Note that this looks exactly like the energy of a single particle!
3. Let Jbe the total angular momentum of the 2-particle system. Show that
J=mq ×˙q
Note that this looks exactly like the angular momentum of a single particle!
At this point we’re back to a problem you’ve already solved: a single particle in a central force.
The only difference is that now qstands for the relative position and mstands for the reduced mass!
So, we instantly conclude that two bodies orbiting each other due to the force of gravity will
both have an orbit that’s either an ellipse, or a parabola, or a hyperbola... when viewed in the
center-of-mass frame.
2 Poisson brackets
Let R2nbe the phase space of a particle in Rn, with coordinates qi, pi(1 ≤i≤n). Let C∞(R2n)
be the set of smooth real-valued functions on R2n, which becomes an commutative algebra using
pointwise addition and multiplication of functions.
We define the Poisson bracket of functions F, G ∈C∞(R2n) by:
{F, G}=
n
X
i=1
∂F
∂pi
∂G
∂qi
−∂G
∂pi
∂F
∂qi
.
4. Show that Poisson brackets make the vector space C∞(R2n) into a Lie algebra. In other
words, check the antisymmetry of the bracket:
{F, G}=−{G, F }
the bilinearity of the bracket:
{F, αG +β H}=α{F , G}+β{F, H}
{αF +βG, H }=α{F, H}+β{G, H }
and Jacobi identity:
{F, {G, H }} ={{F, G}, H}+{G, {F, H }}
for all F, G, H ∈C∞(R2n) and α, β ∈R.
(Note the Jacobi identity resembles the product rule d(GH) = (dG)H+GdH , with bracketing by
Fplaying the role of d. This is no accident!)
5. Show that Poisson brackets and ordinary multiplication of functions make the vector space
C∞(R2n) into a Poisson algebra. This is a Lie algebra that is also a commutative algebra, with
the bracket {F, G}and the product F G related by the Leibniz identity:
{F, GH }={F, G}H+G{F, H}.
(Again this identity resembles the product rule!)
2
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