There is an extremely important problem for which we can obtain the complete solution. It is the *two body problem*, which consists in two interacting particles.
To simplify its approach, we deal with the two particles by considering two motions:
The Lagrangian can be written as
L = (m₁/2)|ṙ₁>² `(m₂/2)|ṙ₂²> - U(||r₁>-|r₂>|) .
The potential energy U depends only on the relative position between the particles (their mutual distance). We can simplify |r₁>-|r₂> into |r>.
We locate the origin at the centre of mass:
m₁|r₁> + m₂|r₂> = |0>.
This allows to write each position as a function of the relative position vector,
m₁|r₁> + m₂(|r₁> - |r>) = |0> |r₁> = m₂/(m₁+m₂) · |r>
and
m₁(|r>+|r₂>) + m₂|r₂> = |0> |r₂> = -m₁/(m₁+m₂) · |r> .
If we substitute these expressions in the Lagrangian, we get
L = (1/2)·m₁·m₂²/(m₁+m₂)·|ṙ>² + (1/2)·m₁²·m₂/(m₁+m₂)·|ṙ>² - U(r) = = (m/2)·|ṙ>² - U(r),
where we define the *reduced mass* m as
m ≡ m₁·m₂ / (m₁+m₂).
Maybe you are like me and prefer to think the reduced mass as
1/m = 1/m₁ + 1/m₂.
This Lagrangian is exactly the one for a single particle under a central field which is symmetrical about a *fixed* origin.
So we reduce the problem of two particles to a problem of one particle under a central field U(r).
Once we obtain |r(t)>, it is very easy to recover |r₁(t)> and |r₂(t)> with the relations we have written.
A system contains one particle of mass M and n particles with equal masses m. Eliminate the motion of the centre of mass in order to reduce the problem to another problem involving n particles instead of n+1.
The radius vector of the mass M is |R> and the positions of the masses m are given by |rₐ>, with a running from 1 to n.
We define the relative positions of the m particles wrt to the M particle as |rₐ> = |Rₐ> - |R>.
The centre of mass is
M|R> + m∑|Rₐ> = |0> => M|R> + m∑(|rₐ> + |R>) = |0> => (M+n·m)|R> + m∑|rₐ> = |0> => |R> = -m/(M+n·m)·∑|rₐ> => |R> = -m/μ·∑|rₐ>,
where μ is the total mass (M+n·m).
The Lagrangian, which initially is
L = (M/2)|Ṙ>² + (m/2)·∑|Ṙₐ>² - U(r),
can be now rewritten as
L = (M/2>|Ṙ>² + (m/2)·∑(|Ṙ>² + 2<Ṙ|ṙₐ> + |ṙₐ>²) - U(r) => => L = ([M+n·m]/2)·(m²/μ²)(∑|vₐ>)² + (m/2)∑|ṙₐ>² - m²/μ(∑|ṙₐ>)·(∑|ṙₐ>) - U(r) => => L = (m/2)∑|ṙₐ>² + m²/(2μ)(∑|vₐ>)² - 2m²/(2μ)(∑|vₐ>)² -U(r) => => L = (m/2)∑|vₐ>² - m²/(2μ)(∑|vₐ>)² -U(r) .