The Variational Principles of Mechanics

Description

Assis 2014 p. 209 mentions this book, regarding Euler forces (ch. IV, §§4-5).

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)

MR0166957

This is an excellent book which covers a great deal of information on variational methods. The brilliant accomplishments as explained in the text would definitely be welcomed by students and teachers of analytical mechanics.
This book is a corrected edition of the previous (1949) edition [MR0034139] with the excellent addition of a chapter on relativity. The author traces the historical development, beginning from the basic ideas of mechanics to the more powerful and advanced methods of D'Alembert, Lagrange, Hamilton, canonical transformations and the Hamilton-Jacobi equation. A number of clear specific examples are presented and terse, clear summaries are given in each section. The fundamental results explained in each section enable the readers to follow the trend and general line of thought in a most clear and lucid fashion. A good illustration is Chapter V (50 pages) entitled, "The Lagrangian Equations of Motion''. Using the principles in the previous chapter entitled "D'Alembert Principle'', the reader is introduced to the "Hamilton's Principle''. This method is the most direct and most natural transformation of D'Alembert into a minimum principle or, more generally, the stationary value of a definite integral. The step-by-step procedure leads from the Hamilton principle to the Lagrange equation, Jacobi's principle and Lagrangian multiplier. The holonomic and scleronomic systems, i.e., the Lagrangian $L$ which does not contain the time explicitly, are explained very clearly. The final portion of this chapter concludes with the application of the principles to the theory of small vibrations. The last chapter (IX) is an excellent introduction to relativistic mechanics. Based upon the previous Hamiltonian formulation, additional information on Einstein's observations plus Minkowski's classic paper on the four-dimensional world and Lorentz transformations, the basic elementary concept of "General Relativity'' is explained in a very simple fashion.
The reviewer regrets that the author has not expanded his work to encompass the variational principles of solid mechanics. The application of Rayleigh-Ritz, Galerkin and Reissner's method would be a most welcome addition to this book. With the advent of digital computers, these variational applications to solid mechanics play a most important part. Reviewed by H. Saunders