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XIII.—On the Application of Hamilton's Characteristic Function to Special Cases of Constraint

Published online by Cambridge University Press:  17 January 2013

Extract

One of the grandest steps which has ever been made in Dynamical Science is contained in two papers, “On a General Method in Dynamics” contributed to the Philosophical Transactions for 1834 and 1835 by Sir W. R. Hamilton. It is there shown that the complete solution of any kinetical problem, involving the action of a given conservative system of forces, and constraint depending upon the reaction of smooth guiding curves or surfaces, also given, is reducible to the determination of a single quantity called the Characteristic Function of the motion. This quantity is to be found from a partial differential equation of the first order, and second degree; and it has been shown that, from any complete integral of this equation, all the circumstances of the motion may be deduced by differentiation. So far as I can discover, this method has not been applied to inverse problems, of the nature of the Brachistochrone for instance, where the object aimed at is essentially the determination of the constraint requisite to produce a given result. It is easy to see, however, that a large class of such questions may be treated successfully by a process perfectly analogous to that of Hamilton; though the characteristic function in such cases is not the same function (of the quantities determining the motion) as that of the Method of Varying Action.

Type
Transactions
Copyright
Copyright © Royal Society of Edinburgh 1865

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References

page 149 note * Thomson, and Tait's, Natural Philosophy, § 323Google Scholar, or Tait, and Steele's, Dynamics of a Particle (2d edition), §§ 252, 253.Google Scholar

page 163 note * Proc. R.S.E. March 1865, or Tait, and Steele's, Dynamics of a Particle (2d edition) § 258.Google Scholar

page 164 note * Cambridge and Dublin Math. Journal, IX., p. 9.Google Scholar

page 166 note * Compare Thomson, and Tait's, Natural Philosophy, §§ 581, 582.Google Scholar