On a new first integral of certain dynamical systems and its applications to recent results concerning first integrals of newtonian systems

1980 ◽  
Vol 29 (9) ◽  
pp. 310-314 ◽  
Author(s):  
F. González-Gascón ◽  
E. Rodriguez-Camino
Keyword(s):  
Dynamical Systems ◽  
First Integral ◽  
First Integrals ◽  
Newtonian Systems

scholarly journals Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

2020 ◽  
Vol 16 (4) ◽  
pp. 637-650
Author(s):  
P. Guha ◽  
◽  
S. Garai ◽  
A.G. Choudhury ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Lax Pair ◽  
First Integrals ◽  
Second Order ◽  
Lax Representation ◽  
Lax Pairs ◽  
Second Order Differential Equations ◽  
Autonomous Version ◽  
Autonomous Differential Equations

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

q-Equivalent Hamiltonian Operators

1972 ◽  
Vol 50 (17) ◽  
pp. 2037-2047 ◽  
Author(s):  
M. Razavy
Keyword(s):  
Quantum Mechanics ◽  
Classical Limit ◽  
First Integral ◽  
Canonical Commutation Relation ◽  
Equation Of Motion ◽  
First Integrals ◽  
Position Operator ◽  
Integrals Of Motion ◽  
Adjoint Operators ◽  
Hamiltonian Operators

From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.

Symmetries, first integrals and splittability of dynamical systems

1978 ◽  
Vol 21 (7) ◽  
pp. 253-257 ◽  
Author(s):  
F. González-Gascón ◽  
F. Moreno-Insertis
Keyword(s):  
Dynamical Systems ◽  
First Integrals
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