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362 Chapter 8 The Hamilton Equations of Motion path of least curvature. By Jacobi’s principle such a path must be a geodesic, and the geometrical property of minimum curvature is one of the well-known characteristics of a geodesic. Tt has been pointed out that variational principles in themselves contain no new physical content, and they rarely simplify the practical solution of a given mechanical problem. Their value lies chiefly as starting points for new formulations of the theoretical structure of classical mechanics. For this purpose, Hamilton’s principle is especially fruitful, and to a lesser extent, so also is the principle of least action. DERIVATIONS 1. (a) Reverse the Legendre transformation to derive the properties of L(g;, 4,,#) from H(q,. Py. 8), treating the g, as independent quantities. and show that 1t leads to the Lagrangian equations of motion. (b) By the same procedure find the equations of motion in terms of the function L'(p, Bt) = ~ Pig, — HG. pot). 2. It has been previously noted thar the total tume derivauve of a function of g, and f can be added to the Lagrangian without changing the equations of motion. What does such an addition do to the canonical momenta and the Hamiltonian? Show that the equations of motion im terms of the new Hamiltonian reduce to the original Hamilton's equations of motion 3. A Hamultonian-like formulation can be set up in which g; and f, are the independent variables with a “Hamiltoman” G(q,, p,, 1). [Here p, is defined in terms of g,, 4, in the usual manner ] Starting from the Lagrangian formulation, show in detail how to construct G(,. p,. 1), and derive the corresponding “Hamulton's equation of motion.” 4, Show that if A, are the eigenvalues of a square matrix, then 1f the reciprocal matnx exists it has the eigenvalues anh. 5, Verify that the matrix J has the properties given in Eqs. (8.38c) and (8.38e) and that its determinant has the value +1. 6 Show that Hamilton’s principle can be written as 3 [ “DHOnt) + falar =0. 7. Venfy that both Hamiltonians, Eq. (8.45) and Eq. (8.47). lead to the same motion as described by Eq. (8.44). 8. Show that the moditied Hamilton’s principle. in the form of Eq. (8.71). leads to Hamil- ton’s equatrons of motion. 9, If the canonical variables are not all independent, but are connected by auxilary con- ditions of the form Vials Pr =0, 
