# 英国代考|MATH0009 Newtonian Mechanics牛顿力学

(i) First, let us write the transformation equation as
$$p_{i}=\frac{\partial F_{2}}{\partial q_{i}}=\frac{\partial S\left(q_{i}, \alpha_{i}, t\right)}{\partial q_{i}}$$
which are $n$ complete equations and at the time $t=0$ these give $n$ values relating $\alpha$ ‘s in terms of initial values $q$ and $p$, as
$$\alpha_{i}=\alpha_{i}\left(q_{i}, p_{i}, c\right) ; \quad P_{i}=P_{i}\left(q_{i}, p_{i}, t\right)$$
It does not indeed satisfy the definition of momenta in terms of Hamilton’s principal functions.

(ii) Further, the second transformation equation which provide the new constants co-ordinates appear as
$$Q_{i}=\frac{\partial F_{2}}{\partial P_{2}}=\frac{\partial S\left(q_{i}, \alpha_{i}, t\right)}{\partial P_{i}}=\frac{\partial S\left(q_{i}, \alpha_{i}, t\right)}{\partial \alpha_{i}}=\beta_{i}$$
which are $n$-equations and at time $t=0$ give the $n$ values of $\beta_{i}$ in terms of the known initial values of $q_{2}$ as $q_{i}=q_{i}\left(\beta_{i}, \alpha_{i}, t\right)$, and so $p_{i}=p_{i}\left(\beta_{i}, \alpha_{i}, t\right)$. This set of equations defines a set of new co-ordinates $Q_{i}$ in terms of $q_{i}, \alpha_{i}, t$.
(iii) The relation between the new Hamiltonian $K$ and old $H$ as
$$K(Q, P, t)=H+\frac{\partial F_{2}}{\partial t}=H+\frac{\partial S}{\partial t}=0$$

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