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#h-a #TODO

DEFINITION

Alternative to Lagrange Equation WHEN $\nexists$ Potential $V$ ⇒ None OR Holonomic Constrain.

Bead on a wire is rotating ($\theta =\omega \cdot t$) on a plain. Due to Angular Acceleration → It's being shoot out

> > 1. **Find Constrains** > - $z=0$ > - $\frac{y}{x}=\tan(\theta)=\tan(\omega\cdot t)$ > 2. **Find Degree of Freedom**: > - Radius $r$ > 3. **Write Transformation Equation** → $\begin{cases} x=r\cdot\cos(\theta)=r\cdot\cos(\omega\cdot t)\\ y=r\cdot\sin(\theta)=r\cdot\sin(\omega\cdot t)\end{cases}$ > 4. **$\exists$ Potential $V$?** No CAUSE no other applied Forces → I can write ELE > 5. **Write ELE** > 1. **Find Lagrangian** → $L=K-\cancel{V}$ = $\frac{1}{2}m\cdot(\dot r^2+r^2\cdot\dot\theta^2)$ = $\frac{1}{2}m\cdot(\dot r^2+r^2\cdot\omega^2)$ > 2. ELE $\frac{d}{dt}\frac{\partial L}{\partial\dot{q}_\alpha}- \frac{\partial L}{\partial q_\alpha}=0$ → $m\cdot\ddot r-m\cdot r\cdot \omega^2=0$ → ==$r(t)=A\cdot e^{\omega\cdot t}+B\cdot e^{-\omega\cdot t}$== > > It's a System that depends on "Initial Conitions"

#ASK

generic Solution
$\sum _{\alpha }{V}_{\gamma ,\alpha }\cdot {\ddot{\eta }}_{\alpha }+V_{\gamma ,\alpha }\cdot {\eta }_{\alpha }=0$

"RAZIO"

0. Take Lagrange Equation $$\frac{d}{dt}\frac{\partial T}{\partial \dot{q}\alpha}- \frac{\partial T}{\partial q\alpha}=Q_\alpha$$
1. IF I can write Applied Forces through Potential Energy $V$ → ${\overrightarrow{F}}_i=-{\overrightarrow{\mathrm{\nabla }}}_i\cdot V$ ⇒ $\mathrm{\exists }$ Generalized Component of Applied Force $Q_{\alpha }$ = $\dots$ = $-\frac{\partial V}{\partial q_{\alpha }}$
2. Substitute it in Lagrange Equation → $\frac{d}{dt}\frac{\partial T}{\partial {\dot{q}}_{\alpha }}-\frac{\partial T}{\partial q_{\alpha }}=-\frac{\partial V}{\partial q_{\alpha }}$ → $\frac{d}{dt}\frac{\partial }{\partial {\dot{q}}_{\alpha }}(T-V)-\frac{\partial }{\partial q_{\alpha }}(T-V)=0$
3. Knowing Lagrangian $L=T-V$ → $\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_{\alpha }}-\frac{\partial L}{\partial q_{\alpha }}=0$

#h-t LINKED