Global Invariants for Variable-Mass Systems

Abstract

We investigate the effect of mass loss on the invariants of single particle motion in potentials having translational or rotational symmetry. These systems are non-Hamiltonian in the physical momenta, but for non-velocity-dependent potentials can be made formally Hamiltonian by treating the velocities as canonical momenta. Applying Noether's theorem to the formal Hamiltonian then yields global invariants corresponding to its symmetries. For velocity-dependent potentials, an exact invariant is constructed from the (non-Hamiltonian) vector field. The results are applied to charged particle motion in a magnetic dipole and agree well with numerical calculations.

Particle dynamics with variable mass and charge

“…There are several other forces to consider, perhaps the most important being solar radiation pressure, which breaks the axisymmetry. The orbits of grains larger than about one micron are gravity dominated and are nearly unchanged by decreasing mass, although their total energy and angular momentum are proportional to m(t) [11]. Very small grains, with radii less than about 100 nm are magnetic dominated and strongly affected by changing q/m.…”

“…In order to demonstrate the usefulness of the quasistatic effective potential, Fig. 7 shows three snapshots of an adiabatic orbit with v = 0.075 and χ 0 = 30 • , from t = 0-100, t = 400-500, and t = 900-1000, along with the corresponding energy boundaries found fromq and withp φ given by the adiabatic formula (11). The shrinking orbit is seen to closely follow the moving confinement boundary given by U e (ρ, z; t) = E, but does not reach ∂ E , owing to magnetic moment conservation.…”

“…As we shall see, having an explicit form for p φ (t) allows definition of a quasistatic effective potential which confines orbits. Our previous method for finding p φ (t) was much more complex, requiring the evaluation of the bounce-averaged drift velocity [11].…”

“…This was done in Ref. [11] by analyzing the equations of motion for gravitational, electrostatic, and magnetostatic fields in both linear and axial symmetry. For axisymmetric gravitational fields the question is essentially trivial, as the orbits are independent of particle mass due to the equality of gravitational and inertial mass.…”

“…In a recent paper [11] we investigated the effects of slow isotropic mass loss on charged particle motion in gravitational E-mail address: jhoward@colorado.edu. and electromagnetic fields having translational or rotational symmetry.…”