A RELAXATION OF THE HOMOGENEITIES IMPOSED ON THE RELATING FUNCTIONS IN THE EXTREMAL PROBLEM FOR THE DERIVATION OF CONSERVED QUANTITIES

In contrast with Noether theorem, we built up a new operative procedure for the derivation of conserved quantities and then applied it to the extremal problem for the integration under constraints in the space of state and control variables. In the problem, conserved quantities were constructed by imposing the homogeneities with respect to the state variables through control variables on relating functions. The purpose of this paper is to release the control variables from the homogeneities and then construct such a conserved quantity in a similar procedure. The quantity is illustratively constructed in a economic model, with the aid of which optimal paths can be determined completely. Introduction. Noether theorem (Noether (10)) concerning with symmetries of the action integral or its generalization (Bessel-Hagan (2)) with those up to divergence plays an effective role for discovering conserved quantity from the Lagrangian or the Hamiltonian structures of considering problem. In contrast with Noether theorem, a new operative procedure for the derivation of conserved quantity was established without using either Lagrangian or Hamiltonian structures (Mimura and Nono (6)). It was discussed for a second- order differential system which was supposed later to be of the Euler-Lagrange system, and also for higher order system (Mimura, Fujiwara and Nono (8)). And the results were applied to various economic growth models (Mimura, Fujiwara and Nono (7), (9); Fujiwara, Mimura and Nono (3), (4), (5)) to discover new economic conserved quantities including non-Noether ones. In the applications, the Euler-Lagrange system was given in the extremal (maxmizing or minimizing) problem for the integration over a finite (0

Keywords:

• Hamiltonian (quantum mechanics)
• Divergence
• State variable
• Economic model
• Derivation
• Noether's theorem
• Mathematical analysis
• Homogeneous space
• Conserved quantity
• Mathematics
• Lagrangian
• Pure mathematics
• Control variable

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