## Abstract

In this note we submit a nonlocal (integral) generalization of the rotational-isotopic symmetries O-circumflex(3) introduced in preceding works for nonlinear and nonhamiltonian systems in local approximation. By recalling that the Lie-isotopic theory naturally admits nonlocal terms when all embedded in the isounit, while the conventional symplectic geometry is strictly local-differential, we introduce the notion of symplectic-isotopic two-forms, which are exact symplectic two-forms admitting a factorization into the Kronecker product of a canonical two-form time the isotopic element of an underlying Euclidean-isotopic space. Topological consistency is then achieved by embedding all nonlocal terms in the isounit of the iso-cotangent bundle, while keeping the local topology for the canonical part. In this way, we identify the symplectic-isotopic geometry as being the natural geometrical counterpart of the Lie-isotopic theory. The results are used for the introduction of the notion of Birkhoffian angular momentum, that is, the generalization of the conventional canonical angular momentum which is applicable to Birkhoffian systems with generally nonlinear, nonlocal and nonhamiltonian internal forces. The generators J (and the parameters {theta}) coincide with the conventional quantities. Nevertheless, the quantity J is defined on the underlying Euclidean-isotopic space, by therefore acquiring a generalized magnitude. The isocommutation rules and isoexponentiation of the
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## Citation Formats

Santilli, R M.
Rotational-isotopic symmetries.
IAEA: N. p.,
1991.
Web.

Santilli, R M.
Rotational-isotopic symmetries.
IAEA.

Santilli, R M.
1991.
"Rotational-isotopic symmetries."
IAEA.

@misc{etde_10113360,

title = {Rotational-isotopic symmetries}

author = {Santilli, R M}

abstractNote = {In this note we submit a nonlocal (integral) generalization of the rotational-isotopic symmetries O-circumflex(3) introduced in preceding works for nonlinear and nonhamiltonian systems in local approximation. By recalling that the Lie-isotopic theory naturally admits nonlocal terms when all embedded in the isounit, while the conventional symplectic geometry is strictly local-differential, we introduce the notion of symplectic-isotopic two-forms, which are exact symplectic two-forms admitting a factorization into the Kronecker product of a canonical two-form time the isotopic element of an underlying Euclidean-isotopic space. Topological consistency is then achieved by embedding all nonlocal terms in the isounit of the iso-cotangent bundle, while keeping the local topology for the canonical part. In this way, we identify the symplectic-isotopic geometry as being the natural geometrical counterpart of the Lie-isotopic theory. The results are used for the introduction of the notion of Birkhoffian angular momentum, that is, the generalization of the conventional canonical angular momentum which is applicable to Birkhoffian systems with generally nonlinear, nonlocal and nonhamiltonian internal forces. The generators J (and the parameters {theta}) coincide with the conventional quantities. Nevertheless, the quantity J is defined on the underlying Euclidean-isotopic space, by therefore acquiring a generalized magnitude. The isocommutation rules and isoexponentiation of the Birkhoffian angular momentum are explicitly computed and shown to characterize the most general known nonlinear and nonlocal realization of the isorotational symmetry. The local isomorphisms between the infinitely possible isotopes O-circumflex(3) and the conventional symmetry O(3) is proved. Finally the isosymmetries O-circumflex(3) are used to characterize the conserved, total, Birkhoffian angular momentum of closed nonselfadjoint systems. (author). 4 refs.}

place = {IAEA}

year = {1991}

month = {Sep}

}

title = {Rotational-isotopic symmetries}

author = {Santilli, R M}

abstractNote = {In this note we submit a nonlocal (integral) generalization of the rotational-isotopic symmetries O-circumflex(3) introduced in preceding works for nonlinear and nonhamiltonian systems in local approximation. By recalling that the Lie-isotopic theory naturally admits nonlocal terms when all embedded in the isounit, while the conventional symplectic geometry is strictly local-differential, we introduce the notion of symplectic-isotopic two-forms, which are exact symplectic two-forms admitting a factorization into the Kronecker product of a canonical two-form time the isotopic element of an underlying Euclidean-isotopic space. Topological consistency is then achieved by embedding all nonlocal terms in the isounit of the iso-cotangent bundle, while keeping the local topology for the canonical part. In this way, we identify the symplectic-isotopic geometry as being the natural geometrical counterpart of the Lie-isotopic theory. The results are used for the introduction of the notion of Birkhoffian angular momentum, that is, the generalization of the conventional canonical angular momentum which is applicable to Birkhoffian systems with generally nonlinear, nonlocal and nonhamiltonian internal forces. The generators J (and the parameters {theta}) coincide with the conventional quantities. Nevertheless, the quantity J is defined on the underlying Euclidean-isotopic space, by therefore acquiring a generalized magnitude. The isocommutation rules and isoexponentiation of the Birkhoffian angular momentum are explicitly computed and shown to characterize the most general known nonlinear and nonlocal realization of the isorotational symmetry. The local isomorphisms between the infinitely possible isotopes O-circumflex(3) and the conventional symmetry O(3) is proved. Finally the isosymmetries O-circumflex(3) are used to characterize the conserved, total, Birkhoffian angular momentum of closed nonselfadjoint systems. (author). 4 refs.}

place = {IAEA}

year = {1991}

month = {Sep}

}