Euler-Lagrange formalism with SymPy












3














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



$$
f = frac{d}{dt} frac{partialmathcal{L}}{partialdot{x}}
- frac{partialmathcal{L}}{partial x}
= mathfrak{m} ddot{x} + mathfrak{m} g.$$



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





euler-lagrange-equation python







share|cite|improve this question





























    3














    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



    $$
    f = frac{d}{dt} frac{partialmathcal{L}}{partialdot{x}}
    - frac{partialmathcal{L}}{partial x}
    = mathfrak{m} ddot{x} + mathfrak{m} g.$$



    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





    euler-lagrange-equation python







    share|cite|improve this question



























      3












      3








      3







      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



      $$
      f = frac{d}{dt} frac{partialmathcal{L}}{partialdot{x}}
      - frac{partialmathcal{L}}{partial x}
      = mathfrak{m} ddot{x} + mathfrak{m} g.$$



      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





      euler-lagrange-equation python







      share|cite|improve this question















      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



      $$
      f = frac{d}{dt} frac{partialmathcal{L}}{partialdot{x}}
      - frac{partialmathcal{L}}{partial x}
      = mathfrak{m} ddot{x} + mathfrak{m} g.$$



      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





      euler-lagrange-equation python




      euler-lagrange-equation python






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      share|cite|improve this question













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      edited Nov 26 at 20:23









      gt6989b

      32.9k22452




      32.9k22452










      asked Nov 26 at 20:21









      Henrique Ferrolho

      1233




      1233






















          1 Answer
          1






          active

          oldest

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          2














          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





          share|cite|improve this answer





















          • I am using Python 3, so the 1 / 2 should not pose a problem. I guess what I did wrong was really the diff(diff(L, xd), 't'). Thank you for your answer!
            – Henrique Ferrolho
            Nov 27 at 0:15










          • @HenriqueFerrolho happy to help
            – caverac
            Nov 27 at 0:43











          Your Answer





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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2














          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





          share|cite|improve this answer





















          • I am using Python 3, so the 1 / 2 should not pose a problem. I guess what I did wrong was really the diff(diff(L, xd), 't'). Thank you for your answer!
            – Henrique Ferrolho
            Nov 27 at 0:15










          • @HenriqueFerrolho happy to help
            – caverac
            Nov 27 at 0:43
















          2














          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





          share|cite|improve this answer





















          • I am using Python 3, so the 1 / 2 should not pose a problem. I guess what I did wrong was really the diff(diff(L, xd), 't'). Thank you for your answer!
            – Henrique Ferrolho
            Nov 27 at 0:15










          • @HenriqueFerrolho happy to help
            – caverac
            Nov 27 at 0:43














          2












          2








          2






          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





          share|cite|improve this 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






          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 26 at 21:02









          caverac

          13.3k21029




          13.3k21029












          • I am using Python 3, so the 1 / 2 should not pose a problem. I guess what I did wrong was really the diff(diff(L, xd), 't'). Thank you for your answer!
            – Henrique Ferrolho
            Nov 27 at 0:15










          • @HenriqueFerrolho happy to help
            – caverac
            Nov 27 at 0:43


















          • I am using Python 3, so the 1 / 2 should not pose a problem. I guess what I did wrong was really the diff(diff(L, xd), 't'). Thank you for your answer!
            – Henrique Ferrolho
            Nov 27 at 0:15










          • @HenriqueFerrolho happy to help
            – caverac
            Nov 27 at 0:43
















          I am using Python 3, so the 1 / 2 should not pose a problem. I guess what I did wrong was really the diff(diff(L, xd), 't'). Thank you for your answer!
          – Henrique Ferrolho
          Nov 27 at 0:15




          I am using Python 3, so the 1 / 2 should not pose a problem. I guess what I did wrong was really the diff(diff(L, xd), 't'). Thank you for your answer!
          – Henrique Ferrolho
          Nov 27 at 0:15












          @HenriqueFerrolho happy to help
          – caverac
          Nov 27 at 0:43




          @HenriqueFerrolho happy to help
          – caverac
          Nov 27 at 0:43


















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