# Duality system in applied mechanics and optimal control by H.L. Dryden and Th. von Kármán (Eds.)

By H.L. Dryden and Th. von Kármán (Eds.)

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A point transformation cannot diagonalize it, so that a linear canonical transformation is looking for. 33) or combined written as where v is the state vector and H is termed a Hamilton matrix, whose characteristic is The Hamilton matrix is asymmetric but instead, the matrix (JH) is symmetric. The Hamilton matrix has the feature of symplectic behavior, which is conceivable. In previous sections, the canonical transformation is used to transform the dual canonical variables q,p , or to transform the state vector v .

2, §6. e. the dual variables of q and p . The corresponding variational principle is derived as follows. Introducing the variable vector s to denote the generalized velocity the Lagrange function becomes L(q,s,t), and becomes a prerequisite condition of the vairational principle. 2) becomes where the three kinds of variables q,p and s are considered varying independently. 2). 3) where the variable s is regarded a function of q,p,t. The function H is termed as the Hamilton function. 7a,b) are called the Hamiltonian canonical equations.

3) where the variable s is regarded a function of q,p,t. The function H is termed as the Hamilton function. 7a,b) are called the Hamiltonian canonical equations. Then the 2n first order ODE substitutes the Lagrange equation. 7b) is the dynamic equation. The formulation of Hamilton function H(q,p,t) explicitly expresses that it is a function of the dual variables q, p and time t . 7), a variational principle with two kinds of dual variables q , p . Hamilton function is very important, that a number of basic equations and fundamental theorems are derived from it.

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