hadronic mathematics, mechanics and chemistry - Institute for Basic ...

HADRONIC MATHEMATICS, MECHANICS AND CHEMISTRY 9
H =
p2 a
2 × m a
+ V (t, r, p), (1.2.2d)
V = U(t, r) ak × v k a + U o (t, r);
(1.2.2e)
where: v and p represent three-vectors; and the convention of the sum of repeated
indices is hereon assumed.
Interior dynamical systems when Newton’s force F ak is partially derivable from
a potential and partially of contact, zero-range, nonpotential types thus admitting
additional interactions that simply cannot be represented with a Lagrangian or
a Hamiltonian. For this reason, Lagrange, Hamilton, Jacobi and other founders
of analytic dynamics presented their celebrated equations with external terms
representing precisely the contact, zero-range, nonpotential forces among extended
particles. Therefore, the treatment of interior systems requires the true Lagrange
and Hamilton analytic equations, those with external terms
d ∂L(t, r, v) ∂L(t, r, v)
dt ∂va
k −
∂ra
k = F ak (t, r, v), (1.2.3a)
dr k a
dt
=
∂H(t, r, p)
∂p ak
,
dp ak
dt
∂H(t, r, p)
= −
∂ra
k + F ak (t, r, p), (1.2.3b)
L = 1 2 × m a × v 2 a − V (t, r, v),
(1.2.3c)
H =
p2 a
2 × m a
+ V (t, r, p), (1.2.3d)
V = U(t, r) ak × v k a + U o (t, r),
F (t, r, v) = F (t, r, p/m).
(1.2.3e)
(1.2.3f)
Comprehensive studies were conducted by Santilli in monographs [9] (including
a vast historical search) on the necessary and sufficient conditions for the existence
of a Lagrangian or a Hamiltonian known as the conditions of variational
selfadjointness. These studies permitted a rigorous separation of all acting forces
into those derivable from a potential, or variationally selfadjoint (SA) forces, and
those not derivable from a potential, or variationally nonselfadjoint (NSA) forces
according to the expression
F ak = Fak SA
NSA
(t, r, v) + Fak (t, r, v, a, ...). (1.2.4)
In particular, the reader should keep in mind that, while selfadjoint forces are of
Newtonian type, nonselfadjoint forces are generally non-Newtonian, in the sense