On the adiabatic invariants

Abstract

Of the canonical systems

$$ \frac{{d{p_i}}}{{dt}} =- \frac{{\partial H}}{{\partial {q_i}}},\;\frac{{d{q_i}}}{{dt}} = \frac{{\partial H}}{{\partial {p_i}}}\;\left( {i = 1,2,...,n} \right) $$

with characteristic function H, independent of t, which contains slowly varying parameters a, two particularly conspicuous types of adiabatic invariants are known:

• 1° (Gibbs-Hertz’s theorems) The volume V enclosed in phase space by a generic isoenergetic manifold

  $$ H = E\;\left( {E\;constant} \right) $$

  which applies to quasi-ergodic systems; systems that do not allow other uniform integrals apart from H = E (see, e.g., nos. 3–5 of this paper).

• 2° (Burgers’s Theorem) Sommerfeld’s n loop integrals

  $$ {J_i} = \oint {{p_i}\;d{q_i}} \;\left( {i = 1,2,...,n} \right) $$

  which are adiabatic invariants for (Stäckel’s) systems that are integrable by means of separation of the variables and that admit in total n integrals, (quadratic in the p’s).

Translated from the Italian “Sugli invarianti adiabatici”, in “Resoconto del Congresso Nazionale dei Fisici”, Como, 11–20 September 1927, on the occasion of the commemoration of the first centenary of Alessandro Volta’s death, vol. II, pp. 475–513 (Zanichelli Editore, Bologna).