物理

運動方程式

md2rdt2=Fm \dfrac{d^2 \boldsymbol{r}}{dt^2} = \boldsymbol{F}

$$
m \dfrac{d^2 \boldsymbol{r}}{dt^2} = \boldsymbol{F}
$$

運動エネルギー

K=12m∥v∥2K = \dfrac{1}{2} m \|\boldsymbol{v}\|^2

$$
K = \dfrac{1}{2} m \|\boldsymbol{v}\|^2
$$

仕事とエネルギーの関係式

12mv2−12mv02=∫t0t1(mg)⋅vdt\dfrac{1}{2}mv^2 - \dfrac{1}{2}mv_0^2 = \int_{t_0}^{t_1} (mg) \cdot vdt

$$
\dfrac{1}{2}mv^2 - \dfrac{1}{2}mv_0^2 
= \int_{t_0}^{t_1} (mg) \cdot vdt
$$

ラグランジュの運動方程式

ddt(∂L∂q˙)−∂L∂q=0\dfrac{d}{dt} \left(\dfrac{\partial \mathcal{L}}{\partial \dot{q}} \right) - \dfrac{\partial \mathcal{L}}{\partial q} = 0

$$
\dfrac{d}{dt} \left(\dfrac{\partial \mathcal{L}}{\partial \dot{q}} \right) 
- \dfrac{\partial \mathcal{L}}{\partial q} 
= 0
$$

単原子分子理想気体

U=32nRTU = \dfrac{3}{2}nRT

$$
U = \dfrac{3}{2}nRT
$$

状態方程式

pV=nRTpV = nRT

$$
pV = nRT
$$

正弦波

y(x,t)=Asin⁡{ω(t−xc)+α}y(x, t) = A \sin \{\omega \left(t - \dfrac{x}{c} \right) + \alpha \}

$$
y(x, t) = A \sin \{\omega \left(t - \dfrac{x}{c}  \right) + \alpha \}
$$

定常波

y(x,t)=2acos⁡(2πxλ+α2−α12)sin⁡(2πft+α2+α12)y(x,t) = 2a \cos \left( 2 \pi \dfrac{x}{\lambda} + \dfrac{\alpha_2 - \alpha_1}{2} \right) \sin \left( 2 \pi ft + \dfrac{\alpha_2 + \alpha_1}{2} \right)

$$
y(x,t) = 2a 
\cos \left( 2 \pi \dfrac{x}{\lambda} + \dfrac{\alpha_2 - \alpha_1}{2} \right) 
\sin \left( 2 \pi ft + \dfrac{\alpha_2 + \alpha_1}{2} \right)
$$

マクスウェル方程式

∇⋅E=ρε0\nabla \cdot \boldsymbol E = \dfrac{\rho}{\varepsilon_{0}}

$$
\nabla \cdot \boldsymbol E 
= \dfrac{\rho}{\varepsilon_{0}}
$$

∇×E=−∂B∂t\nabla \times \boldsymbol E = - \dfrac{\partial \boldsymbol B}{\partial t}

$$
\nabla \times \boldsymbol E 
= - \dfrac{\partial \boldsymbol B}{\partial t}
$$

∇⋅B=0\nabla \cdot \boldsymbol B = 0

$$
\nabla \cdot \boldsymbol B = 0
$$

∇×B=μ0i+1c2∂E∂t\nabla \times \boldsymbol B = \mu_{0}i + \dfrac{1}{c^2} \dfrac{\partial \boldsymbol E}{\partial t}

$$
\nabla \times \boldsymbol B 
= \mu_{0}i + \dfrac{1}{c^2} \dfrac{\partial \boldsymbol E}{\partial t}
$$

ガウスの定理

∭V∇⋅A dV=∬∂VA⋅n dS\iiint_{V} \nabla \cdot \boldsymbol A \, dV = \iint_{ \partial V} \boldsymbol A \cdot \boldsymbol n \,dS

$$
\iiint_{V} \nabla \cdot \boldsymbol A \, dV 
= \iint_{ \partial V} \boldsymbol A \cdot \boldsymbol n \,dS
$$

シュレディンガー方程式

iℏ∂∂tψ(r,t)=(−ℏ22m∇2+V(r,t))ψ(r,t)i \hbar \dfrac{\partial}{\partial t} \psi(r,t) = \left(- \dfrac{\hbar^2}{2m} \nabla^2+ V(r,t) \right) \psi(r,t)

$$
i \hbar \dfrac{\partial}{\partial t} \psi(r,t) 
= \left(- \dfrac{\hbar^2}{2m} \nabla^2+ V(r,t) \right) \psi(r,t)
$$

ナビエ-ストークス方程式

ρ{∂v∂t+(v⋅∇)v}=−∇p+μ∇2v+ρf\rho \left\{\dfrac{\partial{\boldsymbol{v}}}{\partial{t}} + \left(\boldsymbol{v} \cdot \nabla \right) \boldsymbol{v}\right\} = -\nabla p + \mu \nabla^2 \boldsymbol{v} + \rho \boldsymbol{f}

$$
\rho \left\{\dfrac{\partial{\boldsymbol{v}}}{\partial{t}} + \left(\boldsymbol{v} \cdot \nabla \right) \boldsymbol{v}\right\} 
= -\nabla p + \mu \nabla^2 \boldsymbol{v} + \rho \boldsymbol{f}
$$

Einsteinの重力波方程式

□ϕμν=−16πGc4Tμν\Box \phi_{\mu \nu} = - \dfrac{16 \pi G}{c^4} T_{\mu \nu}

$$
\Box \phi_{\mu \nu} = - \dfrac{16 \pi G}{c^4} T_{\mu \nu}
$$