# Some Generalized Variational Principles for Conservative Holonomic Dynamical Systems

Jing-Tang Xing, W. G. Price

## Abstract

For four types of time boundary conditions, some generalized variational principles for conservative holonomic dynamical systems are developed. The traditional Hamilton's principle and its second form as well as Toupin's principle are special cases of the general principles given in this paper. The generalized principles provide several analytical approaches to study dynamical systems formulated in the following spaces: (q$_{i}$, t), (p$_{i}$, t), (q$_{i}$, p$_{i}$, t), (q$_{i}$, v$_{i}$, p$_{i}$, t), (q$_{i}$, Q$_{i}$, p$_{i}$, t) and (q$_{i}$, v$_{i}$, Q$_{i}$, p$_{i}$, t) where q$_{i}$ represents the generalized coordinate, p$_{i}$ the generalized momentum, Q$_{i}$ the generalized force and v$_{i}$ the generalized velocity. In the paper the second form of Hamilton's principle as stated by F. R. Gantmacher is discussed. From the accompanying analysis, new principles are developed and the principal paths as well as the alternative side paths corresponding to these new forms are illustrated and compared with the one originally presented by F. R. Gantmacher.