By Ruggero Maria Santilli
In the previous volume,l I pointed out helpful and enough stipulations for the life of a illustration of given Newtonian structures through a variational precept, the so-called stipulations of variational self-adjointness. a main goal of this quantity is to set up that every one Newtonian structures pleasing yes locality, regularity, and smoothness stipulations, no matter if conservative or nonconservative, should be handled through traditional variational rules, Lie algebra ideas, and symplectic geometrical formulations. This quantity for that reason resolves an argument at the repre sentational services of traditional variational rules that has been 2 lingering within the literature for over a century, as pronounced in Chart 1. three. 1. the first result of this quantity are the next. In bankruptcy 4,3 I turn out a Theorem of Direct Universality of the Inverse challenge. It establishes the lifestyles, through a variational precept, of a illustration for all Newtonian platforms of the category admitted (universality) within the coordinates and time variables of the experimenter (direct universality). The underlying analytic equations change into a generalization of traditional Hamilton equations (those with no exterior phrases) which: (a) admit the main basic attainable motion sensible for first-order platforms; (b) own a Lie algebra constitution within the such a lot basic attainable, commonplace consciousness of the product; and (c) 1 Santilli (1978a). As was once the case for quantity I, the references are indexed on the finish of this quantity, first in chronological order after which in alphabetic order.
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Extra resources for Foundations of Theoretical Mechanics II: Birkhoffian Generalizations of Hamiltonian Mechanics
Sample text
2. Algebraic Profile. The contravariant tensor O)I-'V is called thefundamental Lie tensor (or fundamental cosymplectic tensor) because it characterizes a fundamental realization of the Lie algebra product, that via the conventional Poisson brackets, according to the structure of the time evolution law for functions A(a) in phase space . oA oA voH A(a) = --al' = -0)1-' - v P oa oal-' oa == oA oH _ oA oH ~ [A, BJ. 16) Note that the form of Hamilton's equations characterizing the algebraic profile is contravariant.
More generally, Birkhoff's equations characterize the most general possible local formulation of an exact contact two-form. In turn, this is sufficient to establish that the contravariant Birkhoff's tensor characterizes the most general possible regular realization of the Lie algebra product in mechanics, owing to the known interplay between the Lie algebras and the contact (or symplectic) geometry. 19a) v=O v = 1, 2, ... , 2n. 2. 20) which verifies the following properties: 1. Oz 2. 1 is redundant for a symplectic two-form.
Notice that equations of motion in the second-order form admit only a simple infinity of equivalent forms, those characterized by a regular matrix of multiplicative functions, as in (f16). The reduction to a first-order form therefore doubles the degrees of freedom in writing the equations of motion, with self-evident advantages for the Inverse Problem. 3. The reduction for the case of equations with an arbitrary interpretation in Mechanics, Engineering, Biology, and other branches of science is self-evident, and it will be left to the interested reader.