold through is a subset of .