109
2.7. Hamilton–Jacobi theory
and zero outside this region. This yields the probability density
for N − 1 scattering particles:
W N (f N ρ ) =
1 ∞
dk ρ k ρ [˜ τ (k ρ , 0; 0)]
N −1 J 0 (k ρ f N ρ ), (2.299)
2π 0
where f N ρ is magnitude of transverse component of resultant (net)
trajectory displacement, and we have made use of axial symmetry.
Equations (2.295, 2.297, 2.299) represent the solution. It is shown
in [41] that this solution agrees quantitatively with Monte Carlo
simulation for a particular severe case.
The main result of this section is contained in equations (2.283,
2.291) for the six-vector trajectory displacement in phase space.
In general, the integral in (2.291) must be performed numerically, given an assumed form for the initial distribution coordinates σ 0 (χ 0 ). We have further shown that the dimensionality can
be reduced in a straightforward manner by framing the problem
in terms of Fourier transforms. This has enormous practical significance for extracting quantitative values for the components of
trajectory displacement.
2.7 Hamilton–Jacobi theory
The solution to the general dynamical problem in classical mechanics follows directly from Hamilton’s principle of least action
(2.8). An alternative, but completely equivalent formulation exists, in the theory due to Hamilton and Jacobi. This formulation
will prove useful in the following chapter, where we will explore
the correspondence between the classical and quantum mechanical
descriptions of single-particle motion in the presence of a general
electromagnetic potential. This section closely follows the analysis
of Goldstein, et. al. [35].
2.7. Hamilton–Jacobi theory
and zero outside this region. This yields the probability density
for N − 1 scattering particles:
W N (f N ρ ) =
1 ∞
dk ρ k ρ [˜ τ (k ρ , 0; 0)]
N −1 J 0 (k ρ f N ρ ), (2.299)
2π 0
where f N ρ is magnitude of transverse component of resultant (net)
trajectory displacement, and we have made use of axial symmetry.
Equations (2.295, 2.297, 2.299) represent the solution. It is shown
in [41] that this solution agrees quantitatively with Monte Carlo
simulation for a particular severe case.
The main result of this section is contained in equations (2.283,
2.291) for the six-vector trajectory displacement in phase space.
In general, the integral in (2.291) must be performed numerically, given an assumed form for the initial distribution coordinates σ 0 (χ 0 ). We have further shown that the dimensionality can
be reduced in a straightforward manner by framing the problem
in terms of Fourier transforms. This has enormous practical significance for extracting quantitative values for the components of
trajectory displacement.
2.7 Hamilton–Jacobi theory
The solution to the general dynamical problem in classical mechanics follows directly from Hamilton’s principle of least action
(2.8). An alternative, but completely equivalent formulation exists, in the theory due to Hamilton and Jacobi. This formulation
will prove useful in the following chapter, where we will explore
the correspondence between the classical and quantum mechanical
descriptions of single-particle motion in the presence of a general
electromagnetic potential. This section closely follows the analysis
of Goldstein, et. al. [35].