  Particle-in-cell (PIC) plasma simulations are a productive and valued tool
for the study of nonlinear plasma phenomena, yet there are basic questions
about the simulation methods themselves that remain unanswered. Here we study
one such question: energy and momentum conservation by PIC. We employ both
analysis and simulations of one-dimensional, electrostatic plasmas to
understand why PIC simulations are either energy or momentum conserving but not
both, what the role of numerical stability is in non-conservation, and how do
errors in conservation scale with the numerical parameters. Conserving both
momentum and energy make it possible to model problems such as Jeans' -type
equilibria. Avoiding numerical instability is useful, but so is being able to
identify when its effect on the results may be important. Designing simulations
to achieve the best possible accuracy with the least expenditure of effort
requires results on the scaling of error with the numerical parameters.. Our
results identify the central role of Gauss' law in conservation of both
momentum and energy, and the significant differences in numerical stability and
error scaling between energy-conserving and momentum-conserving simulations.
