经典力学 中一组描写系统运动的一阶微分方程组。 是W.R.哈密顿于1834年提出的，又称 哈密顿方程 或正则方程。 其中带下标的变量视为所有N个该类型的变量。 哈密顿力学的目标是用广义动量（也称为共轭动量）变量取代广义速度。 这样一来，就可能处理特定的系统，例如量子力学的某些方面，否则其表述会更复杂。 在 直角坐标系 中，广义动量就是物理上的线性 动量。 在 极坐标 中，对应角速度的广义动量就是物理上的 角动量。 对于广义坐标的任意选取，可能不能找到共 …

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations. This is sometimes known as form invariance .

Generally, not all transformations preserve the equations of motion. However, the trans-formation is canonical if there exists a new Hamiltonian: For notational purposes let repeated indices be summed over implicitly. Hamilton's principle can be written as: There are a few things to be said about transformations and .

Canonical transformations, de ned here as those that preserve the Poisson brackets or equivalently the symplectic 2-form, also preserve Hamilton's equations. A search for conserved quantities and symmetries is equivalent to a search for a nice coordinate system that preserves Hamilton's equations.

拉格朗日方程是二阶常微分方程组,如果我们把 L 中的广义速度 \dot {q}_ {\alpha} 换为广义动量 p_ {\alpha} 就能使方程组降阶,但得用勒让德变换. 有 n 个变量的函数 f\left ( x_1,x_2,···,x_n \right) ,求其微分. \mathrm {d}f=\frac {\partial f} {\partial x_1}\mathrm {d}x_1+\frac {\partial f} {\partial x_2}\mathrm {d}x_2+···+\frac {\partial f} {\partial x_n}\mathrm {d}x_n\tag {1.1} 把它简单地记为 (函数变量现在还是一群 x )

哈密顿方程（Hamilton's equations）也叫哈密顿正则方程（Hamilton's canonical equations）是一组关于广义坐标和广义动量的微分方程组，共有 $2N$ 条方程：

这里有哈密顿方程的循环积分与从 最小作用量原理 推导正则方程 飯勺：[Analytical mechanics]Hamilton's canonical equations 1.正则变换是啥么. 现在来看满足什么形式的变化才称得上正则变换.

gives us Hamilton’s equations Canonical Eqns dH = X p_ idq i+ X q_ idp i) q_ i= @H @p i; p_ i= @H @q i also known as the “canonical equations”. These are 2Nfirst order differential equations for the 2Nunknown function p i(t) and q i(t), i= 1;:::;N. By contrast, in the Lagrangian formulation we had N2nd order differential equations.

To motivate our next theoretical step, canonical transformations, let’s remind ourselves how we use Lagrangians and Hamiltonians to solve mechanics problems. I’ll use the simple pendulum as a concrete example. From here, you can continue on the Lagrangian path and...

2021年2月10日 · Hamilton derived the canonical equations of motion from his fundamental variational principle, chapter \(9.2\), and made them the basis for a far-reaching theory of dynamics. Hamilton’s equations give \(2s\) first-order differential equations for \(p_{k},q_{k}\) for each of the \(s=n-m\) degrees of freedom.