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I'm given a particle of mass $m$ moving in a circle of radius $a$, under a central force.

Its potential energy, $U(r)=m \lambda r^4$ (where $\lambda>0$).

I'm trying to find the total energy and the specific angular momentum (and the period).

When I try to calculate the total energy using $E=E_0=T_0+U_0=\frac{1}{2}mv_0^2+m \lambda r_0^4$ , I don't know $r_0$ or $v_0$.

Could someone point me in the right direction?

Thanks

beep-boop
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1 Answers1

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In this case the kinetic and potential energy combine to form a lagrangian function $L=T-U$ you can use the Euler-Lagrange equation to define a dynamical system(differential equation). Solving this allows you to find $r(t),v(t)$. You can find the derivation and details of the EL equation online but in your context it is: $\frac{d}{dt}(L_v)-L_r=0$ The total energy is $H = T+U$ and this is valid for all $t$. I suppose the learning objective for this is to understand that this is the truth. Also, The angular momentum, usually denoted $L$, can be defined as ${\bf C} = r\times p= r\times mv=m r\times v$. I used $C$ to distinguish it from the Lagrangian.

There is also an excellent page on circular motion on wikipedia:http://en.wikipedia.org/wiki/Circular_motion you can find the relationships between the radius, velocity, acceleration and momentum for both the agular and radial parts there.

JEM
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