The Variational Principles Of Mechanics by Cornelius Lanczos

Author:Cornelius Lanczos [Cornelius Lanczos]
Language: eng
Format: epub
Publisher: Dover Publications

The function L1 is a homogeneous form of the first order in the n + 1 variables 1, ... ,n+1. Hence, by Euler’s theorem on homogeneous functions,

which shows that

Hence,the canonical integral is reduced to the completely symmetrical form

without any Hamiltonian function. Yet obviously something has to take the place of the function H1 since in the absence of a Hamiltonian function all the qi and pi would become constants, which is obviously impossible.

Actually there is something which takes the place of the Hamiltonian function, namely, an auxiliary condition which must hold between the qi and the Pi. The equations

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