Jacobi's Lectures on Dynamics

Foreword.; 1 Introduction; 2 The Differential Equations of Motion; 3 Conservation of Motion of Centre of Gravity; 4 The Principle of Conservation of 'vis viva'; 5 Conservation of Surface Area.; 6 The Principle of Least Action; 7 Further considerations on the principle of least action - The Lagrange multipliers; 8 Hamilton's Integral and Lagrange's Second Form of Dynamical Equations; 9 Hamilton's Form of the Equations of Motion; 10 The Principle of the Last Multiplier.; 11 Survey of those properties of determinants that are used in the theory of the last multiplier; 12 The multiplier for systems of differential equations with an arbitrary number of variables.; 13 Functional Determinants. Their application in setting up the Partial Differential Equation for the Multiplier; 14 The Second Form of the Equation Defining the Multiplier. The Multipliers of Step Wise Reduced Differential Equations. The Multiplier by the Use of Particular Integrals.; 15 The Multiplier for Systems of Differential Equations with Higher Differential Coefficients. Applications to a System of Mass Points Without Constraints; 16 Examples of the Search for Multipliers. Attraction of a Point by a Fixed Centre in a Resisting Medium and in Empty. Space.; 17 The Multiplier of the Equations of Motion of a System Under Constraint in the first Langrange Form; 18 The Multiplier for the Equations of Motion of a Constrained System in Hamiltonian Form; 19 Hamilton's Partial Differential Equation and its Extension to the Isoperimetric Problem.; 20 Proof that the integral equations derived from a complete solution of Hamilton's partial differential equation actually satisfy the system of ordinary differential equations. Hamilton's equation for free motion; 21 Investigation of the case in which t does not occur explicitly.; 22 Lagrange's method of integration of first order partial differential equations in two independent variables. Application to problems of mechanics which depend only on two defining parameters. The free motion of a point on a plane and the shortest line on a surface; 23 The reduction of the partial differential equation for those problems in which the principle of conservation of centre of gravity holds; 24 Motion of a planet around the sun - Solution in polar coordinates; 25 Solution of the same problem by introducing the distances of the planet from two fixed points; 26 Elliptic Coordinates; 27 Geometric significance of elliptic coordinates on the plane and in space. Quadrature of the surface of an ellipsoid. Rectification of its lines of curvature.; 28 The shortest line on the tri-axial ellipsoid. The problem of map projection; 29 Attraction of a point by two fixed centres; 30 Abel's Theorem; 31 General investigations of the partial differential equations of the first order. Different forms of the integrability conditions; 32 Direct proof of the most general form of the integrability condition. Introduction of the function H, which set equal to an arbitrary constant determines the p as functions of the q; 33 On the simultaneous solutions of two linear partial differential equations; 34 Application of the preceding investigation to the integration of partial differential equations of the first order, and in particular, to the case of mechanics. The theorem on the third integral derived from two given integrals of differential equations of dynamics. 35 The two classes of integrals which one obtains according to Hamilton's method for problems of mechanics. Determination of the value of ([phi], [psi]) for them; 36 Perturbation theory. Supplement.