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XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

Published online by Cambridge University Press:  15 September 2014

A. Gray
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Extract

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 32 , 1913 , pp. 164 - 174
Copyright
Copyright © Royal Society of Edinburgh 1913

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References

page no 164 note * This discussion was suggested by a passage regarding a particular case in Darboux, Théorie des Surfaces, t. ii., § 563.

page no 168 note * The duality represented by this equation and (14), § 1, is one of the first results of the contact transformation theory of dynamics. For the addition of to gives the perfect differential Hence we get and so

page no 170 note * Note added April 18.—Reciprocal dynamical theorems are easily derived by means of the two functions. Thus, let the constants in S, S′ be the as (the initial co-ordinates). Then, for the initial momentum bi, we have and therefore also If the constants in both functions be the bs (the initial momenta), two other theorems are obtained in a similar way, namely,

I find, however, that this mode of deriving (two of) these relations has been anticipated by v. Helmholtz (Crelle, 1886), who also gave physical interpretations. Applications are given by Lamb (Lond. Math. Soc., 1888).