4.  Building a simple N-body code

• physical context
  • few-body problem (N >= 2)
  • planetary systems
  • star clusters
  • galaxies
  • molecular dynamics...

• problem to solve: initial value problem for N particles with specified masses, initial positions and velocities, and accelerations
  or with softening:

• starting code framework: nbody_basic.py
• settable parameters: N, seed, eps, dt, t_end, v0
• units and data structures
  • G = 1, M = 1, R ~ 1, V ~ 1 (convention)
  • optionally scale to virial equilibrium (2T + U = 0)
  • numpy arrays for vector syntax and to speed computation
  • time step

• Exercise 4.1: initialization
  • two-body problem (start here)
    • start with two identical particles (masses m = 0.5) in a bound orbit with initial separation r = 2 and transverse velocity v0 = 0.25
  • many-body problem (later)
    • start with N identical particles (masses m = 1/N) distributed in a homogeneous sphere of unit radius, with speed v = v0 = 0.5 and random direction
    • random numbers: numpy.random.seed(), numpy.random.random()
  • solution

• Exercise 4.2: computation of the potential, kinetic, and total energies, and some simple output diagnostics: initially, just print out the time, the energy E, and the energy error (E - E0)
  • note the use of softening in the expression for the potential energy U:

  • sample code
  • solution

• Exercise 4.3: computation of accelerations
  • add a function to compute and return the acceleration; test it by calling it and printing out the result:

    	acc = get_acc(mass, pos, eps**2)
    	print('acc =', acc)

  • note again the inclusion of softening

  • sample code
  • some numpy trivia -- see numpy primer for details
  • solution

• Exercise 4.4: choice and implementation of integration scheme
  • for now, just use the second-order predictor corrector
  • take a single step and print the energy error
  • solution

• Exercise 4.5: run with diagnostics
  • choose softening eps = 0.05, time step dt = 0.001, and run to time t = 10
  • solution

• Exercise 4.6: run forward and back to the starting point to verify time reversibility
  • print the initial and final positions and velocities, and their difference
  • solution

• diagnostics and visualization: nbody_ex6_plot.py and nbody_ex6_anim.py