First approach to forming equations of motion for a simple particle using Lagrange's method

gistfile1.txt

# Form Lagrangian only with symbols
from sympy import symbols, diff, S, Eq
t, m, g, x, v = symbols('t m g x v')

# Kinetic and Potential energies
KE = 1/S(2)*m*v**2
PE = m*g*x

# Lagrangian
L = KE - PE

# Unforced equation of motion
eom1 = Eq(diff(L, x) - diff(diff(L, v), t), 0)

# Note that derivative of diff(L, v) w.r.t. t is zero since we didn't write KE
# as:
# KE = 1/S(2)*m*v(t)**2
print eom1

# If we write KE as:
# KE = 1/S(2)*m*v(t)**2
# Then, diff(L, v) == 0, and diff(L, v(t)) doesn't work because diff can only
# differentiate w.r.t. to a symbol

# Alternatively, we can make us of the __call__ method of Symbol to make x a
# function of t
v = diff(x(t), t)

# Kinetic and Potential energies
KE = 1/S(2)*m*v**2
PE = m*g*x(t)

# Lagrangian
L = KE - PE

# But this won't work because diff can't differentiate w.r.t to a Symbol
eom1 = Eq(diff(L, x(t)) - diff(diff(L, v), t), 0)