[Math] Euler-Lagrange formalism with SymPy

euler-lagrange-equationpython

I am trying to use SymPy to apply the Euler-Lagrange formalism to the following Lagrangian function:

$$\mathcal{L}(x, \dot{x}) = \mathcal{K}(x, \dot{x}) – \mathcal{P}(x) = \dfrac{1}{2} \mathfrak{m} \dot{x} – \mathfrak{m} g x.$$

The result should be

$$
f = \frac{d}{dt} \frac{\partial\mathcal{L}}{\partial\dot{x}}
– \frac{\partial\mathcal{L}}{\partial x}
= \mathfrak{m} \ddot{x} + \mathfrak{m} g.$$

However, I get $f = \mathfrak{m} g$ with SymPy…

Below is the source of my jupyter notebook.

In [1]: from sympy import *
        from sympy.physics.mechanics import *

In [2]: m,g = symbols('m g', real=True)
        x   = dynamicsymbols('x')
        xd  = dynamicsymbols('x', 1)
        xdd = dynamicsymbols('x', 2)

In [3]: kin_energy = 1/2 * m * xd ** 2
        pot_energy = m * g * x

In [4]: L = kin_energy - pot_energy
        f = diff(L, xdd) - diff(L, x)

In [5]: pprint(f)
Out[5]: g⋅m

Best Answer

There are a couple of errors in your code

1/2

Depending on the version of python you're using, this could be interpreted as either '0' or '1/2'. So To avoid problems just use

kin_energy = 0.5 * m * xd ** 2
pot_energy = m * g * x

${\rm d}/{\rm d}t(\partial L/ \partial \dot{x})$

Note that

$$ \frac{{\rm d}}{{\rm d}t}\frac{\partial \mathcal{L}}{\partial \dot{x}} \not= \frac{\partial \mathcal{L}}{\partial \ddot{x}} $$

If you implement these two things, you should get

m = symbols('m', real = True)
g = symbols('g', real = True)
x   = dynamicsymbols('x')
xd  = dynamicsymbols('x', 1)

kin_energy = 0.5 * m * xd ** 2
pot_energy = m * g * x

L = kin_energy - pot_energy
f = diff(diff(L, xd), 't') - diff(L, x)
pprint(f)

>>>           2      
             d       
g⋅m + 1.0⋅m⋅───(x(t))
              2      
            dt
Related Question