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The frame condition was first described by Richard Duffin and Albert Charles Schaeffer in a 1952 article on nonharmonic Fourier series as a way of computing the coefficients in a linear combination of the vectors of a linearly dependent spanning set (in their terminology, a "Hilbert space frame"). [4]
Since it is undecidable, by Chagrova's theorem, whether an arbitrary modal formula has a first-order correspondent, there are formulas with first-order frame conditions that are not Sahlqvist [Chagrova 1991] (see the examples below). Hence Sahlqvist formulas define only a (decidable) subset of modal formulas with first-order correspondents.
We know a nice sufficient condition: Henrik Sahlqvist identified a broad class of formulas (now called Sahlqvist formulas) such that a Sahlqvist formula is canonical, the class of frames corresponding to a Sahlqvist formula is first-order definable, there is an algorithm that computes the corresponding frame condition to a given Sahlqvist formula.
These rotational conditions follow from the specific construction of the Eckart frame, see Biedenharn and Louck, loc. cit., page 538. Finally, for a better understanding of the Eckart frame it may be useful to remark that it becomes a principal axes frame in the case that the molecule is a rigid rotor , that is, when all N displacement vectors ...
The first Frenet-Serret formula holds by the definition of the normal N and the curvature κ, and the third Frenet-Serret formula holds by the definition of the torsion τ. Thus what is needed is to show the second Frenet-Serret formula. Since T, N, B are orthogonal unit vectors with B = T × N, one also has T = N × B and N = B × T.
The center of mass frame is an inertial frame in which the center of mass of a system is ... Solving this equation for R yields the formula = ... In that condition, ...
Also in some frames not tied to the body can it be possible to obtain such simple (diagonal tensor) equations for the rate of change of the angular momentum. Then ω must be the angular velocity for rotation of that frames axes instead of the rotation of the body. It is however still required that the chosen axes are still principal axes of ...
Top: frame F′ moves at velocity v along the x-axis of frame F. Bottom: frame F moves at velocity −v along the x′-axis of frame F′. [5] The invariant interval can be seen as a non-positive definite distance function on spacetime. The set of transformations sought must leave this distance invariant.