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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 ...
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.
Topspin in ball games is defined as spin about a horizontal axis perpendicular to the direction of travel that moves the top surface of the ball in the direction of travel. Under the Magnus effect, topspin produces a downward swerve of a moving ball, greater than would be produced by gravity alone.
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 vector r(t) has some projection (or, equivalently, some component) r ⊥ (t) on a plane perpendicular to the axis of rotation. Then the angular position of that point is the angle θ from a reference axis (typically the positive x-axis) to the vector r ⊥ (t) in a known rotation sense (typically given by the right-hand rule).
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein , Israel Gohberg and others.
(A very similar problem is the design of a banked turn, where the slope of the turn is set so a car will not slide off the road. The analogy in the case of rotating bucket is that the element of water surface will "slide" up or down the surface unless the normal to the surface aligns with the vector resultant formed by the vector addition F g ...