1.2 Classical Mechanics of Two-Particle Collisions
15
inertial coordinate system. The CM set of coordinates is also said to be barycentric
(center-of-gravity). The inertial system has the important property of bearing a null
total linear momentum.
As described in the related caption, the RHS panels ((a) and (b) of Fig. 1.5) compare a collision of two equal masses occurring on the X,Y plane (as we shall see later
this is a collision under the effect of a repulsive interaction) by giving the evolution
of the r A and r B pair in a Lab frame (panel a) and the evolution of r in a CM frame
having the same orientation as the Lab one in (panel b). The latter illustrates graphically the reduction of complexity associated with the decomposition of the r A and r B
problem into a r and a r C M problem, thanks to the introduction of conservation laws
(in our case, the conservation of CM momentum). The two-center problem is, in fact,
transformed into a one-center problem of a particle of mass μ, equal to the reduced
mass of the system, subject to the potential or interparticle interaction as shown by
the use of the only vector r. Accordingly, the number of Hamilton’s equations to be
integrated is reduced from twelve to six, those relating to the three Cartesian components of the position vector r (r x , r y , r z ) (or their respective components of the
polar representation r , ϑ, ψ (see Fig. 1.5)) and the three components of its conjugated
momentum p r AB .
1.2.3 The Deflection Angle θ
For the central field problem under consideration, it is possible to further decompose
the problem using symmetry properties of the system. As already mentioned, in fact,
in the case of the central field, the interaction potential depends only on the magnitude
of the coordinate r and is preferable for convenience and clarity to use systematically,
as we already do, r and p r instead of r AB and p r AB for the diatomic variables. For the
same reasons, we also assume that V (r ) → 0 when r → ∞ (except when explicitly
said), by setting the zero of energy to the asymptotic value of the potential. Then by
choosing, as is done in Fig. 1.7, the orientation of the axes of the system of reference
so that two of them (for example, z and y) lie in the plane determined by the initial
velocity vector, the system will remain confined to the simple trajectory (dψ/dt = 0),
since the potential depends only on the magnitude of the vector r. Following this
transformation, the classical Hamiltonian can be written explicitly in the following
way
5 :
5 In fact, see Fig. 1.7, we have for the components (z, y) of r, z = −r cos θ and y = r sin θ (θ
−π/2 = ϑ) or by differentiating with respect to time
v z ≡
dz
dt
= −˙ r cos θ + r
dθ
dt
sin θ and v y ≡
dy
dt
= ˙
r sin θ + r
dθ
dt
cos θ.
.
15
inertial coordinate system. The CM set of coordinates is also said to be barycentric
(center-of-gravity). The inertial system has the important property of bearing a null
total linear momentum.
As described in the related caption, the RHS panels ((a) and (b) of Fig. 1.5) compare a collision of two equal masses occurring on the X,Y plane (as we shall see later
this is a collision under the effect of a repulsive interaction) by giving the evolution
of the r A and r B pair in a Lab frame (panel a) and the evolution of r in a CM frame
having the same orientation as the Lab one in (panel b). The latter illustrates graphically the reduction of complexity associated with the decomposition of the r A and r B
problem into a r and a r C M problem, thanks to the introduction of conservation laws
(in our case, the conservation of CM momentum). The two-center problem is, in fact,
transformed into a one-center problem of a particle of mass μ, equal to the reduced
mass of the system, subject to the potential or interparticle interaction as shown by
the use of the only vector r. Accordingly, the number of Hamilton’s equations to be
integrated is reduced from twelve to six, those relating to the three Cartesian components of the position vector r (r x , r y , r z ) (or their respective components of the
polar representation r , ϑ, ψ (see Fig. 1.5)) and the three components of its conjugated
momentum p r AB .
1.2.3 The Deflection Angle θ
For the central field problem under consideration, it is possible to further decompose
the problem using symmetry properties of the system. As already mentioned, in fact,
in the case of the central field, the interaction potential depends only on the magnitude
of the coordinate r and is preferable for convenience and clarity to use systematically,
as we already do, r and p r instead of r AB and p r AB for the diatomic variables. For the
same reasons, we also assume that V (r ) → 0 when r → ∞ (except when explicitly
said), by setting the zero of energy to the asymptotic value of the potential. Then by
choosing, as is done in Fig. 1.7, the orientation of the axes of the system of reference
so that two of them (for example, z and y) lie in the plane determined by the initial
velocity vector, the system will remain confined to the simple trajectory (dψ/dt = 0),
since the potential depends only on the magnitude of the vector r. Following this
transformation, the classical Hamiltonian can be written explicitly in the following
way
5 :
5 In fact, see Fig. 1.7, we have for the components (z, y) of r, z = −r cos θ and y = r sin θ (θ
−π/2 = ϑ) or by differentiating with respect to time
v z ≡
dz
dt
= −˙ r cos θ + r
dθ
dt
sin θ and v y ≡
dy
dt
= ˙
r sin θ + r
dθ
dt
cos θ.
.