Foundations of Theoretical Mechanics II: Birkhoffian by Ruggero Maria Santilli

By Ruggero Maria Santilli

In the previous volume,l I pointed out worthy and adequate stipulations for the lifestyles of a illustration of given Newtonian structures through a variational precept, the so-called stipulations of variational self-adjointness. a chief goal of this quantity is to set up that each one Newtonian platforms pleasant yes locality, regularity, and smoothness stipulations, no matter if conservative or nonconservative, may be taken care of through traditional variational ideas, Lie algebra options, and symplectic geometrical formulations. This quantity for that reason resolves an issue at the repre­ sentational features of traditional variational rules that has been 2 lingering within the literature for over a century, as suggested in Chart 1. three. 1. the first result of this quantity are the next. In bankruptcy 4,3 I end up a Theorem of Direct Universality of the Inverse challenge. It establishes the life, 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 develop into a generalization of traditional Hamilton equations (those with out exterior phrases) which: (a) admit the main normal attainable motion useful for first-order structures; (b) own a Lie algebra constitution within the so much common attainable, ordinary consciousness of the product; and (c) 1 Santilli (1978a). As used to be 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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4 does not ensure the existence of aHamiltonian. The theorem merely provides the integrability conditions for its existence. 4 does not necessarily admit a solution. In fact, if such a solution would always exist, the Indirect Lagrangian Problem (,J',16) always admits a solution, which is not the case (see the Appendix). A moment of reflection is appropriate here. Recall that the existence of a Hamiltonian implies the applicability of an articulated body of established, analytic, algebraic, and geometric tools, ranging from the Hamilton-Jacobi equations (and related quantization) to the canonical realization of Lie's theory (and related symmetries), etc.

Third, Hamilton's equations in the unified notation exhibit in a rather transparent way the interrelation between the analytic, algebraic, and geometrical profiles according to the following lines. 1. Analytic Profile. The well-known derivability of the equations from Hamilton's variational principle in phase space (Section I. 3) can be written in the unified notation as follows. Introduce the action. deE) = i i l2 dt[Pkf' - H(t, r, p)J(E) 11 l2 <)g dt[R~(a)aV - H(t, r, p)J(E) 11 RO = v {Pv, 0, v = 1,2, ...

Characterize the most general possible regular realization of the Lie algebra product via the brackets of a classical time evolution; and 3. admit the most general possible exact symplectic (or contact) structures in local coordinates. 1. In order to study these important properties in the necessary detail, the introduction of the following terminology is advantageous. 1. 4) 19 A linear first-order variational principle occurs when the integrand depends at most on first-order derivatives and the dependence is linear.

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