Foundations of Theoretical Mechanics I

Santilli, Ruggero Maria

doi:10.1007/978-3-642-86757-6

Cited by 105 publications

(111 citation statements)

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“…The first formulation of isomathematics was presented in volumes [20] (written at Harvard University following the delivering of a seminar course in the field) and was based on the lifting of the conventional associative product AB between generic quantities , A B (functions, matrices, operators, etc.) into the associativity-preserving form called isoproduct…”

Section: Isoredshifts and Isoblueshiftsmentioning

confidence: 99%

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Representation of Galactic Dynamics Via Isoshifts Without Universe Expansion, Dark Matter and Dark Energy

Santilli¹

2015

AJMP

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Abstract:In this paper, we outline mathematical, theoretical and experimental studies by the author on the anomalous redshifts (blueshifts) of light passing through cold (hot) gases without relative motion, that are known as Santilli isoredshifts (isoblueshifts); we review the rather vast experimental and visual evidence on the lack of expansion of the universe and related cosmological conjectures, including the lack of acceleration, big bang and dark energy; we recall a number of insufficiencies of the conjecture of dark matter, with particular reference to its lack of quantitative representation of anomalous redshifts of galactic stars;we then present, apparently for the first time, a quantitative and time invariant representation of the anomalous redshift of galactic stars via Santilli isoredshifts and isoblueshifts without dark matter by merely admitting the astrophysical evidence that galaxies are indeed filled up with an actual, physical, detectable, gaseous medium causing deviations from the Doppler shift law; and we point out that the emerging new vistas of the universe do preserve Einstein's axioms of special relativity and merely require broader realizations.

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Section: Isoredshifts and Isoblueshiftsmentioning

confidence: 99%

“…[20], today known as the Lie-Santilli isotheory, with isoalgebra and isogroup for Hermitean generators…”

Section: Isoredshifts and Isoblueshiftsmentioning

confidence: 99%

Representation of Galactic Dynamics Via Isoshifts Without Universe Expansion, Dark Matter and Dark Energy

Santilli¹

2015

AJMP

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“…Thus, the possible motions in a mechanical system form a behavior represented by the Euler-Lagrange equations. We now study the converse problem (called the "inverse problem of the calculus of variations"; see [17]): under which conditions does a linear differential behavior B (typically described by a system of higher-order linear differential equations) consist of the stationary trajectories with respect to some functional interpretable as a Lagrangian (i.e., a functional that represents the difference between kinetic and potential energy in a suitable sense), and how does one construct such a functional on the basis of the equations describing B? It turns out that under mild assumptions, this is the case if and only if B is a Hamiltonian system.…”

Section: Such a Quantity Is Zero If And Only Ifmentioning

confidence: 99%

Linear Hamiltonian Behaviors and Bilinear Differential Forms

Rapisarda¹,

Trentelman²

2004

SIAM J. Control Optim.

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Abstract. We study linear Hamiltonian systems using bilinear and quadratic differential forms. Such a representation-free approach allows us to use the same concepts and techniques to deal with systems isolated from their environment and with systems subject to external influences and allows us to study systems described by higher-order differential equations, thus dispensing with the usual point of view in classical mechanics of considering first-and second-order differential equations only.Key words. linear Hamiltonian systems, two-variable polynomial matrices, bilinear and quadratic differential forms, behavioral system theory AMS subject classifications. 93A10, 93A30, 93C05, 37J99, 70H50 DOI. 10.1137/S03630129024146641. Introduction. This paper aims to give a unified treatment of linear Hamiltonian systems using the formalism of bilinear and quadratic differential forms introduced in [24]. We consider systems with and without external influences, and we deal with both cases using the same techniques and the same concepts. Moreover, we formulate concepts and study the properties of Hamiltonian systems in a representation-free way, thus dispensing with the usual point of view in mechanics and in physics (see, for example, [1]) of concentrating on first-order representations in the (generalized) coordinates and the (generalized) momenta. Instead of postulating the existence of a function (the Lagrangian, or the Hamiltonian) on the basis of physical considerations (conservation of energy, etc.) and deducing from such a function the equations of motion, we proceed by assuming that a set of linear differential equations with constant coefficients describing the system is given, and we deduce the Hamiltonian nature of the system from such equations, by proving the existence of certain bilinear functionals of the variables of the system and of their derivatives satisfying some additional property. Our approach is of a system-theoretic nature rather than derived from the study of mechanics: as happens in optimal control theory for linear systems, we consider the interplay of (quadratic and bilinear) functionals of the system variables and of the equations of motion as the central object of study when dealing with Hamiltonian systems.In this paper we also reconcile our point of view with that of classical mechanics by showing how to construct a "generalized Lagrangian" on the basis of the equations of the system, in the sense that the trajectories of the system are stationary with respect to such a quadratic functional of the variables of the system and their derivatives. In this context, the concept of internal force also arises naturally from the equations describing the system: in this paper we show that generalized internal forces can be defined which depend on higher-order derivatives of the external variables and not only first-order ones at most, as happens in classical mechanics.

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“…The reason is that there is not yet a consistent Lagrangian and Hamiltonian formulation for dissipative systems. The problem of obtaining the Lagrangian and Hamiltonian from the equations of motion of a mechanical system is a particular case of "The Inverse problem of the Calculus of Variations" [6,7]. This topic has been studied by many mathematicians and theoretical physicists since the end of the last century.…”

Section: Introductionmentioning

confidence: 99%