ハミルトニアン力学 (Hamiltonian mechanics)

一般運動量 (generalized momenta)

pi=∂L∂q˙i p_i = \frac{ \partial L }{ \partial \dot{q}_i }

ハミルトニアン (Hamiltonian)

H=∑ipiq˙i−LdH=∑i[(Fi−p˙i)dqi+q˙idpi]−∂L∂tdt=∑i[∂H∂qidqi+∂H∂pidpi]+∂H∂tdt\begin{aligned} H & = \sum_i p_i \dot{q}_i - L \\ dH & = \sum_i \left[ \left( F_i - {\dot p}_i \right) d q_i + {\dot q_i} d p_i \right] - \frac{ \partial L }{\partial t} d t \\ & = \sum_i \left[ \frac{ \partial H }{ \partial q_i } d q_i + \frac{ \partial H }{ \partial p_i } d p_i \right] + \frac{ \partial H }{ \partial t } d t \end{aligned}

ハミルトンの正準方程式 (Hamilton's canonical equations)

∂H∂qi=Fi−p˙i∂H∂pi=q˙i∂H∂t=−∂L∂t\begin{aligned} \frac{ \partial H }{ \partial q_i } & = F_i - \dot{p}_i \\ \frac{ \partial H }{ \partial p_i } & = \dot{q}_i \\ \frac{ \partial H }{ \partial t } & = - \frac{ \partial L }{ \partial t } \end{aligned}

ポアソンの括弧式 (Poisson bracket)

{X,H}=∑i(∂X∂qi∂H∂pi+∂X∂pi∂H∂qi)dXdt=∑i(∂X∂qi∂H∂pi+∂X∂pi∂H∂qi)+∂X∂t={X,H}+∂X∂t\begin{aligned} \left\{ X , H \right\} & = \sum_i \left( \frac{\partial X}{\partial q_i}\frac{\partial H}{\partial p_i} + \frac{\partial X}{\partial p_i}\frac{\partial H}{\partial q_i} \right) \\ \frac{dX}{dt} & = \sum_i \left( \frac{\partial X}{\partial q_i}\frac{\partial H}{\partial p_i} + \frac{\partial X}{\partial p_i}\frac{\partial H}{\partial q_i} \right) + \frac{\partial X}{\partial t} \\ & = \left\{ X , H \right\} + \frac{\partial X}{\partial t} \\ \end{aligned}
qi˙={qi,H}pi˙={pi,H}\begin{aligned} \dot{q_i} & = \left\{ q_i , H \right\} \\ \dot{p_i} & = \left\{ p_i , H \right\} \\ \end{aligned}