Newtonian Mechanics — formula sheet

\[ \mathbf{F}' = m\,\mathbf{a}' \quad\Longleftrightarrow\quad \mathbf{F} = m\,\mathbf{a} \]

\[ W_{\text{net}} = \int_{A}^{B}\vec{F}\cdot d\vec{r} = \frac{1}{2}mv_B^{2}-\frac{1}{2}mv_A^{2} = \Delta K \]

\[ \oint_{\mathcal{C}}\vec{F}\cdot d\vec{r}=0 \iff \nabla\times\vec{F}=\vec{0} \iff \vec{F}=-\nabla U,\qquad W_{A\to B}=U(A)-U(B) \]

\[ E = T + U = \tfrac{1}{2}mv^2 + U(\vec{r}) = \text{const}, \qquad \frac{dE}{dt}=0 \]

\[ \frac{d\vec{P}}{dt}=\vec{F}^{\text{ext}}\qquad\xrightarrow{\;\vec{F}^{\text{ext}}=\vec{0}\;}\qquad \vec{P}=\sum_{i} m_i\vec{v}_i=\text{const} \]

\[ M\,\ddot{\vec{R}}=\vec{F}^{\,\text{ext}} \]

\[ \frac{d\vec{L}}{dt}=\vec{\tau}=\vec{r}\times\vec{F},\qquad \vec{F}\ \text{central}\ \Longrightarrow\ \vec{L}=\vec{r}\times\vec{p}=\text{const} \]

\[ M\ddot{\mathbf{R}} = \mathbf{0}, \qquad \mu\,\ddot{\mathbf{r}} = -\nabla_{\mathbf{r}}V(\mathbf{r}), \qquad \mu=\frac{m_1 m_2}{m_1+m_2} \]

\[ \tfrac12\mu\dot r^2+\underbrace{U(r)+\frac{L^2}{2\mu r^2}}_{U_{\text{eff}}(r)}=E,\qquad \frac{d^2u}{d\theta^2}+u=-\frac{\mu}{L^2u^2}\,F\!\left(\tfrac1u\right),\ \ u=\frac1r. \]

\[ r(\theta)=\frac{a(1-e^2)}{1+e\cos\theta} \quad\Big|\quad \frac{dA}{dt}=\frac{L}{2\mu} \quad\Big|\quad T^2=\frac{4\pi^2 a^3}{G(M+m)} \]

\[ \ddot q=-\omega^2 q,\qquad \omega=\sqrt{\frac{U''(x_0)}{m}},\qquad T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{U''(x_0)}} \]

\[ A(\omega)=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}},\qquad \tan\delta=\frac{\gamma\omega}{\omega_0^2-\omega^2},\qquad \omega_{\text{res}}=\sqrt{\omega_0^2-\tfrac12\gamma^2},\qquad Q=\frac{\omega_0}{\gamma} \]

\[ \mathbf{K}\,\mathbf{a}^{(r)}=\omega_r^2\,\mathbf{M}\,\mathbf{a}^{(r)},\qquad \mathbf{q}(t)=\sum_{r=1}^{n}\mathbf{a}^{(r)}\big(A_r\cos\omega_r t+B_r\sin\omega_r t\big),\qquad \big(\mathbf{a}^{(r)}\big)^{\mathsf T}\mathbf{M}\,\mathbf{a}^{(s)}=\delta_{rs} \]

\[ \Delta v = v_{\mathrm{ex}}\,\ln\!\left(\frac{m_0}{m_f}\right) \]

\[ \vec{L}=\mathbf{I}\,\vec{\omega},\qquad I_{jk}=\sum_i m_i\big(r_i^2\,\delta_{jk}-x_{i,j}x_{i,k}\big)=\int \rho(\vec{r})\big(r^2\,\delta_{jk}-x_j x_k\big)\,dV,\qquad \mathbf{I}=\mathbf{I}^{\mathsf T} \]

\[ I_i\dot{\omega}_i-(I_j-I_k)\,\omega_j\omega_k=\tau_i \qquad \Omega=\frac{I_3-I_1}{I_1}\,\omega_3 \qquad K=\frac{(I_3-I_1)(I_1-I_2)}{I_2 I_3}\,\omega_0^{2} \]

\[ m\,\vec{a}_R = \vec{F} \;\underbrace{-\,2m\,\vec{\omega}\times\vec{v}_R}_{\text{Coriolis}} \;\underbrace{-\,m\,\vec{\omega}\times(\vec{\omega}\times\vec{r})}_{\text{centrifugal}} \;\underbrace{-\,m\,\dot{\vec{\omega}}\times\vec{r}}_{\text{Euler}} \]

\[ \boxed{\;\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot q_j}\right) - \frac{\partial L}{\partial q_j} = 0, \qquad L = T - V, \qquad j=1,\dots,n\;} \]

\[ y(x) = x\tan\theta - \frac{g\,x^2}{2v_0^2\cos^2\theta},\qquad T = \frac{2v_0\sin\theta}{g},\qquad H = \frac{v_0^2\sin^2\theta}{2g},\qquad R = \frac{v_0^2\sin 2\theta}{g} \]

\[ a_c = \frac{v^2}{R} = \omega^2 R, \qquad F_c = \frac{m v^2}{R} = m\omega^2 R, \qquad T = \frac{2\pi R}{v} \]

\[ \begin{aligned} &\text{Static:} \quad \tan\theta \le \mu_s \\ &\text{Sliding down:} \quad a = g(\sin\theta - \mu_k\cos\theta) \\ &\text{Sliding up:} \quad a = -g(\sin\theta + \mu_k\cos\theta) \\ &\text{Work by friction:} \quad W_f = -\mu_k mg\cos\theta\, d \end{aligned} \]

\[ \vec{g} = -\frac{GM}{r^2}\,\hat{\vec{r}},\quad \phi = -\frac{GM}{r},\quad U = -\frac{GMm}{r},\quad v_{\text{circ}} = \sqrt{\frac{GM}{r}},\quad v_{\text{esc}} = \sqrt{\frac{2GM}{R}} \]