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Hilbert’s sixth problem was a proposal to expand the axiomatic method outside the existing mathematical disciplines, to physics and beyond. This expansion requires development of semantics of physics with formal analysis of the notion of physical reality that should be done. [ 9 ]
This can be demonstrated by the following experiment: Hold a tennis racket at its handle, with its face being horizontal, and throw it in the air such that it performs a full rotation around its horizontal axis perpendicular to the handle (ê 2 in the diagram), and then catch the handle. In almost all cases, during that rotation the face will ...
A solution of the falling cat problem is a curve in the configuration space that is horizontal with respect to the connection (that is, it is admissible by the physics) with prescribed initial and final configurations. Finding an optimal solution is an example of optimal motion planning. [11] [12]
The simplest non-trivial examples are the exponential growth model/decay (one unstable/stable equilibrium) and the logistic growth model (two equilibria, one stable, one unstable). The phase space of a two-dimensional system is called a phase plane , which occurs in classical mechanics for a single particle moving in one dimension, and where ...
Noteworthy examples of vacuum solutions, electrovacuum solutions, and so forth, are listed in specialized articles (see below). These solutions contain at most one contribution to the energy–momentum tensor, due to a specific kind of matter or field. However, there are some notable exact solutions which contain two or three contributions ...
Whittaker suggests that line AK be selected parallel to the axis of the given rotation, with K the foot of a perpendicular from B. The appropriate screw displacement is about an axis parallel to AK such that K is moved to B. In Whittaker's terms, "A rotation about any axis is equivalent to a rotation through the same angle about any axis ...
For example, the first solution of Domokos and Várkonyi closely resembled a sphere, with a shape deviation of only 10 −5. It was dismissed as it was tough to test experimentally. [ 8 ] The first physically produced example is less sensitive; yet it has a shape tolerance of 10 −3 , that is 0.1 mm for a 10 cm size.
The inverse scattering or inverse spectral problem associated to the Cauchy problems for 1+1 dimensional partial differential equations on the line, or to periodic problems, or even to initial-boundary value problems (Fokas (2002)), can be stated as a Riemann–Hilbert problem.