Physics 🍊⭐
Published:
Equations of fundamental physics.
Analytical Mechanics / 解析力学
Newtonian Mechanics / Newton力学
Equation of Motion / 運動方程式
\[\mathbf{F} = \frac{d}{dt}\left(m\mathbf{v}\right) \\ \mathbf{F}_{\alpha\beta} = -\mathbf{F}_{\beta\alpha}\]
Momentum / 運動量
\[\frac{d\mathbf{P}}{dt} = \mathbf{F} \\ \begin{aligned} & \mathbf{P} = m\mathbf{v} && \mathbf{F} \\ & \mathbf{P} = M\mathbf{V} && \mathbf{F} = \sum_{\alpha}\mathbf{F}_\alpha^{(e)} \end{aligned}\]
\[\frac{d\mathbf{P}}{dt} = \mathbf{F} \\ \begin{aligned} \mathbf{P} &= \sum_\alpha m_\alpha\left(\mathbf{V} + \mathbf{v'}_\alpha\right) \\ &= \sum_\alpha m_\alpha\mathbf{V} + \sum_\alpha m_\alpha\mathbf{v'}_\alpha \\ &= M\mathbf{V} \end{aligned} \\ \begin{aligned} \mathbf{F} &= \sum_\alpha\left(\mathbf{F}_\alpha^{(e)} + \sum_{\beta\neq\alpha}\mathbf{F}_{\alpha\beta}\right) \\ &= \sum_\alpha\mathbf{F}_{\alpha}^{(e)} + \sum_{\beta>\alpha}\left(\mathbf{F}_{\alpha\beta} + \mathbf{F}_{\beta\alpha}\right) \\ &= \sum_\alpha\mathbf{F}_\alpha^{(e)} \end{aligned}\]
Angular Momentum / 角運動量
\[\frac{d\mathbf{M}}{dt} = \mathbf{N} \\ \begin{aligned} & \mathbf{M} = m\mathbf{r}\times\mathbf{v} && \mathbf{N} = \mathbf{r}\times\mathbf{F} \\ & \mathbf{M} = M\mathbf{R}\times\mathbf{V} + \sum_\alpha m_\alpha\mathbf{r'}_\alpha\times\mathbf{v'}_\alpha && \mathbf{N} = \sum_\alpha\mathbf{r}_\alpha\times\mathbf{F}_\alpha^{(e)} \end{aligned}\]
\[\frac{d\mathbf{M}}{dt} = m\mathbf{v}\times\mathbf{v} + m\mathbf{r}\times\mathbf{a} = \mathbf{r}\times\mathbf{F} = \mathbf{N} \\ \begin{aligned} \mathbf{M} &= \sum_\alpha m_\alpha\left(\mathbf{R}+\mathbf{r'}_\alpha\right)\times\left(\mathbf{V}+\mathbf{v'}_\alpha\right) \\ &= \sum_\alpha m_\alpha\mathbf{R}\times\mathbf{V} + \mathbf{R}\times\left(\sum_\alpha m_\alpha\mathbf{v}'_\alpha\right) \\ &+ \left(\sum_\alpha m_\alpha\mathbf{r'}_\alpha\right)\times\mathbf{V} + \sum_\alpha m_\alpha\mathbf{r'}_\alpha\times\mathbf{v'}_\alpha \\ &= M\mathbf{R}\times\mathbf{V} + \sum_\alpha m_\alpha\mathbf{r'}_\alpha\times\mathbf{v'}_\alpha \end{aligned} \\ \begin{aligned} \mathbf{N} &= \sum_\alpha \mathbf{r}_\alpha\times\left(\mathbf{F}_\alpha^{(e)} + \sum_{\beta\neq\alpha}\mathbf{F_{\alpha\beta}}\right) \\ &= \sum_\alpha\mathbf{r}_\alpha\times\mathbf{F}_\alpha^{(e)} + \sum_{\beta>\alpha}\left(\mathbf{r}_\alpha\times\mathbf{F}_{\alpha\beta} + \mathbf{r}_\beta\times\mathbf{F}_{\beta\alpha}\right) \\ &= \sum_\alpha\mathbf{r}_\alpha\times\mathbf{F}_\alpha^{(e)} + \sum_{\beta>\alpha}\left(\mathbf{r}_\alpha-\mathbf{r}_\beta\right)\times\mathbf{F}_{\alpha\beta} \\ &= \sum_\alpha\mathbf{r}_\alpha\times\mathbf{F}_\alpha^{(e)} \end{aligned}\]
Energy / エネルギー
\[\frac{dE}{dt} = 0 \\ \begin{aligned} E &= \frac{1}{2} mv^2 + U \\ E &= \frac{1}{2} MV^2 + \sum_\alpha \frac{1}{2} m_\alpha v_\alpha'^2 + U \\ \end{aligned}\]
\[\frac{dE}{dt} = m\mathbf{v} \cdot \mathbf{a} + (\nabla U) \cdot \mathbf{v} + \frac{\partial U}{\partial t} = (\mathbf{F} + \nabla U) \cdot \mathbf{v} = 0 \\ \begin{aligned} E &= \sum_\alpha \frac{1}{2} m_\alpha (\mathbf{V} + \mathbf{v}_\alpha') \cdot (\mathbf{V} + \mathbf{v}_\alpha') \\ &= \sum_\alpha \frac{1}{2} m_\alpha V^2 + \left(\sum_\alpha m_\alpha \mathbf{v}_\alpha'\right) \cdot \mathbf{V} + \sum_\alpha \frac{1}{2} m_\alpha v_\alpha'^2 \\ &= \frac{1}{2} MV^2 + \sum_\alpha \frac{1}{2} m_\alpha v_\alpha'^2 \end{aligned}\]
Lagrangian Mechanics / Lagrange力学
Lagrange’s Equation / Langrange方程式
\[\delta S = \delta\int_{t_1}^{t_2} L dt = 0 \quad L = T - U\\ \begin{aligned} & \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}_i}} - \frac{\partial{L}}{\partial{q_i}} = 0 \\ & \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}_i}} - \frac{\partial{L}}{\partial{q_i}} + \sum_j \lambda_j \frac{\partial{f_j}}{\partial{q_i}} = 0 \end{aligned}\]
\[\begin{aligned} \delta S &= \int_{t_1}^{t_2} \sum_i \left( \frac{\partial{L}}{\partial{q_i}} \delta q_i + \frac{\partial{L}}{\partial{\dot{q}_i}} \delta \dot{q}_i \right) dt \\ &= \int_{t_1}^{t_2} \sum_i \frac{\partial{L}}{\partial{q_i}} \delta q_i dt + \left[ \sum_i \frac{\partial{L}}{\partial{\dot{q}_i}} \delta q_i \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} \sum_i \frac{d}{dt} \left( \frac{\partial{L}}{\partial{\dot{q_i}}} \right) \delta q_i dt \\ &= \int_{t_1}^{t_2} \sum_i \left( \frac{\partial{L}}{\partial{q_i}} - \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}_i}} \right) \delta q_i dt \\ &= 0 \end{aligned}\]
Conservation Law / 保存則
\[\begin{aligned} E &= \sum_i \dot{q}_i \frac{\partial{L}}{\partial{\dot{q}_i}} - L = const. \\ \mathbf{P} &= \sum_\alpha \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} = const. \\ \mathbf{M} &= \sum_\alpha \mathbf{r}_\alpha \times \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} = const. \end{aligned}\]
\[\begin{aligned} & \quad\; \frac{d}{dt} \left( \sum_i \dot{q}_i \frac{\partial{L}}{\partial{\dot{q}_i}} - L \right) \\ &= \sum_i \left[ \ddot{q}_i \frac{\partial{L}}{\partial{\dot{q}_i}} + \dot{q}_i \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{q}_i}} - \frac{\partial{L}}{\partial{q_i}} \dot{q}_i - \frac{\partial{L}}{\partial{\dot{q}_i}} \ddot{q}_i \right] \\ &= 0 \\ \delta L &= \sum_\alpha \frac{\partial{L}}{\partial{\mathbf{r}_\alpha}} \cdot \mathbf{\epsilon} \\ &= \mathbf{\epsilon} \cdot \sum_\alpha \frac{d}{dt} \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} \\ &= 0 \\ \delta L &= \sum_\alpha \left( \frac{\partial{L}}{\partial{\mathbf{r}_\alpha}} \cdot \mathbf{\theta} \times \mathbf{r}_\alpha + \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} \cdot \mathbf{\theta} \times \mathbf{v}_\alpha \right) \\ &= \sum_\alpha \left[ \mathbf{\theta} \cdot \mathbf{r}_\alpha \times \frac{d}{dt}\frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} + \mathbf{\theta} \cdot \mathbf{v}_\alpha \times \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} \right] \\ &= \mathbf{\theta} \cdot \sum_\alpha \frac{d}{dt} \left( \mathbf{r}_\alpha \times \frac{\partial{L}}{\partial{\mathbf{v}_\alpha}} \right) \\ &= 0 \end{aligned}\]
Galilean Transformation / ガリレイ変換
\[\begin{aligned} \mathbf{r}' &= \mathbf{r} + \mathbf{V}t \\ t' &= t \end{aligned}\]
\[\begin{aligned} L' &= \frac{1}{2} m \left( \mathbf{v}' \right)^2 - U'(\mathbf{r}') \\ &= \frac{1}{2} m \mathbf{v}^2 + m \mathbf{v} \cdot \mathbf{V} + \frac{1}{2} m \mathbf{V}^2 - U(\mathbf{r}) \\ &= L + \frac{d}{dt} \left( m \mathbf{r} \cdot \mathbf{V} + \frac{1}{2} m \mathbf{V}^2 t \right) \\ \delta S' &= \delta \int_{t_1}^{t_2} L' dt' \\ &= \delta \int_{t_1}^{t_2} L dt + \delta \left[ m \mathbf{r} \cdot \mathbf{V} + \frac{1}{2} m \mathbf{V}^2 t \right]_{t_1}^{t_2} \\ &= \delta S \end{aligned}\]
Virial Theorem / Virial定理
\[U(\alpha \mathbf{r}) = \alpha ^k U(\mathbf{r}) \\ \frac{t'}{t} = \left( \frac{l'}{l} \right)^{1-\frac{k}{2}} \\ \langle T \rangle = \frac{k}{2}\langle U \rangle\]
\[l' = \alpha l \quad t' = \beta t \\ \frac{T'}{T} = \alpha^2 \beta^{-2} = \frac{U'}{U} = \alpha^k \\ \beta = \alpha^{1-\frac{k}{2}} \\ \begin{aligned} 2T &= \sum_\alpha\mathbf{P}_\alpha\cdot\mathbf{v}_\alpha \\ &= \frac{d}{dt}\left(\sum_\alpha\mathbf{P}_\alpha\cdot\mathbf{r}_\alpha\right) - \sum_\alpha\mathbf{F}_\alpha\cdot\mathbf{r}_\alpha \end{aligned} \\ \begin{aligned} \langle 2T \rangle &= \lim_{\tau\to\infty}\frac{1}{\tau}\int_o^\infty\frac{d}{dt}\left(\sum_\alpha\mathbf{P}_\alpha\cdot\mathbf{r}_\alpha\right)dt - \langle\sum_\alpha\mathbf{F}_\alpha\cdot\mathbf{r}_\alpha\rangle \\ &= \lim_{\tau\to\infty}\frac{1}{\tau}\left[\sum_\alpha\mathbf{P}_\alpha\cdot\mathbf{r}_\alpha\right]_0^\infty + \langle\sum_\alpha\frac{\partial U}{\partial\mathbf{r}_\alpha}\cdot\mathbf{r}_\alpha\rangle \\ &= k\langle U \rangle \end{aligned}\]
Hamiltonian Mechanics / Hamilton力学
Hamilton’s Equation / Hamilton方程式
\[H = \sum_i p_i \dot{q}_i - L \quad p_i = \frac{\partial{L}}{\partial{\dot{q}_i}} \\ \dot{q}_i = \frac{\partial{H}}{\partial{p}_i} \quad \dot{p}_i = -\frac{\partial{H}}{\partial{q}_i}\]
\[\begin{aligned} dH &= \sum_i \left(\dot{q}_i dp_i + p_i d\dot{q}_i \right) - \sum_i \left(\frac{\partial{L}}{\partial{q_i}} dq_i + \frac{\partial{L}}{\partial{\dot{q}_i}}d\dot{q_i}\right) - \frac{\partial{L}}{\partial{t}}dt \\ &= \sum_i \left(\dot{q}_i dp_i - \dot{p}_i dq_i\right) - \frac{\partial{L}}{\partial{t}} dt \end{aligned} \\ \dot{q}_i = \frac{\partial{H}}{\partial{p_i}} \quad \dot{p}_i = -\frac{\partial{H}}{\partial{q_i}} \quad \frac{\partial{L}}{\partial{t}} = -\frac{\partial{H}}{\partial{t}}\]
Poisson Bracket / Poisson括弧
\[\{F, G\} = \sum_i \left(\frac{\partial{F}}{\partial{q_i}} \frac{\partial{G}}{\partial{p_i}} - \frac{\partial{F}}{\partial{p_i}} \frac{\partial{G}}{\partial{q_i}}\right) \\ \frac{dF}{dt} = \{F, H\} + \frac{\partial{F}}{\partial{t}}\]
\[\begin{aligned} \frac{dF}{dt} &= \sum_i \left(\frac{\partial{F}}{\partial{q_i}}\dot{q}_i + \frac{\partial{F}}{\partial{p_i}}\dot{p}_i \right) + \frac{\partial{F}}{\partial{t}} \\ &= \sum_i \left(\frac{\partial{F}}{\partial{q_i}} \frac{\partial{H}}{\partial{p_i}} - \frac{\partial{F}}{\partial{p_i}} \frac{\partial{H}}{\partial{q_i}}\right) + \frac{\partial{F}}{\partial{t}} \\ &= \{F, H\} + \frac{\partial{F}}{\partial{t}} \end{aligned}\]
Canonical Transformation / 正準変換
\[\begin{aligned} & W(q,Q,t): && p = \frac{\partial{W}}{\partial{q}} && P = -\frac{\partial{W}}{\partial{Q}} && H' = H + \frac{\partial{W}}{\partial{t}} \\ & W(q,P,t): && p = \frac{\partial{W}}{\partial{q}} && Q=\frac{\partial{W}}{\partial{P}} && H' = H + \frac{\partial{W}}{\partial{t}} \\ & W(p,Q,t): && q = -\frac{\partial{W}}{\partial{p}} && P = -\frac{\partial{W}}{\partial{Q}} && H' = H + \frac{\partial{W}}{\partial{t}} \\ & W(p,P,t): && q = -\frac{\partial{W}}{\partial{p}} && Q = \frac{\partial{W}}{\partial{P}} && H' = H + \frac{\partial{W}}{\partial{t}} \end{aligned}\]
\[\delta \int_{t_1}^{t_2} \left(\sum_i p_i \dot{q}_i - H\right) dt = 0 \quad \delta \int_{t_1}^{t_2} \left(\sum_i P_i \dot{Q}_i - H'\right) dt = 0 \\ \sum_i p_i dq_i - H dt = \sum_i P_i dQ_i - H' dt + dW_1(q,Q,t) \\ dW_1 = \sum_i p_i dq_i - \sum_i P_i dQ_i + (H' - H) dt \\ p_i = \frac{\partial{W_1}}{\partial{q_i}} \quad P_i = -\frac{\partial{W_1}}{\partial{Q_i}} \quad H' = H + \frac{\partial{W_1}}{\partial{t}} \\ W_2 = W_1 + \sum_i P_i Q_i \\ dW_2 = \sum_i p_i dq_i + \sum_i Q_i dP_i + (H' - H) dt \\ p_i = \frac{\partial{W_2}}{\partial{q_i}} \quad Q_i = \frac{\partial{W_2}}{\partial{P_i}} \quad H' = H + \frac{\partial{W_2}}{\partial{t}} \\ W_3 = W_1 - \sum_i p_i q_i \\ dW_3 = -\sum_i q_i dp_i - \sum_i P_i dQ_i + (H' - H) dt \\ q_i = -\frac{\partial{W_3}}{\partial{p_i}} \quad P_i = -\frac{\partial{W_3}}{\partial{Q_i}} \quad H' = H + \frac{\partial{W_3}}{\partial{t}} \\ W_4 = W_1 - \sum_i p_i q_i + \sum_i P_i Q_i \\ dW_4 = - \sum_i q_i dp_i + \sum_i Q_i dP_i + (H' - H) dt \\ q_i = -\frac{\partial{W_4}}{\partial{p_i}} \quad Q_i = \frac{\partial{W_4}}{\partial{P_i}} \quad H' = H + \frac{\partial{W_4}}{\partial{t}}\]
Hamilton–Jacobi Equation / Hamilton–Jacobi方程式
\[\frac{\partial{S}}{\partial{t}} + H\left(q,\frac{\partial{S}}{\partial{q}},t\right) = 0\]
\[H' = H + \frac{\partial{W}}{\partial{t}} = 0 \\ \dot{P}_i = -\frac{\partial{H'}}{\partial{Q_i}} = 0 \quad P_i = \alpha_i \\ \begin{aligned} \frac{dW}{dt} &= \sum_i \frac{\partial{W}}{\partial{q_i}} \dot{q}_i + \frac{\partial{W}}{\partial{t}} \\ &= \sum_i p_i \dot{q}_i - H \\ &= L \end{aligned} \\ W = \int L dt = S\]
Liouville’s Theorem / Liouvilleの定理
\[\int_R \prod_i dq_i dp_i = \int_{R'} \prod_i dQ_i dP_i\]
\[\prod_i dQ_i dP_i = \frac{\partial{(Q,P)}}{\partial{(q,p)}} \prod_i dq_i dp_i \\ \begin{aligned} \frac{\partial{(Q,P)}}{\partial{(q,p)}} &= \frac{\partial{(Q,P)}}{\partial{(q,P)}} \frac{\partial{(q,P)}}{\partial{(q,p)}} = \frac{\partial{(Q)}}{\partial{(q)}} \frac{\partial{(P)}}{\partial{(p)}} \\ &= \frac{\frac{\partial{(Q)}}{\partial{(q)}}}{\frac{\partial{(p)}}{\partial{(P)}}} = \frac{\det\left(\frac{\partial^2 W}{\partial q_j \partial P_i}\right)}{\det\left(\frac{\partial^2 W}{\partial P_j \partial q_i}\right)} \\ &= 1 \end{aligned}\]
Oscillation / 振動
Small Oscillation / 微小振動
\[\ddot{x} + \omega_0^2 x = 0 \\ x = \tilde{A} e^{i \omega_0 t}\]
\[U'(x_0) = 0 \quad U''(x_0) = k \\ L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} k x^2 \\ m \ddot{x} + k x = 0 \\ \ddot{x} + \omega_0^2 x = 0 \\ x = \tilde{A} e^{i \omega_0 t}\]
Damped Oscillation / 減衰振動
\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = 0 \\ \begin{aligned} x &= e^{-\beta t} \cdot \tilde{A} e^{i \sqrt{\omega_0^2 - \beta^2} t} \\ x &= e^{-\beta t} (A + B t) \\ x &= e^{-\beta t} \left( A_1 e^{\sqrt{\beta^2 - \omega_0^2} t} + A_2 e^{-\sqrt{\beta^2 - \omega_0^2} t} \right) \end{aligned}\]
\[m \ddot{x} + b \dot{x} + k x = 0 \\ \ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = 0 \\ x = e^{\gamma t} \\ \gamma^2 + 2 \beta \gamma + \omega_0^2 = 0 \\ \gamma = -\beta \pm \sqrt{\beta^2 - \omega_0^2} \\ x = e^{-\beta t} \left( A_1 e^{\sqrt{\beta^2 - \omega_0^2} t} + A_2 e^{-\sqrt{\beta^2 - \omega_0^2} t} \right)\]
Forced Oscillation / 強制振動
\[\ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = \alpha e^{i \omega t} \\ x_p = \tilde{A} e^{i \omega t} \\ |\tilde{A}| = \frac{\alpha}{\sqrt{(\omega^2 - \omega_0^2)^2 + 4 \beta^2 \omega^2}} \\ \text{arg}(\tilde{A}) = \tan^{-1} \left( \frac{2 \beta \omega}{\omega^2 - \omega_0^2} \right)\]
\[m \ddot{x} + b \dot{x} + k x = f\cos(\omega t) \\ \ddot{x} + 2 \beta \dot{x} + \omega_0^2 x = \alpha e^{i \omega t} \\ x_p = \tilde{A} e^{i \omega t} \\ (-\omega^2 + 2 \beta \omega i + \omega_0^2) \tilde{A} = \alpha \\ \tilde{A} = \frac{\alpha}{(\omega_0^2 - \omega^2) + 2 \beta \omega i}\]
Coupled Oscillation / 連成振動
\[M \ddot{\mathbf{x}} + K \mathbf{x} = 0 \\ \left( -\omega_n^2 M + K \right) \mathbf{A_n} = 0 \\ \mathbf{x} = \sum_n \tilde{c}_n \mathbf{A_n} e^{i \omega_n t} \\ \mathbf{x} = \sum_n q_n \mathbf{A_n} \quad \ddot{q}_n + \omega_n^2 q_n = 0\]
\[\begin{aligned} L &= \frac{1}{2} \sum_{i,j} m_{ij} \dot{x}_i \dot{x}_j - \frac{1}{2} \sum_{i,j} k_{ij} x_i x_j \\ &= \frac{1}{2} \dot{\mathbf{x}}^T M \dot{\mathbf{x}} - \frac{1}{2} \mathbf{x}^T K \mathbf{x} \end{aligned} \\ M \ddot{\mathbf{x}} + K \mathbf{x} = 0 \quad \mathbf{x} = \mathbf{A} e^{i \omega t} \\ |-\omega_n^2 M + K| = 0 \quad (-\omega_n^2 M + K) \mathbf{A}_n = 0 \\ \mathbf{x} = \sum_n \hat{c}_n \mathbf{A}_n e^{i \omega_n t} \\ \mathbf{A}_n^T (K \mathbf{A}_m) - (K \mathbf{A}_n)^T \mathbf{A}_m = (\omega_n^2 - \omega_m^2) \mathbf{A}_n^T M \mathbf{A}_m = 0 \\ \mathbf{A}_n^T M \mathbf{A}_m = \delta_{nm} \quad \mathbf{A}_n^T K \mathbf{A}_m = \delta_{nm} \omega_m^2 \\ \mathbf{x} = \sum_n q_n \mathbf{A}_n \\ \begin{aligned} L &= \frac{1}{2} \sum_{n,m} \dot{q}_n (\mathbf{A}_n^T M \mathbf{A}_m) \dot{q}_m - \frac{1}{2} \sum_{n,m} q_n (\mathbf{A}_n^T K \mathbf{A}_m) q_m \\ &= \frac{1}{2} \sum_n \dot{q}_n^2 - \frac{1}{2} \sum_n \omega_n^2 q_n^2 \end{aligned} \\ \ddot{q}_n + \omega_n^2 q_n = 0\]
Wave / 波動
Wave Equation / 波動方程式
\[\frac{\partial}{\partial t} \left( \frac{\delta L}{\delta (\partial_t \phi)} \right) + \frac{\partial}{\partial x} \left( \frac{\delta L}{\delta (\partial_x \phi)} \right) - \frac{\delta L}{\delta \phi} = 0 \\ \rho \frac{\partial^2 \phi}{\partial t^2} - k \frac{\partial^2 \phi}{\partial x^2} = 0\]
\[L = \int \mathcal{L} \left( \phi, \frac{\partial \phi}{\partial t}, \frac{\partial \phi}{\partial x}, t \right) dx \\ \begin{aligned} \delta S &= \int dt \int dx \left[ \frac{\delta L}{\delta \phi} \delta \phi + \frac{\delta L}{\delta (\partial_t \phi)} \delta (\partial_t \phi) + \frac{\delta L}{\delta (\partial_x \phi)} \delta (\partial_x \phi) \right] \\ &= \int dt \int dx \left[ \frac{\delta L}{\delta \phi} - \frac{\partial}{\partial t} \left( \frac{\delta L}{\delta (\partial_t \phi)} \right) - \frac{\partial}{\partial x} \left( \frac{\delta L}{\delta (\partial_x \phi)} \right) \right] \delta \phi \end{aligned} \\ \frac{\partial}{\partial t} \left( \frac{\delta L}{\delta (\partial_t \phi)} \right) + \frac{\partial}{\partial x} \left( \frac{\delta L}{\delta (\partial_x \phi)} \right) - \frac{\delta L}{\delta \phi} = 0 \\ L = \int \left[ \frac{1}{2} \rho (\partial_t \phi)^2 - \frac{1}{2} k (\partial_x \phi)^2 \right] dx \\ \delta L = \int \left[ \rho (\partial_t \phi) \delta (\partial_t \phi) - k (\partial_x \phi) \delta (\partial_x \phi) \right] dx \\ \frac{\delta L}{\delta (\partial_t \phi)} = \rho \frac{\partial \phi}{\partial t} \quad \frac{\delta L}{\delta (\partial_x \phi)} = -k \frac{\partial \phi}{\partial x} \\ \rho \frac{\partial^2 \phi}{\partial t^2} - k \frac{\partial^2 \phi}{\partial x^2} = 0\]
Bounded Wave / 有界波
\[\frac{\partial^2 \phi}{\partial t^2} - v^2 \frac{\partial^2 \phi}{\partial x^2} = 0 \\ \phi = \sum_n \tilde{A}_n e^{i (\omega_n t - k_n x)} \\ \begin{aligned} &\phi(0) = \phi(L) = 0: && k_n = \frac{n \pi}{L} \quad \tilde{A}_{-n} = -\tilde{A}_n \\ &\frac{\partial \phi}{\partial x}(0) = \frac{\partial \phi}{\partial x}(L) = 0: && k_n = \frac{n \pi}{L} \quad \tilde{A}_{-n} = \tilde{A}_n \\ &\phi(0) = \phi(L),\; \frac{\partial \phi}{\partial x}(0) = \frac{\partial \phi}{\partial x}(L): && k_n = \frac{2n \pi}{L} \end{aligned}\]
\[\phi(x,t) = \psi(x) e^{i \omega t} \\ \psi(x) = \tilde{A} e^{-ikx} \\ \phi = \sum_n \tilde{A}_n e^{i(\omega_n t - k_n x)} \quad k_n = \pm \frac{\omega_n}{v} \\ \phi(0) = \phi(L) = 0 \\ \begin{aligned} &\tilde{A}_n e^{i \omega_n t} + \tilde{A}_{-n} e^{i \omega_n t} = 0 && \tilde{A}_{-n} = -\tilde{A}_n \\ &\tilde{A}_n e^{i \omega_n t} \left( e^{-ik_n L} - e^{ik_n L} \right) = 0 && k_n L = n \pi \end{aligned} \\ \frac{\partial \phi}{\partial x}(0) = \frac{\partial \phi}{\partial x}(L) = 0 \\ \begin{aligned} & -\tilde{A}_n k_n e^{i \omega_n t} - \tilde{A}_{-n} (-k_n) e^{i \omega_n t} = 0 && \tilde{A}_{-n} = \tilde{A}_n \\ & -\tilde{A}_n k_n e^{i \omega_n t} \left( e^{-ik_n L} - e^{ik_n L} \right) = 0 && k_n L = n \pi \end{aligned} \\ \phi(0) = \phi(L) \quad \frac{\partial \phi}{\partial x}(0) = \frac{\partial \phi}{\partial x}(L) \\ \begin{align*} \left( \tilde{A}_n + \tilde{A}_{-n} \right) e^{i\omega_n t} &= \left(\tilde{A}_n e^{-i k_n L} + \tilde{A}_{-n} e^{i k_n L} \right) e^{i\omega_n t} \\ \left( -\tilde{A}_n k_n + \tilde{A}_{-n} k_n \right) e^{i\omega_n t} &= \left( -\tilde{A}_n k_n e^{-i k_n L} + \tilde{A}_{-n} k_n e^{i k_n L} \right) e^{i\omega_n t} \end{align*} \\ e^{ik_n L} = e^{-ik_n L} = 1 \quad k_n L = 2n \pi\]
Free Wave / 自由波
\[\frac{\partial^2 \phi}{\partial t^2} - v^2 \frac{\partial^2 \phi}{\partial x^2} = 0 \\ \phi = \int \tilde{A}(k) e^{i (\omega t - k x)} \, dk \\ v_p = \frac{\omega}{k} \quad v_g = \frac{d\omega}{dk}\]
\[\phi(x, t) = \psi(x) e^{i \omega t} \\ \psi(x) = \tilde{A} e^{-i k x} \\ \phi = \int \tilde{A}(k) e^{i (\omega t - k x)} \, dk \quad k = \frac{\omega}{v} \\ \omega = \omega_0 + \left( \frac{d\omega}{dk} \right)_0 (k - k_0) \\ \omega t - k x = (\omega_0 t - k_0 x) + (k - k_0)(\omega_0' t - x) \\ \phi = \int \tilde{A}(k) e^{i(\omega_0 t - k_0 x)} e^{i (k - k_0) (\omega_0' t - x)} \, dk \\ v_p = \frac{\omega}{k} \quad v_g = \frac{d\omega}{dk}\]
Central Force Motion / 中心力運動
Reduced Mass / 換算質量
\[\begin{aligned} \mu &= \frac{m_1 m_2}{m_1 + m_2} \\ M &= \mu r^2 \dot{\theta} \\ E &= \frac{1}{2} \mu \dot{r}^2 + \frac{1}{2}\frac{M^2}{\mu r^2} + U(r) \end{aligned}\]
\[\begin{aligned} L &= \frac{1}{2} m_1 \left| \frac{m_2}{m_1 + m_2} \dot{\mathbf{r}} \right|^2 + \frac{1}{2} m_2 \left| \frac{m_1}{m_1 + m_2} \dot{\mathbf{r}} \right|^2 - U(r) \\ &= \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (\dot{r}^2 + r^2 \dot{\theta}^2) - U(r) \end{aligned} \\ M = \frac{\partial L}{\partial\dot{\theta}} = \mu r^2 \dot{\theta} \\ \begin{aligned} E &= \frac{1}{2} \mu \dot{r}^2 + \frac{1}{2} \mu r^2 \dot{\theta}^2 + U(r) \\ &= \frac{1}{2} \mu \dot{r}^2 + \frac{1}{2}\frac{M^2}{\mu r^2} + U(r) \end{aligned}\]
Orbit / 軌道
\[U_{\text{eff}}(r) = U(r) + \frac{M^2}{2\mu r^2} \\ t = \int \frac{dr}{\sqrt{\frac{2}{\mu} (E - U_{\text{eff}}(r))}} \\ \theta = \int \frac{M dr}{r^2 \sqrt{2\mu (E - U_{\text{eff}}(r))}}\]
\[E = \frac{1}{2} \mu \dot{r}^2 + U_{\text{eff}}(r) \\ \frac{dr}{dt} = \sqrt{\frac{2}{\mu} (E - U_{\text{eff}}(r))} \\ t = \int \frac{dr}{\sqrt{\frac{2}{\mu} (E - U_{\text{eff}}(r))}} \\ \begin{aligned} \frac{d\theta}{dr} &= \frac{d\theta}{dt} \frac{dt}{dr} \\ &= \frac{M}{\mu r^2} \frac{1}{\sqrt{\frac{2}{\mu} (E - U_{\text{eff}}(r))}} \end{aligned} \\ \theta = \int \frac{M dr}{r^2 \sqrt{2\mu (E - U_{\text{eff}}(r))}}\]
Kepler Problem / Kepler問題
\[U = -\frac{\alpha}{r} \quad p = \frac{M^2}{\mu\alpha} \quad e = \sqrt{1 + \frac{2EM^2}{\mu \alpha^2}} \\ r = \frac{p}{1 + e \cos \theta} \\ \tau = 2\pi \sqrt{\frac{\mu}{\alpha}}a^{3/2}\]
\[\begin{aligned} \theta &= \int \frac{\frac{M}{r^2} \, dr}{\sqrt{2\mu \left( E + \frac{\alpha}{r} - \frac{M^2}{2\mu r^2} \right)}} \\ &= \int \frac{\frac{1}{r^2} \,dr}{\sqrt{ -\left( \frac{1}{r} - \frac{\mu \alpha}{M^2}\right)^2 + \frac{\mu^2 \alpha^2}{M^4}\left(1 + \frac{2EM^2}{\mu \alpha^2}\right)}} \\ &= \int \frac{\frac{1}{r^2} \,dr}{\sqrt{ -\left( \frac{1}{r} - \frac{1}{p}\right)^2 + \frac{e^2}{p^2}}} \\ &= \int \frac{\frac{e}{p}\sin\phi\,d\phi}{\frac{e}{p}\sin \phi} \\ &= \phi + C \quad \left( \frac{1}{r} - \frac{1}{p} = \frac{e}{p} \cos \phi \right) \end{aligned} \\ r = \frac{p}{1 + e \cos \theta} \quad a = \frac{p}{1 - e^2} \quad b = \frac{p}{\sqrt{1 - e^2}} \\ \tau = \frac{\pi ab}{\dot{S}} = \frac{\pi a \sqrt{ap}}{\frac{M}{2\mu}}= 2\pi \sqrt{\frac{\mu}{\alpha}}a^{3/2}\]
Collision / 衝突
\[\begin{aligned} \mathbf{v}_1 &= \frac{m_2 \mathbf{v}}{m_1 + m_2} + \frac{m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2}{m_1 + m_2} \\ \mathbf{v}_2 &= \frac{-m_1 \mathbf{v}}{m_1 + m_2} + \frac{m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2}{m_1 + m_2} \end{aligned} \\ \begin{aligned} \tan{\theta_1} &= \frac{\sin{\theta}}{\cos{\theta} + \frac{m_1}{m_2}} \\ \tan{\theta_2} &= \frac{\sin{\theta}}{\cos{\theta} - 1} \end{aligned}\]
\[m_1 \mathbf{v}_1' + m_2 \mathbf{v}_2' = 0 \\ \mathbf{v}_1' - \mathbf{v}_2' = \mathbf{v} \\ \begin{aligned} \mathbf{v}_1 &= \mathbf{v}_1' + \mathbf{V} = \frac{m_2 \mathbf{v}}{m_1 + m_2} + \frac{m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2}{m_1 + m_2} \\ \mathbf{v}_2 &= \mathbf{v}_2' + \mathbf{V} = \frac{-m_1 \mathbf{v}}{m_1 + m_2} + \frac{m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2}{m_1 + m_2} \end{aligned} \\ u_2 = 0 \quad u_1 = v \\ \begin{aligned} \tan{\theta_1} &= \frac{m_2 \mathbf{v} \sin{\theta}}{m_2 \mathbf{v} \cos{\theta} + m_1 \mathbf{v}} = \frac{\sin{\theta}}{\cos{\theta} + \frac{m_1}{m_2}} \\ \tan{\theta_2} &= \frac{-m_1 \mathbf{v} \sin{\theta}}{-m_1 \mathbf{v} \cos{\theta} + m_1 \mathbf{v}} = \frac{\sin{\theta}}{\cos{\theta} - 1} \end{aligned}\]
Scattering / 散乱
\[\begin{aligned} \frac{d\sigma}{d\Omega} &= \frac{R^2}{4} \\ \frac{d\sigma}{d\Omega} &= \left(\frac{\alpha}{4T}\right)^2 \frac{1}{\sin^4\frac{\theta}{2}} \end{aligned}\]
\[b = R \sin{\varphi} = R \sin{\left(\frac{\pi}{2} - \frac{\theta}{2}\right)} = R \cos{\frac{\theta}{2}} \\ \begin{aligned} \frac{d\sigma}{d\Omega} &= \frac{b}{\sin{\theta}} \left| \frac{db}{d\theta} \right| \\ &= \frac{R \cos{\frac{\theta}{2}}}{2 \sin{\frac{\theta}{2}} \cos{\frac{\theta}{2}}} \cdot \frac{R}{2} \sin{\frac{\theta}{2}} \\ &= \frac{R^2}{4} \end{aligned} \\ \frac{1}{r_\infty} + \frac{1}{p} = \frac{e}{p} \cos{\varphi} \quad \quad \cos{\varphi} = \frac{1}{e} \\ e = \sqrt{1 + \frac{2EM^2}{m \alpha^2}} = \sqrt{1 + \frac{4T^2 b^2}{\alpha^2}} \\ \tan{\varphi} = \sqrt{e^2 - 1} = \frac{2Tb}{\alpha} \\ b = \frac{\alpha}{2T} \tan{\varphi} = \frac{\alpha}{2T} \tan{\left(\frac{\pi}{2} - \frac{\theta}{2}\right)} = \frac{\alpha}{2T} \cot{\frac{\theta}{2}} \\ \begin{aligned} \frac{d\sigma}{d\Omega} &= \frac{b}{\sin{\theta}} \left| \frac{db}{d\theta} \right| \\ &= \frac{\frac{\alpha}{2T} \cot{\frac{\theta}{2}}}{2 \sin{\frac{\theta}{2}} \cos{\frac{\theta}{2}}} \cdot \frac{\alpha}{4T} \frac{1}{\sin^2{\frac{\theta}{2}}} \\ &= \left(\frac{\alpha}{4T}\right)^2 \frac{1}{\sin^4{\frac{\theta}{2}}} \end{aligned}\]
Gravity / 重力
\[\begin{aligned} \Phi &= \frac{U}{m} = -\frac{GM}{r} \\ \mathbf{g} &= -\nabla \Phi = -\frac{GM}{r^2} \hat{\mathbf{e}}_r \\ \mathbf{F} &= m\mathbf{g} = -\frac{GMm}{r^2} \hat{\mathbf{e}}_r \end{aligned}\]
Rigid Body Motion / 剛体運動
Non-Inertial Frame / 非慣性系
\[\begin{aligned} \mathbf{r} &= \mathbf{R} + \mathbf{r}' \\ \mathbf{v} &= \mathbf{V} + \mathbf{v}' + \boldsymbol{\Omega} \times \mathbf{r}' \\ \mathbf{a} &= \mathbf{A} + \mathbf{a}' + \dot{\boldsymbol{\Omega}} \times \mathbf{r}' + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r}') + 2 \boldsymbol{\Omega} \times \mathbf{v}' \end{aligned}\]
\[\frac{d\mathbf{Q}}{dt} = \frac{d'\mathbf{Q}}{dt} + \boldsymbol{\Omega} \times \mathbf{Q} \\ \mathbf{r} = \mathbf{R} + \mathbf{r}' \\ \begin{aligned} \frac{d\mathbf{r}}{dt} &= \frac{d\mathbf{R}}{dt} + \frac{d\mathbf{r}'}{dt} \\ &= \frac{d\mathbf{R}}{dt} + \frac{d'\mathbf{r}'}{dt} + \boldsymbol{\Omega} \times \mathbf{r}' \end{aligned} \\ \mathbf{v} = \mathbf{V} + \mathbf{v}' + \boldsymbol{\Omega} \times \mathbf{r}' \\ \begin{aligned} \frac{d\mathbf{v}}{dt} &= \frac{d\mathbf{V}}{dt} + \frac{d\mathbf{v}'}{dt} + \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{r}' + \boldsymbol{\Omega} \times \frac{d\mathbf{r}'}{dt} \\ &= \frac{d\mathbf{V}}{dt} + \frac{d'\mathbf{v}'}{dt} + \boldsymbol{\Omega}\times\mathbf{v}' + \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{r}' + \boldsymbol{\Omega} \times \left( \frac{d'\mathbf{r}'}{dt} + \boldsymbol{\Omega} \times \mathbf{r}' \right) \end{aligned} \\ \mathbf{a} = \mathbf{A} + \mathbf{a}' + \dot{\boldsymbol{\Omega}} \times \mathbf{r}' + \boldsymbol{\Omega} \times (\boldsymbol{\Omega} \times \mathbf{r}') + 2 \boldsymbol{\Omega} \times \mathbf{v}'\]
Euler Angle / Euler角
\[\begin{aligned} \Omega_1 &= \dot{\phi} \sin \theta \sin \psi + \dot{\theta} \cos \psi \\ \Omega_2 &= \dot{\phi} \sin \theta \cos \psi - \dot{\theta} \sin \psi \\ \Omega_3 &= \dot{\phi} \cos \theta + \dot{\psi} \end{aligned} \\\]
Inertia Tensor / 慣性テンソル
\[\begin{aligned} I_{ij} &= \sum_{\alpha} m_{\alpha} (\delta_{ij} r_{\alpha}^2 - r_{\alpha ,i} r_{\alpha ,j}) \\ T &= \sum_{i,j} \frac{1}{2} I_{ij} \Omega_i \Omega_j \\ M_i &= \sum_j I_{ij} \Omega_j \end{aligned}\]
\[\begin{aligned} T &= \sum_{\alpha} \frac{1}{2} m_{\alpha} (\boldsymbol{\Omega} \times \mathbf{r}_{\alpha}) \cdot (\boldsymbol{\Omega} \times \mathbf{r}_{\alpha}) \\ &= \sum_{\alpha} \frac{1}{2} m_{\alpha} [\Omega^2 r_{\alpha}^2 - (\boldsymbol{\Omega}\cdot\mathbf{r}_{\alpha})^2] \\ &= \sum_{i,j}\sum_{\alpha} \frac{1}{2} m_{\alpha} (\delta_{ij} r_{\alpha}^2 - r_{\alpha ,i} r_{\alpha ,j}) \Omega_i \Omega_j \\ \end{aligned} \\ \begin{aligned} \mathbf{M} &= \sum_{\alpha} m_{\alpha} \mathbf{r}_{\alpha} \times (\mathbf{\Omega} \times \mathbf{r}_{\alpha}) \\ &= \sum_{\alpha} m_{\alpha} [\mathbf{\Omega} (r_{\alpha}^2) - \mathbf{r}_{\alpha} (\mathbf{r}_{\alpha} \cdot \Omega)] \\ M_i &= \sum_j \sum_{\alpha} m_{\alpha} (\delta_{ij} r_{\alpha}^2 - r_{\alpha ,i} r_{\alpha ,j}) \Omega_j \end{aligned}\]
Euler’s Equation / Eulerの運動方程式
\[\begin{aligned} I_1 \dot{\Omega}_1 + (I_3 - I_2) \Omega_2 \Omega_3 &= N_1 \\ I_2 \dot{\Omega}_2 + (I_1 - I_3) \Omega_3 \Omega_1 &= N_2 \\ I_3 \dot{\Omega}_3 + (I_2 - I_1) \Omega_1 \Omega_2 &= N_3 \end{aligned}\]
\[\frac{d\mathbf{M}}{dt} = \frac{d'\mathbf{M}}{dt} + \boldsymbol{\Omega} \times \mathbf{M} = \mathbf{N} \\ \begin{bmatrix} I_1 \dot{\Omega}_1 \\ I_2 \dot{\Omega}_2 \\ I_3 \dot{\Omega}_3 \end{bmatrix} + \begin{bmatrix} \Omega_1 \\ \Omega_2 \\ \Omega_3 \end{bmatrix} \times \begin{bmatrix} I_1 \Omega_1 \\ I_2 \Omega_2 \\ I_3 \Omega_3 \end{bmatrix} = \begin{bmatrix} N_1 \\ N_2 \\ N_3 \end{bmatrix}\]
Symmetrical Top (Free) / 対称コマ（自由）
\[\begin{aligned} \theta &= \text{const.} \\ \dot{\phi} &= \frac{M_z}{I_1} \\ \dot{\psi} &= M_z \cos \theta \left( \frac{1}{I_3} - \frac{1}{I_1} \right) \end{aligned}\]
\[L = \frac{1}{2} I_1 (\dot{\phi}^2 \sin^2 \theta + \dot{\theta}^2) + \frac{1}{2} I_3 (\dot{\phi} \cos \theta + \dot{\psi})^2 \\ \begin{aligned} \frac{\partial L}{\partial \dot{\psi}} &= I_3 (\dot{\phi} \cos \theta + \dot{\psi}) = M_3 = M_z \cos \theta \\ \frac{\partial L}{\partial \dot{\phi}} &= I_1 \dot{\phi} \sin^2 \theta + I_3 (\dot{\phi} \cos \theta + \dot{\psi}) \cos \theta = M_z \end{aligned} \\ \begin{aligned} \dot{\phi} &= \frac{M_z - M_z \cos^2 \theta}{I_1 \sin^2 \theta} = \frac{M_z}{I_1} \\ \dot{\psi} &= \frac{M_z \cos \theta}{I_3} - \dot{\phi} \cos \theta = M_z \cos \theta \left( \frac{1}{I_3} - \frac{1}{I_1} \right) \end{aligned} \\ \begin{aligned} \frac{\partial L}{\partial \theta} &= I_1 \dot{\phi}^2 \sin \theta \cos \theta - I_3 (\dot{\phi} \cos \theta + \dot{\psi}) \dot{\phi} \sin \theta \\ &= \frac{M_z^2}{I_1} \sin \theta \cos \theta - \frac{M_z^2}{I_1} \sin \theta \cos \theta = 0 \\ &= \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) = \frac{d}{dt} (I_1 \dot{\theta}) \quad (\dot{\theta} = \text{const.}) \end{aligned} \\ E = \frac{1}{2} I_1 \left( \frac{M_z^2}{I_1^2} \sin^2 \theta + \dot{\theta}^2 \right) + \frac{1}{2} I_3 \frac{M_z^2}{I_3^2} \cos^2 \theta \\ \quad \theta = \text{const.}\]
Symmetrical Top (Gravity) / 対称コマ（重力）
\[\begin{aligned} \theta_1 &\leq \theta \leq \theta_2 \\ \dot{\phi}& = \frac{M_z - M_3 \cos \theta}{(I_1 + Ma^2) \sin^2 \theta} \\ \dot{\psi} &= \frac{M_3}{I_3} - \frac{(M_z - M_3 \cos \theta) \cos \theta}{(I_1 + Ma^2) \sin^2 \theta} \end{aligned}\]
\[L = \frac{1}{2} (I_1 + Ma^2) (\dot{\phi}^2 \sin^2 \theta + \dot{\theta}^2) + \frac{1}{2} I_3 (\dot{\phi} \cos \theta + \dot{\psi})^2 - Mga \cos \theta \\ \begin{aligned} \frac{\partial L}{\partial \dot{\psi}} &= I_3 (\dot{\phi} \cos \theta + \dot{\psi}) = M_3 \\ \frac{\partial L}{\partial \dot{\phi}} &= (I_1 + Ma^2) \dot{\phi} \sin^2 \theta + I_3 (\dot{\phi} \cos \theta + \dot{\psi}) \cos \theta = M_z \end{aligned} \\ \begin{aligned} \dot{\phi} &= \frac{M_z - M_3 \cos \theta}{(I_1 + Ma^2) \sin^2 \theta} \\ \dot{\psi} &= \frac{M_3}{I_3} - \frac{(M_z - M_3 \cos \theta) \cos \theta}{(I_1 + Ma^2) \sin^2 \theta} \end{aligned} \\ E = \frac{1}{2} (I_1 + Ma^2) \dot{\theta}^2 + \frac{1}{2} \frac{(M_z - M_3 \cos \theta)^2}{(I_1 + Ma^2) \sin^2 \theta} + \frac{1}{2} \frac{M_3^2}{I_3} + Mga \cos \theta \\ u = \cos \theta \quad \dot{\theta}^2 = \frac{\dot{u}^2}{1 - u^2} \\ E = \frac{1}{2} (I_1 + Ma^2) \frac{\dot{u}^2}{1 - u^2} + \frac{1}{2} \frac{(M_z - M_3 u)^2}{(I_1 + Ma^2)(1 - u^2)} + \frac{1}{2} \frac{M_3^2}{I_3} + Mga u \\ \dot{u}^2 = (1 - u^2) \left( A - B u \right) - \frac{(M_z - M_3 u)^2}{(I_1 + Ma^2)^2} \geq 0 \\ f(\cos \theta_1) = f(\cos \theta_2) = 0 \\ \theta_1 \leq \theta \leq \theta_2\]
Asymmetrical Top (Free) / 非対称コマ（自由）
\[\sqrt{2EI_1} \leq M \leq \sqrt{2EI_3} \\ \Omega_1: \text{stable} \quad \Omega_2: \text{unstable} \quad \Omega_3: \text{stable}\]
\[M^2 = M_1^2 + M_2^2 + M_3^2 \\ E = \frac{M_1^2}{2I_1} + \frac{M_2^2}{2I_2} + \frac{M_3^2}{2I_3} \\ \sqrt{2EI_1} \leq M \leq \sqrt{2EI_3} \\ \delta \Omega_1 = a e^{\lambda t} \quad \delta \Omega_2 = b e^{\lambda t} \quad \delta \Omega_3 = c e^{\lambda t} \\ \begin{aligned} \Omega_1 : \quad \lambda^2 &= \frac{-\Omega_1^2(I_3 - I_1)(I_2 - I_1)}{I_2 I_3} < 0, &&\text{stable} \\ \Omega_2 : \quad \lambda^2 &= \frac{\Omega_2^2(I_3 - I_2)(I_2 - I_1)}{I_1 I_3} > 0, &&\text{unstable} \\ \Omega_3 : \quad \lambda^2 &= \frac{-\Omega_3^2(I_3 - I_2)(I_3 - I_1)}{I_1 I_2} < 0, &&\text{stable} \end{aligned}\]
Continuum Mechanics / 連続体力学
Fundamental Equation / 基礎方程式
\[\frac{DQ}{Dt} = \frac{\partial Q}{\partial t} + \mathbf{v} \cdot \nabla Q \\ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \\ \rho \frac{D \mathbf{v}}{Dt} = \mathbf{f} + \nabla \cdot \sigma\]
\[\frac{d}{dt} \int_V \rho \, dV = \int_V \frac{\partial \rho}{\partial t} \, dV = -\oint_{\partial V} \rho \mathbf{v} \cdot d\mathbf{S} = -\int_V \nabla \cdot (\rho \mathbf{v}) \, dV \\ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \\ \begin{aligned} \frac{d}{dt} \int_V \rho v_i \, dV &= \int_V \frac{\partial (\rho v_i)}{\partial t} \, dV \\ &= - \oint_{\partial V} \rho v_i \mathbf{v} \cdot d\mathbf{S} + \int_V f_i \, dV + \oint_{\partial V} \boldsymbol{\sigma}_i \cdot d\mathbf{S} \\ &= - \int_V \nabla \cdot (\rho v_i \mathbf{v}) \, dV + \int_V f_i \, dV + \int_V \nabla \cdot \boldsymbol{\sigma}_i \, dV \end{aligned} \\ \begin{aligned} \frac{\partial (\rho v_i)}{\partial t} + \nabla \cdot (\rho v_i \mathbf{v}) &= \frac{\partial \rho}{\partial t} v_i + \rho \frac{\partial v_i}{\partial t} + (\nabla \cdot \rho \mathbf{v}) v_i + \rho (\mathbf{v} \cdot \nabla) v_i \\ &= \rho \left( \frac{\partial v_i}{\partial t} + (\mathbf{v} \cdot \nabla) v_i \right) + \left( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) \right) v_i \\ &= f_i + \nabla \cdot \boldsymbol{\sigma}_i \end{aligned} \\ \rho \frac{D \mathbf{v}}{Dt} = \mathbf{f} + \nabla \cdot \sigma\]
Elastic Body / 弾性体
\[E_{ij} = \frac{1}{2} \left( \frac{\partial r_j}{\partial x_i} + \frac{\partial r_i}{\partial x_k} \right) \\ {\sigma}_{ij} = \lambda \delta_{ij} E_{kk} + 2 \mu E_{ij} \\ \rho \frac{D \mathbf{v}}{Dt} = \mathbf{f} + (\lambda + \mu) \nabla (\nabla \cdot \mathbf{r}) + \mu \nabla^2 \mathbf{r}\]
\[\begin{aligned} \rho \frac{Dv_i}{Dt} &= f_i + \frac{\partial {\sigma}_{ij}}{\partial x_j} \\ &= f_i + \lambda \frac{\partial}{\partial x_i} \left( \frac{\partial r_k}{\partial x_k} \right) + \mu \frac{\partial}{\partial x_j} \left( \frac{\partial r_j}{\partial x_i} + \frac{\partial r_i}{\partial x_j} \right) \\ &= f_i + (\lambda + \mu) \frac{\partial}{\partial x_i} (\nabla \cdot \mathbf{r}) + \mu \nabla^2 r_i \end{aligned}\]
Fluid / 流体
\[\dot{E}_{ij} = \frac{1}{2} \left( \frac{\partial v_j}{\partial x_i} + \frac{\partial v_i}{\partial x_j} \right) \\ {\sigma}_{ij} = - p \delta_{ij} + \lambda \delta_{ij} \dot{E}_{kk} + 2 \eta \dot{E}_{ij} \\ \rho \frac{D \mathbf{v}}{Dt} = \mathbf{f} - \nabla p + (\lambda + \eta) \nabla (\nabla \cdot \mathbf{v}) + \eta {\nabla}^2 \mathbf{v}\]
\[\begin{aligned} \rho \frac{D v_i}{Dt} &= f_i + \frac{\partial {\sigma}_{ij}}{\partial x_j} \\ &= f_i - \frac{\partial p}{\partial x_i} + \lambda \frac{\partial}{\partial x_i} \left( \frac{\partial v_k}{\partial x_k} \right) + \eta \frac{\partial}{\partial x_j} \left( \frac{\partial v_j}{\partial x_i} + \frac{\partial v_i}{\partial x_j} \right) \\ &= f_i - \frac{\partial p}{\partial x_i} + (\lambda + \eta) \frac{\partial}{\partial x_i} (\nabla \cdot \mathbf{v}) + \eta {\nabla}^2 v_i \end{aligned}\]
Bernoulli’s Principle / Bernoulliの定理
\[\rho \mathbf{v} \cdot \mathbf{s} = \text{const.} \\ p + \frac{1}{2} \rho v^2 + \rho gz = \text{const.}\]
\[\lambda = \eta = 0 \quad \frac{\partial \rho}{\partial t} = \frac{\partial \mathbf{v}}{\partial t} = 0 \\ \begin{aligned} \int_{V} \frac{\partial \rho}{\partial t} \, dV &= \int_{V} \nabla \cdot (\rho \mathbf{v}) \, dV \\ &= \oint_{\partial V} \rho \mathbf{v} \cdot d\mathbf{S} \\ &= \int_{S_1} \rho \mathbf{v} \cdot d\mathbf{S} - \int_{S_2} \rho \mathbf{v} \cdot d\mathbf{S} = 0 \end{aligned} \\ \rho \mathbf{v} \cdot \mathbf{s} = \text{const.} \\ \begin{aligned} \rho \frac{D\mathbf{v}}{Dt} &= \rho \left( \mathbf{v} \cdot \nabla \right) \mathbf{v} \\ &= \rho \nabla \left(\frac{v^2}{2}\right) - \rho \nabla \times \left( \nabla \times \mathbf{v} \right) \\ &= -\nabla \left( \rho gz \right) - \nabla p \end{aligned} \\ \nabla \left( p + \frac{1}{2} \rho v^2 + \rho gz \right) = \rho \mathbf{v} \times \left( \nabla \times \mathbf{v} \right) \\ \mathbf{v} \cdot \nabla \left( p + \frac{1}{2} \rho v^2 + \rho gz \right) = 0 \\ p + \frac{1}{2} \rho v^2 + \rho gz = \text{const.}\]
Special Relativity / 特殊相対性理論
Lorentz Transformation / Lorentz変換
\[g_{\mu \nu} dx^{\mu} dx^{\nu} = \text{const.} \\ \begin{aligned} t' &= \frac{t - \frac{v}{c^2} x}{\sqrt{1 - \frac{v^2}{c^2}}} \\ x' &= \frac{x - vt}{\sqrt{1 - \frac{v^2}{c^2}}} \\ y' &= y \\ z' &= z \end{aligned}\]
\[g = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \quad X = \begin{pmatrix} ct \\ x \\ y \\ z \end{pmatrix} \\ X' = L X \\ (dX')^T g (dX') = (L dX)^T g (L dX) = (dX)^T g (dX) \\ L^T g L = g \\ L = \begin{pmatrix} \cosh \omega & \sinh \omega & 0 & 0 \\ \sinh \omega & \cosh \omega & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \\ \begin{pmatrix} ct' \\ 0 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} \cosh \omega & \sinh \omega & 0 & 0 \\ \sinh \omega & \cosh \omega & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct \\ vt \\ 0 \\ 0 \end{pmatrix} \\ \tanh \omega = -\frac{v}{c} \quad \cosh \omega = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \quad \sinh \omega = \frac{-\frac{v}{c}}{\sqrt{1 - \frac{v^2}{c^2}}}\]
Consequence of Transformation / 変換の帰結
\[\begin{aligned} \tau &= \frac{\tau_0}{\sqrt{1 - \frac{v^2}{c^2}}} \\ \ell &= \ell_0 \sqrt{1 - \frac{v^2}{c^2}} \\ v_x &= \frac{v_x' + v}{1 + \frac{v}{c^2} v_x'} \\ v_y &= \frac{\sqrt{1 - \frac{v^2}{c^2}} v_y'}{1 + \frac{v}{c^2} v_x'} \end{aligned}\]
\[\Delta x' = 0 \quad \Delta t' = \tau_0 \\ \Delta t = \frac{\Delta t' - \frac{v}{c^2} \Delta x'}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{\Delta t'}{\sqrt{1 - \frac{v^2}{c^2}}} \\ \Delta t = 0 \quad \Delta x' = \ell_0 \\ \Delta x' = \frac{\Delta x - v \Delta t}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{\Delta x}{\sqrt{1 - \frac{v^2}{c^2}}} \\ v_x = \frac{dx}{dt} = \frac{dx' + v dt'}{dt' + \frac{v}{c^2} dx'} = \frac{v_x' + v}{1 + \frac{v}{c^2} v_x'} \\ v_y = \frac{dy}{dt} = \frac{\sqrt{1 - \frac{v^2}{c^2}} \, dy'}{dt' + \frac{v}{c^2} dx'} = \frac{\sqrt{1 - \frac{v^2}{c^2}} v_y'}{1 + \frac{v}{c^2} v_x'}\]
Relativistic Dynamics / 相対論的力学
\[\delta S = \delta \int -mc \sqrt{g_{\mu \nu} dx^{\mu} dx^{\nu}} = 0 \\ \begin{aligned} \mathbf{p} &= \frac{m\mathbf{v}}{\sqrt{1 - \frac{v^2}{c^2}}} \\ E &= \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} = \sqrt{p^2 c^2 + m^2 c^4} \end{aligned}\]
\[\begin{aligned} S &= \int -mc \sqrt{c^2 dt^2 - dx^2 - dy^2 - dz^2} \\ &= \int -mc^2 \sqrt{1 - \frac{v^2}{c^2}} \, dt \end{aligned} \\ L = -mc^2 \sqrt{1 - \frac{v^2}{c^2}} \\ \mathbf{p} = \frac{\partial L}{\partial \mathbf{v}} = \frac{m\mathbf{v}}{\sqrt{1 - \frac{v^2}{c^2}}} \\ \begin{aligned} E &= \mathbf{p} \cdot \mathbf{v} - L \\ &= \frac{m v^2}{\sqrt{1 - \frac{v^2}{c^2}}} + mc^2 \sqrt{1 - \frac{v^2}{c^2}} \\ &= \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}} \end{aligned} \\ p^2 c^2 + m^2 c^4 = \frac{m^2 v^2 c^2}{1 - \frac{v^2}{c^2}} + \frac{m^2 c^4 - m^2 c^2 v^2}{1 - \frac{v^2}{c^2}} = E^2\]
Four-Vector / ４元ベクトル
\[\mathbf{A'} = L \mathbf{A} \\ \begin{aligned} \mathbf{X} &= (ct, x, y, z) \\ \mathbf{V} &= (\frac{c}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_x}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_y}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_z}{\sqrt{1 - \frac{v^2}{c^2}}}) \\ \mathbf{P} &= (\frac{E}{c}, p_x, p_y, p_z) \end{aligned}\]
\[\begin{aligned} d\mathbf{X} &= (cdt, dx, dy, dz) \\ d\tau &= \sqrt{dt^2 - \frac{dx^2 + dy^2 + dz^2}{c^2}} = dt \sqrt{1 - \frac{v^2}{c^2}} = \text{const.} \end{aligned} \\ \begin{aligned} \mathbf{V} &= \frac{d\mathbf{X}}{d\tau} = (\frac{c}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_x}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_y}{\sqrt{1 - \frac{v^2}{c^2}}}, \frac{v_z}{\sqrt{1 - \frac{v^2}{c^2}}}) \\ \mathbf{P} &= m\mathbf{V} = (\frac{E}{c}, p_x, p_y, p_z) \end{aligned}\]