By William R. Hamilton

William R. Hamilton published two celebrated papers on Dynamics in the Philosophical Transactions of the Royal Society of London. They represent an adaption to dynamics of methods that Hamilton had developed for the study of optical systems. These are

- On a General Method in Dynamics; by which the Study of the Motions of all free Systems of attracting or repelling Points is reduced to the Search and Differentiation of one central Relation, or characteristic Function,
- Second Essay on a General Method in Dynamics.

He described his method at meeting of the British Association for the Advancement of Science in a short report entitled

Hamilton's first paper on dynamics is entitled `On a General Method in Dynamics; by which the Study of the Motions of all free Systems of attracting or repelling Points is reduced to the Search and Differentiation of one central Relation, or characteristic Function'. This was published in the Philosophical Transactions of the Royal Society (part II for 1834, pp. 247-308).

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Hamilton associates to a dynamical system of attracting and repelling points a characteristic function. The value of this characteristic function is the action determined by the evolution of the dynamical system from an initial to a final configuration. (This action is defined to be twice the integral of the kinetic energy of the particles with respect to time as the system evolves from its initial to its final state.) The characteristic function is considered as a function of coordinates determining the initial and final configurations and of the energy associated with the motion. The characteristic function satisfies an equation corresponding to the Hamilton-Jacobi equation satisfied by a closely related function, the principal function introduced briefly by Hamilton at the end of his first paper on his general method in dynamics.

Hamilton refined his approach in his second paper on dynamics, entitled `Second Essay on a General Method in Dynamics'. This was published in the Philosophical Transactions of the Royal Society (part I for 1835, pp. 95-144).

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The Second Essay describes systems of attracting and repelling points by means of Hamilton's Equations of Motion, and shows how the evolution of dynamical systems is characterised by a single function, Hamilton's Principal Function which satisfies an equation now referred to as the Hamilton-Jacobi equation.

Hamilton outlined his general method in dynamics at a meeting of the British Association for the Advancement of Science, held in Edinburgh in 1834. His report in the proceedings of the meeting is entitled `On the Application to Dynamics of a General Mathematical Method previously applied to Optics'.

This report is available in the following formats:

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D.R. Wilkins(

School of Mathematics

Trinity College, Dublin