Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics

Description

The conditions under which a function admits a Lagrangian or Hamiltonian are known as "conditions of variational self-adjointness."

The author states that this monograph grew out of his uneasiness in teaching classical Newtonian mechanics. So many techniques (analytical, variational, algebraic) are applicable only to systems with forces derivable from a potential. The usual Lagrangian representation of a physical system is at best a crude approximation of reality. The physical reality is full of nonconservative systems with nonholonomic constraints.
The author attempts to reformulate various problems of physics from the so-called inverse calculus of variations point of view. Given a complete set of solutions of ordinary differential equations of order r:Fk(x,y,y′,y″,⋯,y(r))=0r\colon F_k(x,y,y',y'',\cdots,y^{(r)})=0, determine whether a functional A(y)=∫x1x2L(x,y,y′⋯yr−1)dxA(y)=\int_{x_1}^{x_2}L(x,y,y'\cdots y^{r-1})\,dx exists which admits these solutions as extremals. The author offers a careful discussion of the selfadjointness conditions, their algebraic and variational significance. In many respects this monograph is unique. The relations between the properties of the Lagrangians and the solutions of Jacobi's equations have not been investigated before to the best of the reviewer's knowledge. The Morse-Feshbach Lagrangian is introduced and some rather puzzling symmetry breaking phenomena under the gauge transformation φ→φeiωe\varphi\rightarrow\varphi e^{i\omega e} are pointed out in a footnote referring to the author's research.
The central theme concerns the theorem on necessary and sufficient conditions for the existence of a Lagrangian, and of a Hamiltonian. The author makes no claim to originality in the introduction. In this respect he is much too modest. The monograph surveys many known facts, but does present them from a new point of view, and a large section of the material presented here originated with the author's research.

Reviewed by Vadim Komkov