We can learn a lot about a system (without doing much work) if the potential energy is a
homogenous function of the coordinates. This is when the potential satisfies the relationship
$V(\alpha r_1, \alpha r_2, ..., \alpha r_N) = \alpha^k V(r_1, r_2, ..., r_N)$.
(1)
In the above equation, the potential is called a homogenous function \"of degree k.\" Through-
out this problem, let us assume the potential does not depend on velocity and is homoge-
neous of degree k in the coordinates. We will keep standard Cartesian coordinates $r_i$ as our
generalized coordinates and assume we have no constraints.
1. Let us imagine that we take this system and multiply all the positions of the bodies
by a constant $\alpha$, that is $r_i \to \alpha r_i$. Then, we take the time and scale it by a factor $\beta$,
that is $t \to \beta t$. By what factor does the kinetic energy T scale? By what factor does
the potential energy V scale?
2. What is the relationship between $\alpha$, $\beta$, and k such that this transformation leaves the
equations of motion unchanged?
3. Let us denote the scaled system by primes, that is $t' = \beta t$ and $r'_i = \alpha r_i$, where $\alpha$ and $\beta$
are related by the answer above in 1.2. We can conclude that any path length $l'$ taken
in the scaled system is longer than in the original system path length $l$ by a factor $\alpha$.
Compute the ratio of the times between the scaled and unscaled systems $t'/t$ in terms
of the ratio $l'/l$ and k only.
