Add derivations of n-body jacobians

Specifically, partial derivatives of the state gradient (so velocities
and accelerations) w.r.t. the state variables (so displacements and
velocities).

_code/notes/diffsim/derivations.py

-from sympy import *
-from sympy.assumptions.assume import global_assumptions
+import sympy as sp
+#from sympy.assumptions.assume import global_assumptions
 
 def print_expr(name, expr):
     print(name + ": ")
-    print(latex(expr))
+    print(sp.latex(expr))
     print("")
 
-t = Symbol('t', real=True)
-dt = Symbol('\Delta t', real=True)
-x = Symbol('x', real=True)
-xn = Function('x_n', real=True)
-f = Function('f', real=True)
+t = sp.Symbol('t', real=True)
+dt = sp.Symbol('\Delta t', real=True)
+x = sp.Symbol('x', real=True)
+xn = sp.Function('x_n', real=True)
+f = sp.Function('f', real=True)
+
+x_1, y_1, x_2, y_2 = sp.symbols("x_1 y_1 x_2 y_2", real=True)
+G, m_2 = sp.symbols("G m_2", real=True)
 
 def euler():
     expr = xn(x) + dt * f(t, xn(x))
-    return simplify(Derivative(expr, x))
+    return sp.simplify(sp.diff(expr, x))
 
 def rk4():
     k1 = dt * f(t, xn(x))
@@ -22,7 +25,24 @@ def rk4():
     k3 = dt * f(t + dt/2, xn(x) + k2/2)
     k4 = dt * f(t + dt, xn(x) + k3)
     expr = xn(x) + k1/6 + k2/3 + k3/3 + k4/6
-    return simplify(Derivative(expr, x))
+    return sp.simplify(sp.diff(expr, x))
+
+def nbody():
+    distance = sp.sqrt((x_2 - x_1)**2 + (y_2 - y_1)**2)
+    f12_x = G * (x_2 - x_1) / distance**3
+    f12_y = G * (y_2 - y_1) / distance**3
+    df12_x_x1 = sp.simplify(sp.diff(f12_x, x_1))
+    df12_x_y1 = sp.simplify(sp.diff(f12_x, y_1))
+    df12_y_x1 = sp.simplify(sp.diff(f12_y, x_1))
+    df12_y_y1 = sp.simplify(sp.diff(f12_y, y_1))
+    print_expr("f12_x", f12_x)
+    #print_expr("f12_x", f12_y)
+    print_expr("df12_x_x1", df12_x_x1)
+    print_expr("df12_x_y1", df12_x_y1)
+    print_expr("df12_y_x1", df12_y_x1)
+    print_expr("df12_y_y1", df12_y_y1)
+
 
-print_expr("euler", euler())
-print_expr("rk4", rk4())
+#print_expr("euler", euler())
+#print_expr("rk4", rk4())
+nbody()