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E =
m| q˙|^2 + V (|q|).
Recall that the the kinetic energy of the i-th particle is given by
Ti =
m q˙^2 i
and that the total kinetic energy for the system is T = T 1 + T 2. Bearing this in mind, we calculate:
m q˙^2 =
m 1 m 2 m 1 + m 2
( ˙q 1 − q˙ 2 )^2
m 1 + m 2
(m 2 T 1 + m 1 T 2 − m 1 m 2 q˙ 1 q˙ 2 ). (2)
Note that (1) implies that
q˙i = −
mj mi
q ˙j (3)
for qi equal to q 1 or q 2. Hence
1 2
m 1 m 2 q˙ 1 q˙ 2 = −
m^2 i q˙ i^2 = −miTi
for i = 1, 2. Thus (2) becomes
m 1 + m 2
(m 2 T 1 + m 1 T 2 − m 1 m 2 q˙ 1 q˙ 2 ) =
m 1 + m 2
(m 2 T 1 + m 1 T 2 + (m 1 T 1 + m 2 T 2 )) = T 1 + T 2.
Whence
1 2
m q˙^2 = T 1 + T 2 = T,
and the energy of the system is given by
E = T + V (|q 1 − q 2 |) =
m q˙^2 + V (|q|).
- Establish
J = mq × q˙
where J is the total angular momentum. We work from the right hand side:
mq × q˙ =
m 1 m 2 m 1 + m 2
(q 1 − q 2 ) × ( ˙q 1 − q˙ 2 )
= m 1 m 2 m 1 + m 2
(q 1 × q˙ 1 − q 1 × q˙ 2 − q 2 × q˙ 1 + q 2 × q˙ 2 )
m 1 m 2 m 1 + m 2
q 1 × q˙ 1 +
m 1 m 2
q 1 × q˙ 1 +
m 2 m 1
q 2 × q˙ 2 + q 2 × q˙ 2
m 1 m 2 m 1 + m 2
m 1 + m 2 m 2
q 1 × q˙ 1 +
m 1 + m 2 m 1
q 2 × q˙ 2
= m 1 q 1 × q˙ 1 + m 2 q 2 × q˙ 2 = J 1 + J 2 = J;
where the third equality follows from (3).
Poisson brackets
Let R^2 n^ be the phase space of a particle in Rn, with coordinates qi, pi (1 ≤ i ≤ n). Let C∞(R^2 n) be the set of smooth real-valued functions on R^2 n, which becomes an commutative algebra using pointwise addition and multiplication of functions.
We define the Poisson bracket of functions F, G ∈ C∞(R^2 n) by:
{F, G} =
∑^ n
i=
∂F
∂pi
∂G
∂qi
∂G
∂pi
∂F
∂qi
- Show that Poisson brackets make the vector space C∞(R^2 n) into a Lie algebra. In other words, check the antisymmetry of the bracket:
{F, G} = −{G, F }
the bilinearity of the bracket:
{F, αG + βH} = 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∞(R^2 n) and α, β ∈ R.
To make life a whole lot easier on ourselves we will impose the following conventions:
- Lower indices indicate differentiation with respect to pi, for instance:
Fi =
∂F
∂pi
- Upper indices indicate differentiation with respect to qi, that is:
Gi^ =
∂G
∂qi
