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GR0177 #74
Problem
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Mechanics$\Rightarrow$}Lagrangians

The Lagrangian equation of motion is given by $\frac{\partial L}{\partial q} = \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}$ for the generalized coordinate q.

Chunking out the derivatives, one finds that

$\frac{\partial L}{\partial q}=4bq^3$

$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}=2a\ddot{q}$

Setting the two equal to each other as in the Lagrangian equations of motion given above (without undetermined multipliers), one finds that $2bq^3=a\ddot{q}$, which gives choice (D).

Alternate Solutions
 insertphyspun2011-02-21 14:39:38 Say one does not remember the Lagrangian equation of motion (maybe you should memorize it now). One can still get the answer by conservation of energy.rnrn$\frac{dE}{dt}=0$rnrn$\frac{dE}{dt}=2a\dot{q}\ddot{q}-4bq^3\dot{q}$rnrnSolve for $\ddot{q}$ and find $\ddot{q}=2\frac{b}{a}q^3$. Answer D.Reply to this comment
 antonis2013-09-14 09:29:15 Just another way - I think - recalling that the Lagrangian usually takes the form $L=T-V$. This Lagrangian could bring us in mind a body with kinetic energy $T=\frac{1}{2}m\dot{q}^2=a \dot{q}^2$ in a potential well $V=-bq^4$, meaning that the force exerted to the body would be $F=-\frac{\partial V}{\partial q}=4bq^3} \Rightarrow \ddot{q}=\frac{4bq^3}{m}=\frac{2b}{a}q^3$Reply to this comment insertphyspun2011-02-21 14:39:38 Say one does not remember the Lagrangian equation of motion (maybe you should memorize it now). One can still get the answer by conservation of energy.rnrn$\frac{dE}{dt}=0$rnrn$\frac{dE}{dt}=2a\dot{q}\ddot{q}-4bq^3\dot{q}$rnrnSolve for $\ddot{q}$ and find $\ddot{q}=2\frac{b}{a}q^3$. Answer D.Reply to this comment

Just another way - I think - recalling that the Lagrangian usually takes the form $L=T-V$. This Lagrangian could bring us in mind a body with kinetic energy $T=\frac{1}{2}m\dot{q}^2=a \dot{q}^2$ in a potential well $V=-bq^4$, meaning that the force exerted to the body would be $F=-\frac{\partial V}{\partial q}=4bq^3} \Rightarrow \ddot{q}=\frac{4bq^3}{m}=\frac{2b}{a}q^3$
LaTeX syntax supported through dollar sign wrappers $, ex.,$\alpha^2_0$produces $\alpha^2_0$. type this... to get...$\int_0^\infty$$\int_0^\infty$$\partial$$\partial$$\Rightarrow$$\Rightarrow$$\ddot{x},\dot{x}$$\ddot{x},\dot{x}$$\sqrt{z}$$\sqrt{z}$$\langle my \rangle$$\langle my \rangle$$\left( abacadabra \right)_{me}$$\left( abacadabra \right)_{me}$$\vec{E}$$\vec{E}$$\frac{a}{b}\$ $\frac{a}{b}$