Classical Mechanics

Euler-Lagrange Equation

$$\delta I = \int_a^b f(z, \dot{z}, x) dx = 0$$

$$z \rightarrow z + \epsilon n$$

$$\frac{dI}{d\epsilon} = \int_a^b \bigg( \frac{\partial f}{\partial... ...ot{z}}{d\epsilon} + \frac{\partial f}{\partial x}\frac{dx}{d\epsilon} \bigg) dx$$

$$\frac{dI}{d\epsilon} \vert _{\epsilon = 0} = \int_a^b \bigg( \fra... ...artial f}{\partial z} \eta + \frac{\partial f}{\partial\dot{z}} \eta' \bigg) dx$$

Integrate by parts;

$$\int_a^b \bigg( \frac{\partial f}{\partial z} - \frac{d}{dx} \fra... ...\eta dx + \bigg[\frac{\partial f}{\partial\dot{z}}\eta \bigg]^b_a \rightarrow 0$$ if stationary.

$$0 = \int_a^b \bigg(\frac{\partial f}{\partial z} = \frac{d}{dx} \frac{\partial f}{\partial\dot{z}}\eta \bigg) dx$$

$$\boxed{\frac{\partial f}{\partial z} = \frac{d}{dx}\frac{\partial f}{\partial\dot{z}} = 0}$$

A differentiable functional is stationary at a local maximum or minimum.

Lagrange Multipliers
Lagrange Multipliers allow optimization without explicit parameterization. Look for the stationary points of a function $f(x)$ subject to the constraint $g(x) = c$

$$\mathcal{L}(x,\lambda) = f(x) + \lambda g(x)$$ (1)

This can also be formed by modifying Lagrange's equations, rather than the lagrangian itself.

$$\frac{d}{dt} \frac{\partial\mathcal{L}}{\partial\dot{x}} - \frac{\partial\mathcal{L}}{\partial x} = \lambda \frac{\partial f}{\partial x}$$ (2)

Example: Atwood's Machine

$$\mathcal{L} = \frac{1}{2}m_1\dot{x}_1^2 + \frac{1}{2}m_2\dot{x}_2^2 + m_1 g x_1 + m_2 g x_2$$ (3)

$$x_1 + x_2 = l =$$ (4) Turn the E-L crank.

$$m_1 \ddot{x_1} - m_1 g = \lambda$$

$$m_2 \ddot{x_2} - m_2 g = \lambda$$

$$\ddot{x}_1 = - \ddot{x}_2;$$ (5)

$$m_1 \ddot{x}_1 - m_1 g = -m_2 \ddot{x}_1 - m_2 g$$

$$\rightarrow \boxed{\ddot{x}_1 = \frac{(m_1 - m_2) g}{m_1 + m_2}}$$ (6)

This is also of extreme use in statistical mechanics, where lagrange multipliers often appear in the form of the chemical potential $\mu$.

Hamiltonian Mechanics
One can obtain the Hamiltonian via a Legendre transform of the Lagrangian:

$$\mathcal{H}(q,p) = p\dot{q} - \mathcal{L}$$ (7)

The equations of motion are

• $\frac{\partial\mathcal{H}}{\partial p} = \dot{q}$
• $\frac{\partial\mathcal{H}}{\partial q} = -\dot{p}$
• $\frac{\partial\mathcal{H}}{\partial t} = - \frac{\partial\mathcal{L}}{\partial t}$

The Hamiltonian is the cotangent bundle over t.

Poisson Bracket

$$\{A,B\}_{q,p} = \frac{\partial A}{\partial q}\frac{\partial B}{\partial p} - \frac{\partial A}{\partial p}\frac{\partial B}{\partial q}$$ (8)

$$\frac{df}{dt} = \{f,\mathcal{H} \} + \frac{\partial f}{\partial t}$$ (9)

There is a strong relation between the poisson bracket of classical mechanics and the commutator of quantum mechanics.

Moment of Inertia

$$I = \int r^2 dm = \int r^2 \rho(x) dV$$ (10)

$$I_{ij} = \int \rho(x) [\delta^i_j \left\vert\mathbf{x}\right\vert^2 - x_ix_j] d^3x$$ (11)

$$I_{parallel} = I_{cm} + MR^2$$ (12)

• point mass: $I = mr^2$
• rod through center: $I = \frac{1}{12} mL^2$
• rod through end: $I = \frac{1}{3}mL^2$
• thin hoop: $I = \frac{mr^2}{2}$ in x,y
• thin hoop (symmetric): $I = mr^2$
• sphere: $I = \frac{2}{5} mr^2$

Euler Equations
For rigid body rotations:

$$\boxed{\frac{d}{dt} \bigg\vert _I = \frac{d}{dt} \bigg\vert _B + \mathbf{\omega} \times ( \ \ \ ) }$$ (13)

$$\frac{d}{dt} \bigg\vert _I \mathbf{L} = \frac{d}{dt}\bigg\vert _B \mathbf{L} + \mathbf{\omega}\times \mathbf{L}$$

$$\rightarrow \mathbf{\Gamma}^{(e)} = \mathbf{I}\cdot \frac{\mathbf{\omega}}{dt} + \mathbf{\omega} \times \mathbf{I}' \mathbf{\omega}'$$

$$\Gamma_1^{(e)} = I_1 \frac{d\omega_1}{dt} + \omega_2 I_3 \omega_3 - \omega_3 I_2 \omega_2$$

$$\Gamma_1^{(e)} = I_1 \dot{\omega_1} + \omega_2 \omega_3 (I_3 - I_2)$$

$$\boxed{I_i \dot{\omega_i} = \omega_j \omega_k (I_j - I_k) + \Gamma_i^{(e)} }$$ (14)

This is especially useful when motion is constrained to be torque free; the $\Gamma$ dependence drops out and the equations are much simpler.

Euler Angles TODO: PICTURE

• Rotate about $\hat{z}$ by $\alpha$; $y \neq y'$ after rotation.
• Rotate about $\hat{y}'$ by $\beta$
• Rotate around $\hat{z}'$ by $\gamma$

The dashed line is the line of nodes.

$$\boxed{\omega_1 = -\dot{\alpha} \sin{\beta} \cos{\gamma} + \dot{\beta} \cos{\gamma} }$$ (15)

$$\boxed{\omega_2 = \dot{\alpha} \sin{\beta} \sin{\gamma} + \dot{\beta} \cos{\gamma} }$$ (16)

$$\boxed{\omega_3 = \dot{\alpha} \cos{\beta} + \dot{\gamma}}$$ (17)

Note: The angle projections always depend on the previous angle's rate of change, never a later one; i.e., $\dot{\alpha} \cos{\beta}$ will occur, but NOT $\dot{\beta}\cos{\alpha}$

Differential Scattering Cross Section

$$\frac{d\sigma}{d\Omega} = \frac{b}{\sin{\theta}} \left\vert\frac{db}{d\theta}\right\vert$$ (18)

$$d\sigma = 2\pi b db$$

$$d\Omega = 2 \pi \sin{\theta} d\theta$$

$$\textbf{TODO: DIAGRAM}$$

Gauss' Law for Newtonian Gravity

$$\mathcal{L}_{g} = -\rho(\mathbf{x},t)\phi(\mathbf{x},t) - \frac{1}{8\pi G}(\nabla\mathbf{\phi}(\mathbf{x},t))^2$$ (19)

$$\rightarrow -\rho = \frac{2}{8\pi G} \nabla^2{\phi}$$

$$\boxed{\nabla^2\phi = 4\pi G \rho}$$ (20)

$$\boxed{\mathbf{g} = -\nabla\mathbf{\phi}}$$ (21)

$$\nabla\cdot\mathbf{\mathbf{g}} = -4\pi G \rho$$ (22)

$$\oint \nabla\cdot\mathbf{\mathbf{g}} dV$$ (23)

$$= \oint \mathbf{g} \cdot d\mathbf{\sigma} = -4\pi G M$$

Relativistic Doppler Shift

$$p^{\mu} = \{ E, p^x, p^y, 0 \}$$ (24)

$$v_{obs}^{\mu} = \{ \frac{dt}{d\tau}, \frac{dx}{d\tau} \} = \{ \frac{dt}{d\tau}, \gamma v_x \}$$ (25)

$$E = p^{\mu}V_{\mu} = \omega \frac{dt}{d\tau} - \omega \cos{\theta}v_x\gamma$$ (26)

$$\omega' = \omega(\gamma - \cos{\theta}\gamma v_x)$$ (27)

$$\boxed{ \frac{\omega'}{\omega} = \frac{1 - v\cos{\theta}}{\sqrt{1-v^2}} }$$ (28)

$$\boxed{ \frac{\omega'}{\omega} = \sqrt{ \frac{1 - v}{1 + v}}}$$ (29)

Schwarzschild Metric

$$ds^2 = -\bigg( 1 - \frac{2GM}{r} \bigg) dt^2 + \bigg(1 + \frac{2GM}{r} \bigg)^{-1} dr^2 + r^2 d\Omega^2$$ (30)

Gravitational field outside a spherical mass; no charge/ angular momentum. Universal cosmological constant = 0.

FLRW Metric
Friedmann-Lemaître-Robertson-Walker metric for the curvature of the universe:

$$ds^2 = -dt^2 + \frac{dr^2}{1-kr^2} + r^2 d\Omega^2$$ (31)

• k = -1 $\rightarrow$ Hyperbolic curvature (negative). Substitute $r=\sinh{\theta}$.
• k = 0 $\rightarrow$ Flat space. (No need to sub to integrate.)
• k = 1 $\rightarrow$ Elliptical curvature (positive). Sub $r=\sin{\theta}$.