上海交通大学：《常微分方程》课程教学资源（讲座稿1）第十八讲 前沿探索——Hamilton系统初步

第十八讲、前沿探索：Hamilton系统初步 张祥 xzhang@sjtu.edu.cn 答疑时间：周三晚上6：30-8：20点 答疑地点：老图书馆数学楼301 张祥：上海交通大学数学系 第十八讲、前沿探索：Hamilton系统初步

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本讲教学目的与目标 ●前沿介绍：Hamilton系统的可积与应用。 温故： n维空间中首次积分、可积的概念和性质等。 日+4艺”4主12月双0 张样：上将交通大学数学系第十八讲，的沿探装：Hamilton系统初步

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Hamilton系统的可积 可积Hamilton系统具有极其丰富的内容，它 ·不仅联系到经典分析， ●还联系到代数几何，Riemann几何和代数拓扑，辛几何和辛 拓扑等等， 本讲介绍其中一些最浅显的基础知识。 4口6·4之·4生+280 张样：上海交通大学数学系 第十八讲、前沿探索：Hamilton系统初步

Hamilton X⁄å» å» Hamilton X⁄‰k4Ÿ¥LSN, ß ÿ=ÈX²;©¤, ÑÈXìÍA¤, Riemann A¤⁄ì͡¿, "A¤⁄" ˇ¿. ˘0Ÿ•ò Åfwƒ:£. ‹å: ˛°œåÆÍÆX 1õl˘!c˜&¢µHamiltonX⁄–⁄

定义： 设H(q,p)是2n维空间中的连续可微函数，其中 q=(q1,,qn),p=(p1,,Pn) 微分方程组 dqi(t)）∂H ∂H i=1, dt Opi 0= 称为 。n自由度的Hamilton系统， 。H称为Hamilton函数. Hamilton函数总是相应的Hamilton系统的首次积分. 张样：上海交通大学数学系 第十八讲，館沿探索：Hamilton系统初步

½¬µ  H(q,p) ¥ 2n ëòm•ÎYåáºÍ, Ÿ• q = (q1,...,qn), p = (p1,...,pn). á©êß| dqi(t) dt = ∂H ∂pi , dpi(t) dt = − ∂H ∂qi , i = 1,..., °è n gd› Hamilton X⁄, H °è Hamilton ºÍ. Hamilton ºÍo¥ÉA Hamilton X⁄ƒg»©. ‹å: ˛°œåÆÍÆX 1õl˘!c˜&¢µHamiltonX⁄–⁄

上述Hamilton系统用矩阵的形式可写成 (8)=a() (1) 其中0是n阶零矩阵，E是n阶单位矩阵，VH表示H的梯度，即 VH=(Ha....Han Hp...,Hpa). 以上定义的是R2m中标准的Hamilton系统. 口0·4之·4生+2刀a0四 张样：上海交通大学数学系 第十八讲、前沿探索：Hamilton系统初步

˛„ Hamilton X⁄^› /™å§ q˙ p˙ ! = J∇H, J = 0 E −E 0 ! , (1) Ÿ• 0 ¥ n "› , E ¥ n ¸†› , ∇H L´ H F›, = ∇H = (Hq1 ,...,Hqn ,Hp1 ,...,Hpn ) T . ±˛½¬¥ R 2n •IO Hamilton X⁄. ‹å: ˛°œåÆÍÆX 1õl˘!c˜&¢µHamiltonX⁄–⁄

一般的R2m中的Hamilton系统是通过 辛矩阵代替矩阵J或用Poisson括号 来定义Hamilton系统.这些知识超出本书的范围，不在此叙述. 有兴趣的读者可参见 o Abraham R and Marsden JE,Foundations of Mechanics, 1978 o Arnold VI.Mathematical Methods of Classical Mechanics, 1989 日+4艺”4主12月双0 张样：上海交通大学数学系 第十八讲，馆沿探索：Hamilton系统初步

òÑ R 2n • Hamilton X⁄¥œL "› ìO› J ½^ Poisson )“ 5½¬ Hamilton X⁄. ˘ £á—÷âå, ÿ3dQ„. k,÷ˆåÎÑ Abraham R and Marsden JE, Foundations of Mechanics, 1978 Arnold VI, Mathematical Methods of Classical Mechanics, 1989 ‹å: ˛°œåÆÍÆX 1õl˘!c˜&¢µHamiltonX⁄–⁄

o Hamiltonian systems are studied in Hamiltonian mechanics. o Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. o Hamiltonian mechanics arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange(1736-1813)in 1788: ●born in Turin,Italy .lived part of his life in Prussia(Germany)and part in France Lagrange is alternatively considered a French and an Italian scientist 张样：上海交通大学数学系 第十八讲、前沿探索：Hamilton系统初步

Hamiltonian systems are studied in Hamiltonian mechanics. Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Hamiltonian mechanics arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange (1736–1813) in 1788: born in Turin, Italy lived part of his life in Prussia (Germany) and part in France Lagrange is alternatively considered a French and an Italian scientist ‹å: ˛°œåÆÍÆX 1õl˘!c˜&¢µHamiltonX⁄–⁄

Sir William Rowan Hamilton(1805-1865)was o an Irish physicist,astronomer,and mathematician, o made important contributions to classical mechanics, geometric optics,and algebra. His greatest contribution is perhaps the reformulation of Newtonian mechanics,now called Hamiltonian mechanics. o This work has proven central to the modern study of classical field theories such as electromagnetism,and to the development of quantum mechanics. o In mathematics,he is perhaps best known as the inventor of quaternions. 口9+二+生42刀风0 张样：上海交通大学数学系 第十八讲、前沿探索：Hamilton系统初步

Sir William Rowan Hamilton(1805–1865) was an Irish physicist, astronomer, and mathematician, made important contributions to classical mechanics, geometric optics, and algebra. His greatest contribution is perhaps the reformulation of Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In mathematics, he is perhaps best known as the inventor of quaternions. ‹å: ˛°œåÆÍÆX 1õl˘!c˜&¢µHamiltonX⁄–⁄

Hamilton系统大量地出现在实际力学系统中，是当前动力系统的 主要研究对象之一. 例题： 1.自由落体运动方程 x(t)=-8 通过变换q=x,p=mr转化为Hamilton系统 aH aH 4=ap' 巾=一 aq 其中Hamilton函数H(q,p)=玩p2+mgq由动能和重力势能 构成 张样：上海交通大学数学系 第十八讲、前沿探索：Hamilton系统初步

Hamilton X⁄å˛/—y3¢SÂÆX⁄•, ¥cƒÂX⁄ ÃáÔƒÈñÉò. ~K: 1. gd·N$ƒêß x¨(t) = −g œLCÜ q = x, p = mx˙ =zè Hamilton X⁄ q˙ = ∂H ∂p , p˙ = − ∂H ∂q , Ÿ• Hamilton ºÍ H(q,p) = 1 2m p 2 +mgq dƒU⁄­Â³U §. ‹å: ˛°œåÆÍÆX 1õl˘!c˜&¢µHamiltonX⁄–⁄

2.单摆运动方程 (t)=-a2sinx,a2=g/1 通过变换q=x,p=mr转化为Hamilton系统 aH ∂H 9= p=-0q 其中Hamilton函数H(q,p=六p2-a2 mcosq.. 日+4艺”4主12月双0 张样：上将交通大学数学系第十八讲，的沿探装：Hamilton系统初步

2. ¸{$ƒêß x¨(t) = −a 2 sinx, a 2 = g/l œLCÜ q = x, p = mx˙ =zè Hamilton X⁄ q˙ = ∂H ∂p , p˙ = − ∂H ∂q , Ÿ• Hamilton ºÍ H(q,p) = 1 2m p 2 −a 2mcosq. ‹å: ˛°œåÆÍÆX 1õl˘!c˜&¢µHamiltonX⁄–⁄