**M. Crâ \c sm\u areanu**

## Adjoint variables:
a common way to first
integrals and inverse problems

**abstract:**

We obtain a type of first integrals for ODE by using a generalization of the
method of adjoint variables to higher-order systems. A generalization of the
harmonic oscillator and a classical spinning particle is completely discussed
in order to solve the associated inverse problem.

**Mathematics Subject Classification:**
34A34, 34A30, 70D99.

**Key words and phrases:**
First integral, adjoint system, system in
kinematical form, inverse problem, harmonic oscilator, spinning particle.