Lagrangians

$$L=T-U$$

• $T=\frac{m({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2)}{2}$ | Kinetisk energi
• $U=U(x,y,z,t)$ | Lägesenergi

Path minimizes the action (integral of $L$)

Euler-Legrange

$$J={\int }_{x_1}^{x_2}f(y,\dot{y},x)dx$$

$\dot{y}=\frac{dy}{dx}$

Euler-Legrange

$\frac{\partial f}{\partial y}-\frac{d}{dt}\frac{\partial f}{\partial \dot{y}}=0$

Lagrangian Mechanics

• Pick coordinates
• Find $T$, $U$ - from that get $L$
• Use Euler-Legrange for $L$
• Get stuff? $\frac{\partial L}{\partial q}-\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}=0$

Ex. Motion of ball thrown vertically

$T=\frac{1}{2}m{\dot{y}}^2$

$U=mgy$

$L=T-U=\frac{1}{2}m{\dot{y}}^2-mgy$

$\frac{\partial L}{\partial y}=-mg$

$\frac{\partial L}{\partial \dot{y}}=m\dot{y}$

$\frac{d}{dt}\frac{\partial L}{\partial \dot{y}}=m\ddot{y}$

i Euler-Legrange ger

$-mg-m\ddot{y}=0$

$\Rightarrow m\ddot{y}=-mg\Rightarrow \ddot{y}=-g$

Ex pendulum

$y=-l\cos \theta$

$x=l\sin \theta$

$U=mgy=-mgl\cos \theta$

$T=\frac{1}{2}m({\dot{x}}^2+{\dot{y}}^2)$

$\dot{x}=\frac{d}{dt}l\sin \theta =l\cos (\theta )\dot{\theta }$

$\dot{y}=\frac{d}{dt}l\cos \theta =-l\sin (\theta )\dot{\theta }$

$T=\frac{1}{2}m(l^2{\cos }^2(\theta ){\dot{\theta }}^2+l^2{\sin }^2(\theta ){\dot{\theta }}^2)=\frac{1}{2}m{l}^2{\dot{\theta }}^2$

$L=\frac{1}{2}m{l}^2{\dot{\theta }}^2+mgl\cos \theta$

$\frac{\partial L}{\partial \theta }=-mgl\sin \theta$

$\frac{\partial L}{\partial \dot{\theta }}=m{l}^2\dot{\theta }$

$\frac{d}{dt}\frac{\partial L}{\partial \dot{\theta }}=m{l}^2\ddot{\theta }$

Euler-Legrange: $\frac{\partial L}{\partial \theta }-\frac{d}{dt}\frac{\partial L}{\partial \dot{\theta }}=0$

$\Rightarrow -mgl\sin \theta -m{l}^2\ddot{\theta }=0$

$\Rightarrow \ddot{\theta }=-\frac{g}{l}\sin \theta$