By Felix Klein, George Gavin Morrice
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Extra resources for Lectures on the Icosahedron
Sample text
2) would allow o n e particle at most t o be referred t o an inertial frame. T h e very foundations of Newtonian mechanics would thus appear t o dissolve in confusion as soon as local action of G is allowed. W e a r e thus on t h e horns of a dilemma, for local action of G provides t h e additional degrees of freedom that model dislocations and disclinations but it destroys t h e foundation of Newtonian mechanics. Drastic, and possibly unfamiliar remedies are required. T h e situation just described is not particular to elasticity theory of material bodies.
If Ρ is a particle that is identified by the point in R with the coordinates {X\Χ , X , 0 } , we define its reference orbit to be the line { X \ Χ , Χ , τ | - oo < < oo} parallel to the Γ-axis. T h u s , if B is a 3-dimensional subset of the hyperplane T = 0 in R , then B χ R is the 4-dimensional cylinder in R that is the history of the body in the reference configuration history space. T h e history of a body in the reference configuration history space is thus trivial for nothing changes in the course of time - the 3-dimensional reference configuration at any o n e time is the same as that at any other time.
A n y point Ρ in R is thus uniquely defined by its four coordinates {X | 1 < a < 4 } , where the first t h r e e are the usual Cartesian spatial coordinates and the fourth is the time of t h e event labeled by the point P. T h e space R is assumed to be populated by a known system of fields Ψ(Χ ), or just Ψ for short, that are the state variables for a physical system. We assume that these state variables may be organized as the c o m p o n e n t s of a column matrix Ψ with a finite n u m b e r of entries.