正则变换 生成函数

传统相空间中 \(q, p = q_1, q_2, \dots, q_D, p_1, \dots, p_D\). 变换方程:

构成正则变换的充要条件为: \[[Q_k, Q_l]_{qp} = 0,\;[Q_k, P_l]_{qp} = \delta_{kl},\;[P_k, P_l]_{qp} = 0.\] 由于

故得: \(\left( ab^T \right)_{kl} = \delta_{kl}\), 即 \(ab^T = I_D\) 为单位阵时, 变换为正则变换.

• 哈密顿方程: \[ [\dot{\gamma}] = s \left[ \frac{\partial H}{\partial \gamma} \right]. \]
• 泊松括号: \[[f, g]_{qp} = \left[ \frac{\partial f(\gamma)}{\partial \gamma} \right]^T s \left[ \frac{\partial g(\gamma)}{\partial\gamma} \right].\]

由前述, 正则变换满足: \[[Q_k, Q_l]_{qp} = 0,\;[Q_k, P_l]_{qp} = \delta_{kl},\;[P_k, P_l]_{qp} = 0.\] 即 \[ [\Gamma_k, \Gamma_l]_{qp} = s_{kl}. \] 辛形式下的泊松括号有: \[ [\Gamma_k, \Gamma_l]_{qp} = \left[ \frac{\partial \Gamma_k}{\partial \gamma} \right]^T s \left[ \frac{\partial \gamma_l}{\partial \gamma} \right] = \left[ \frac{\partial \Gamma_k}{\partial \gamma_i} \right]^T s_j^i \left[ \frac{\partial \gamma_l}{\partial \gamma^j} \right] = \left( JsJ^T \right)^T_{kl}\] 故得: \[JsJ^T = s.\]

由前述, 有: \(\left| JsJ^T \right| = |s|\), 可得 \({|J|}^2 = 1 \ne 0\). 故\(J\)可逆, 易得逆矩阵 \(J^{-1}\) 有: \[J^{-1} = \begin{bmatrix} \partial_Q q & \partial_P q \\ \partial_Q p & \partial_P p \end{bmatrix}\]

存在 \(J^{-1}\) 使得 \(J^{-1} = -sJ^Ts\), 则变换为正则变换. 此时有: \[\left(\frac{\partial q}{\partial Q}\right)=\left(\frac{\partial P}{\partial p}\right)^T, \;\left(\frac{\partial q}{\partial P}\right)=-\left(\frac{\partial Q}{\partial p}\right)^T, \;\left(\frac{\partial p}{\partial Q}\right)=-\left(\frac{\partial P}{\partial q}\right)^T, \;\left(\frac{\partial p}{\partial P}\right)=\left(\frac{\partial Q}{\partial q}\right)^T,\]

从泊松括号出发: \(JsJ^T = s\), 且 \(ss = -I_n\).

1. 左乘 \(sJ^{-1}\): \((sJ^{-1})JsJ^T = (sJ^{-1})s\Rightarrow -J^T = sJ^{-1}s\)
2. 右乘 \(sJ\): \(J^Tsj = -sJ^{-1}ssJ = s\).

Vice versa. 充要条件.

由此得到正则变换三大判据:

1. 泊松括号判据: \(JsJ^T = s\),
2. 直接判据: \(J^{-1} = -sJ^Ts\),
3. 拉格朗日括号判据: \(J^TsJ = s\).

群:定义了乘法且满足以下性质的集合.

1. 封闭性:两个元素之积仍在集合内.
2. 存在单位元:对应于作用前后不做任何改变.
3. 存在逆元素:一个元素与其逆元素之积为单位元素.
4. 乘法满足结合律(但不一定满足交换律).

正则变换构成一个群

1. 封闭性: 若 \(A, B\) 为正则变换, 则 \(C = AB\) (先做 \(B\) 变换, 再做\(A\)变换) 也是正则变换.
2. 存在单位元 \(Q_k = q_k,\;P_k = p_k,\;J = I_n\), 显然 \(I_n sI_n = s\).
3. 存在逆元素:正则变换可逆,逆变换也为正则变换.
4. 乘法结合律:由矩阵乘法易证.

泊松括号具有形式不变性 \([f, g]_{QP} = [f, q]_{qp}\), 等价于相空间变换为正则变换 \(\left[ \frac{\partial f}{\partial\gamma} \right] = J^T \left[ \frac{\partial f}{\partial\Gamma} \right]\) . 以下给出证明:

• 形式不变性⇒ 正则变换: 取 \(f = Q_i\), \(g = Q_j\), 易得 \[[Q_i, Q_j]_{QP} = [Q_i, Q_j]_{qp} = 0.\] 同理可得构成正则变换的其余充要条件, 故成立.
• 正则变换⇒ 形式不变性: \[[f, g]_{qp} = \left[ \frac{\partial f}{\partial \gamma} \right]^T s \left[ \frac{\partial g}{\partial\gamma} \right], \;\left[ \frac{\partial f}{\partial\gamma} \right] = J^T \left[ \frac{\partial f}{\partial\Gamma} \right].\] 可得: \[[f, g]_{qp} = \left[ \frac{\partial f}{\partial \Gamma} \right]^T JsJ^T \left[ \frac{\partial g}{\partial\Gamma} \right],\;JsJ^T = s.\] 即: \[[f, g]_{qp} = \left[ \frac{\partial f}{\partial \Gamma} \right]^T s \left[ \frac{\partial g}{\partial\Gamma} \right] = [f, g]_{QP}.\blacksquare\]

在 \(q, p\)-系中, 成立: \[\dot{q}_k = \frac{\partial h(q, p, t)}{\partial p_k}, \;\dot{p}_k = -\frac{\partial h(q, p, t)}{\partial q_k}.\] 即 \[[\dot{\gamma}] = s \left[ \frac{\partial h}{\partial \gamma} \right].\] 有正则变换

则哈密顿量有 \(H(Q, P, t) = h \left[ q(Q, P), p(Q, P), t \right]\). 在 \(Q, P\)-系中, 哈密顿方程有相同形式: \[\dot{Q}_k = \frac{\partial H(Q, P, t)}{\partial P_k}, \;\dot{P}_k = -\frac{\partial H(Q, P, t)}{\partial P_k}.\]

以下给出证明. 利用变换辛形式: \[\left[\dot{\Gamma}\right] = J[\dot{\gamma}], \;\left[ \frac{\partial f}{\partial\gamma} \right] = J^T \left[ \frac{\partial f}{\partial\Gamma} \right].\] 且 \([\dot{\gamma}] = s \left[ \partial_{\gamma}h \right]\), 可得: \[\left[ \dot{\Gamma} \right] = JsJ^T \left[ \frac{\partial H}{\partial\gamma} \right] = s \left[ \frac{\partial H}{\partial\gamma} \right].\blacksquare\]

注意当正则变换显含时间时, \(\left[ \dot{\Gamma} \right]=J[\dot{\gamma}]\) 不成立. 此时有: \[\frac{\mathrm{d}\Gamma_k}{\mathrm{d}t} = \frac{\partial\Gamma_k}{\partial t} + \frac{\partial\Gamma_k}{\partial\gamma_l}\dot{\gamma}^l.\] 当变换显含时间且时间不参与变换时,哈密顿量需要修正. \[\dot{Q}_k(q, p, t) = [Q_k, h]_{qp} + \frac{\partial Q_k(q, p, t)}{\partial t}.\] 设 \(H(q, p, t) = h(q, p, t) + g(q, p, t)\), \(H(Q, P, t) = H[q(Q, P), p(Q,P), t]\), 此时有:

同理可得: \[ \frac{\partial P_k(q, p, t)}{\partial t} = [P_k, g]_{qp}, \;\dot{P}_k = -\frac{\partial H(Q, P, t)}{\partial Q_k}.\] 故: 变换显含时间时, 需要 \(H = h + g\), 其中 \[ g = \begin{cases}\frac{\partial Q_k}{\partial t} = [Q_k, g]_{qp}\\ \frac{\partial P_k}{\partial t} = [P_k, g]_{qp}\end{cases}. \]

扩展理论:扩展理论将时间\(t\)作为变换坐标与其他坐标一起变换.

由变分原理知,使得作用量积分取驻值的相空间路线满足哈密顿方程. \[\int_{t(1)}^{t(2)}\left[ p_k\dot{q}^k - H(q, p, t)\right]\mathrm{d}t\stackrel{Q,\;P}{\Longrightarrow} \int_{t(1)}^{t(2)}\left[ P_k\dot{Q}^k - K(Q, P, t) \right]\mathrm{d}t.\]

正则变换(变分原理判据;条件;定义)

满足下式的可逆变换 \(Q_k = Q_k(q, p, t)\), \(P_k = P_k(q, p, t)\) 称为正则变换: \[P_k\dot{Q}^k - K(Q, P, t) = p_k\dot{q}^k - H(q, p, t) - \frac{\mathrm{d}U}{\mathrm{d}t}.\] 即 \[p_k\mathrm{d}q^k - P_k\mathrm{d}Q^k + (K - H)\mathrm{d}t = \mathrm{d}U.\] 即上式左式是某个多元函数 \(U\) 的全微分. 此定义与泊松括号定义一致. \(U\) 的含义即下一节所述生成函数.

由前述, 函数 \(U(q, Q, t)\) 决定了正则变换, 称为正则变换的生成函数. 其偏微分有: \[p_k = \frac{\partial U}{\partial q_k},\;P_k =-\frac{\partial U}{\partial Q_k}, \;K - H = \frac{\partial U}{\partial t}.\] 变换可逆, 可求得 \(q_k, p_k\), \(Q_k, P_k\) 关系. \[K(Q, P, t) = H[q(Q, P), p(Q, P), t] + \frac{\partial U(q, Q, t)}{\partial t}\Big{|}_{q_k = q_k(Q, P, t)}.\] 只有\(U\)不显含时间时,哈密顿量不变.

第一类生成函数

记上述 \(U(q, Q, t)\) 为 \(U_1(q, Q, t)\) 即第一类生成函数.

第二类生成函数

\(U_2(q, P, t) = U_1 + P_kQ^k\). \[ p_k = \frac{\partial U_2}{\partial q_k},\;Q_k = \frac{\partial U_2}{\partial P_k}, \;K = \frac{\partial U_2}{\partial t} + H. \]

第三类生成函数

\(U_3(p, Q, t) = U_1(q, Q, t) - p_kq^k\). \[q_k = -\frac{\partial U_3}{\partial p_k},\;P_k = -\frac{\partial U_3}{\partial Q_k}, \;K = \frac{\partial U_3}{\partial t} + H.\]

第四类生成函数

\(U_4(p, P, t) = U_3+P_kQ^k\). \[q_k = -\frac{\partial U_4}{\partial p_k},\;Q_k = \frac{\partial U_4}{\partial P_k}, \;K = \frac{\partial U_4}{\partial t} + H.\]

除了这四种生成函数,还有混合型生成函数.

无论哪一种生成函数,哈密顿量总是以下列形式变化: \[K = H + \frac{\partial U}{\partial t}.\] 做正则变换使得: \[K = H + \frac{\partial U}{\partial t} = 0.\] 则有:

取第二类生成函数 \(U(q, P, t)\). \[p_k = \frac{\partial U_2}{\partial q_k},\;Q_k = \frac{\partial U_2}{\partial P_k}.\] 可得:

\[K = H(q_1, q_2,\dots, q_D, \frac{\partial U_2}{\partial q_1}, \frac{\partial U_2}{\partial q_2},\dots \frac{\partial U_2}{\partial q_D}, t) + \frac{\partial U_2}{\partial t} = 0.\] 上式称为 哈密顿-雅克比方程, 可用于求解生成函数 \(U_2(q, P, t)\). 哈-雅方程的解称为哈密顿作用量函数. (也称哈密顿主函数)

• 哈-雅方程为一阶非线性偏微分方程, 有 \(D+1\) 个自变量 \(q_1, q_2, \dots, q_D, t\).
• 哈-雅方程的解含有 \(D+1\) 个积分常数, 最后一个常数不重要, 直接略去.

故方程解含有 \(D\) 个积分常数: \(\alpha_k,\;k=1, 2, 3\dots,D\). 认定 \(P_k = \alpha_k\) (注意到 \(P_k\) 为常数), 即可构造生成函数 \(U_2(q, P, t)\).

归纳出哈雅方程求解方法:

1. \(p_j = \frac{\partial U_2}{\partial q_j}\) 代入哈密顿量, 得到哈-雅方程: \[K = H \left( q, \frac{\partial U_2}{\partial q}, t \right) + \frac{\partial U_2}{\partial t} = 0.\]
2. 求解 \(U_2(q1,q2\dots,q_D, \alpha_1,\dots \alpha_D, t)\), 认定积分常数 \(\alpha_j = P_j,\;j=1, 2, \dots, D\). 偏微分方程仅对简单问题可解. (3.2法)
3. 利用生成函数求 \(Q\): \[Q_j = \frac{\partial U_2(q, \alpha, t)}{\partial \alpha_j} = \beta_j = \text{const}.\] 进而计算出 \(q_j(\alpha, \beta, t),\;j=1, 2, \dots, D\).
4. 计算 \(p_j = \frac{\partial U_2(q, \alpha, t)}{\partial q_j}\).
5. 由初始条件确定 \(\alpha, \beta\).

求偏微分方程特解最常用的方法: 分离变量. 这里将变量分离成和的形式,把循环坐标(可遗坐标)变量分离出来.

完全可分离体系

一个哈密顿量不显含时间的体系, 若存在一组广义坐标 \(q_1,\dots, q_D\), 使得其不含时哈-雅方程 \[H \left( q, \frac{\partial W}{\partial q} \right) = E.\] 存在以下形式解: \[W = W(q_1, q_2, \dots, q_D, \alpha_1, \alpha_2, \dots, \alpha_D) = W_k(q^k, \alpha_1, \dots, \alpha_D).\] 则称该体系为 完全可分离体系.

对于完全可分离体系, \(p_k = \partial_{q_k}W_k = p_k(q_k, \alpha)\).

多周期体系

完全可分离 哈密顿体系投影在每个 \(q_k, p_k\) 相平面,

1. \(p_k = p_k(q_k, \alpha)\) 闭合, 此事称为 天平动.
2. \(p_k(q_k, \alpha) = p_k \left[ \left( q_k + q_k^T \right), \alpha \right]\), 但 \(q_k\) 不一定为时间的周期函数, 此时称为 转动.

对所有\(k\)都满足以上两性质,则称之为 多周期体系.

角变量\(\phi_k\) 在一个周期的变化 \[\Delta\phi_k = \oint \mathrm{d}\phi_k = 2\pi.\] 设 \(\Delta \phi_k = \omega_k\tau_k\), \(\tau_k\) 为 \(q_k\) 运动一周期时间. 可得: \[\omega_k = \phi_k = \frac{\partial K}{\partial P_k} = \frac{\partial E(J_1, \dots, J_D)}{\partial J_k}.\]

不同相平面周期不尽相同.

简单周期体系

各相平面周期之比为有理数.

条件周期体系

有两个或更多相平面内运动周期之比不是有理数.

条件周期体系运动状态永远不能完全复原,简单周期函数反之.

简单周期体系有: \(n_1\omega_1 = n_2\omega_2 = \cdots = n_{D-1}\omega_{D-1} = n_D\omega_D\). 因此存在 \(D-1\) 个以下形式独立关系式: \[n_k^{(l)}\omega^k = 0,\;n_k^{(l)}\in \mathbb{N},\;l=1, \dots, D-1.\]

简单周期体系完全简并. 简并条件可用于减少作用量数目.

\(m\)重简并体系

如果一个体系存在 \(m\) 个以上形式的关系时, 称其为 \(m\) 重简并体系.