A Treatise on the Analytic Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies

Description

ch. 10 (DjVu pp. 281ff.) is on integral invariants (Integralinvarianten)

Santilli mentions this along with Goldstein and another. It's the first book of its kind.

from Santilli 1978 (pp. 1-2):

1. analytic formulations, e.g., Lagrange's and Hamilton's equations, Hamilton-Jacobi theory, etc.
   Whittaker (1904) , Goldstein (1950), Pars (1965)
2. variational formulations, e.g., variational problems, variational principles, etc.
   Lanczos (1949), Rund (1966)
3. algebraic formulations, e.g., infinitesimal and finite canonical transformations, Lie algebras and Lie groups, symmetries and conservation laws, etc.
   Saletan and Cromer (1971), Sudarshan and Mukunda (1974).
4. geometric formulations, e.g., symplectic geometry, canonical structure, etc.
   Jost (1964), Abraham and Marsden (1967), Guillemin and Sternberg (1977)
5. statistical formulations, e.g., Liouville's theorem, equilibrium and nonequilibrium statistical mechanics, etc.
   Gibbs (1948), Katz (1967)
6. thermodynamic formulations, e.g., irreversible processes, entropy, etc.
   Sommerfeld (1956), Tisza (1966)
7. many-body formulations, e.g., stability of orbits, quadrature problems, etc.
   Wintner (1941), Khilmi (1961), Hagihara (1970)