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Hirzel, Stuttgart, 95-105. 11. Gelfand, I. , and Fomin, S. V. (1963). Calculus oj Variations. , Englewood Cliffs, NJ. 12. Pars, L. A. (1962). An Introduction to the Calculus oj Variations, Heinemann, London. 13. Bliss, G. A. (1944). Calculus of Variations, Carus Mathematical Monograph, 4th ed. Math. Assoc. , La Salle, IL. 1. Introduction Systems of differential equations that are derivable from Lagrangians are called Lagrangian systems; they have some very desirable properties. The Lagrangian formulation possesses an inherent symmetry that allows for an immediate reduction to canonical form, where the canonical variables often contain the essence of the solution.

33). 2), we will study the same conservation laws from a different point of view, namely, we will use the invariant properties of Hamilton's variational principle. As an example, consider the motion of a thin, rigid, rectangular plate OABC that is connected to a spherical joint at 0 and moves under the force of gravity in such a way that the side OA = a slides without friction on the Oxy plane. The mass of the plate is M. Initially, at t = 0, the plate is in the vertical position in the Oxz plane and at rest.

1. Motion of a quadratic plate in a gravity field. 7. A Brief Analysis of the Lagrangian Equations 25 coordinate system Oxyz, let us introduce the Cartesian coordinate system Of,,,t; rigidly connected to the plate. The position of an arbitrary point K with mass dm can be expressed in terms of the generalized coordinates in the following way: x = f, sin e sin '1/ + f/ cos '1/, Y = - f, sin Z = f, cos ecos '1/ + n sin '1/, (1. 34) e. Thus, the velocity components of K are f,0 cos e sin '1/ + f,lit sin e cos '1/ - f/lit sin '1/, f,0 cos e cos '1/ + f,lit sin e sin '1/ + f/lit cos '1/, = -f,0 sin e.

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