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1 Galilean Symmetry and its Conserved Quantity
Last time we discovered there was a symmetry called Galilean symmetry, but we did not know a corresponding conserved quantity. Given n particles in R^3 interacting via central forces, if qi: R → R^3 is a solution of Newton’s 2nd^ law, we get a new solution
q˜i(t) = qi(t) + tv
where v ∈ R^3. This is called Galilean symmetry; Galilean symmetries form a group, R^3. What are the conserved quantities? Our system of particles has a total mass:
m =
∑^ n
i=
mi
and a center of mass
q(t) =
miqi(t) m
We have also discussed the total momentum
p(t) =
∑^ n
i=
pi(t)
which is also conserved. Note: p(t) = m q˙(t)
so the center of mass moves at a constant velocity, so:
q(t) + q(0) + tv
for some v ∈ R^3. So q(t) − tv ∈ R^3
is a conserved quantity! This is “center of mass at time zero” - this is the conserved quantity corresponding to Galilean symmetry.
q(t) − tv =
miqi(t) m
t
mi q˙i(t) m
Compare this to total momentum: p(t) =
mi q˙i(t).
Note: the center of mass at time zero has “explicit time dependence” - not just a function of qi(t) and ˙qi(t).
2 Hamilton’s Equations
Let’s just consider a single particle in Rn, with position
q: R → Rn
satisfying newton’s 2nd^ law:
m q¨i(t) =
∂V
∂qi
(q(t))
for some potential V : Rn^ → R. This equation is 2nd-order, so you an rewrite it as a pair of 1st-order equations:
q˙i(t) =
m pi(t) (∗∗)
p˙i(t) =
∂V
∂qi
(q(t))
describing the rate of position and momentum - these are “equal partners” in the Hamiltonian approach. The right-hand side is related to energy
E =
m q˙^2 + V (q)
p^2 2 m
The Hamiltonian H: Rn^ × Rn^ → R is the energy as a function of p ∈ Rn, q ∈ Rn:
H(q, p) =
p^2 2 m
Note: ∂H ∂pi
(q, p) = pi m ∂H ∂qi
(q, p) =
∂V
∂qi
So, (**) are equivalent to Hamilton’s equations:
d dt
qi(t) =
∂H
∂pi
(q(t), p(t))
d dt
pi(t) = −
∂H
∂qi
(q(t), p(t))
This pattern reminds of us rotating by 90 degrees in the plane or multiplying by i. This is the secret expanation of what is going on!
3 Poisson Brackets
We call Rn^ the phase space of a particle in n-dimensions - a point in it specifies the particles position and momentum (q, p) ∈ Rn^ × Rn.
