9.1: Introduction to Hamilton's Action Principle
Hamilton’s principle of stationary 稳定的 action
provide the foundation for building Lagrangian mechanicsHamilton’s principle of least action哈密顿最小作用量原理
underlies theoretical physics and many other disciplines in mathematics and ecomics
In 1834, Hamilton was seeking a theory of optics.
$$ S=\int_{t_i}^{t_f}L(\mathbf{q},\mathbf{\dot{q}},t)dt $$
action functional 泛函 allows for more general Lagrangians than the standard Lagrangian $L(\mathbf{q},\mathbf{\dot{q}},t)=T(\mathbf{\dot{q}},t)-U(\mathbf{q},t)$
Hamilton stated that actual trajectory 轨迹 of a mechanical system is that given by requiring that the action fucntional is stationary with respect to change of the variables. The action functional is stationary (when the variational principle 变分原理 can written in terms of a virtual infinitessimal displacement, $\delta$, )to be
$$ \delta S=\delta\int_{t_i}^{tf}L(\mathbf{q},\mathbf{\dot{q}},t)dt=0 $$
Typically the stationary point 驻点 corresponds to a minimun of the action fucntional.
Consider the acton $S_A$ for the extremum极值 path of a system in configuration space, that is, along path $A$ for $j=1,2,...,n$ coordinates $q_j(t_i)$ at initial time $t_i$ to $q_j(t_f)$ at a final time $t_f$. Then the action $S_A=\int_{t_i}^{t_f}L(\mathbf{q}(t),\mathbf{\dot{q}}(t),t)dt$
A family of neighboring path is defined by $q_j(t,\epsilon)=q_j(t,0)+\epsilon\eta_j(t)$, adding an infinitessimal fraction 无穷小分数$\epsilon$ of a continuous, well-behaved neighboring function $\eta_j$. The variational 变分path used here does not assume that the functions$\eta_j(t)$ vanish at the end points. Assume that the neighboring path $B$ has an action $S_B=\int_{t_i+\Delta t}^{t_f+\Delta t}L(\mathbf{q}(t)+\delta\mathbf{q}(t),\mathbf{\dot{q}}(t)+\delta\mathbf{\dot{q}}(t))dt$ {non-standard Lagrangian→Lagrangian }