Enrico Fermi pp 85-114 | Cite as

On the adiabatic invariants

  • Tullio Levi-Civita


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. 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. 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).


Phase Space Canonical System Adiabatic Invariant Uniform Integral Total Differential 
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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Tullio Levi-Civita
    • 1
  1. 1.RomeItaly

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