Why is Newton's second law only second-order in position? For instance, could there exist higher-order masses m i with F ( x ) = m x ¨ + ∑ i = 3 ∞ m i x ( i ) ?

Question

Why is Newton&#039;s second law only second-order in position? For instance, could there exist higher-order masses       m    i   with  F  (  x  )  =  m            x      ¨        +      ∑          i      =      3              ∞            m    i        x          (      i      )        ?

kregde84 · Accepted Answer

I think Newton&#039;s first law, linearity of motion, and the time-reversal symmetry strongly constrain the form of Newton&#039;s second law.Generically, the linear relation between the position and the force is given by  F  =      β    0    x  +      β    1              d      x              d      t        +      β    2                      d        2            x              d              t        2              +      β    3                      d        3            x              d              t        3              +  ⋯  .The time-reversal symmetry demands that all odd derivatives vanish,       β          2      k      +      1        =  0 for   k  =  0  ,  1  ,  2  ⋯. The spatial homogeneity demands       β    0    =  0. Therefore,  F  =      β    2                      d        2            x              d              t        2              +      β    4                      d        4            x              d              t        4              +  ⋯  =      β    2    a  +      β    4                      d        2            a              d              t        2              +  ⋯  .Here a is the acceleration. The first law demands that, when F=0, the only solution is the trivial solution a=0. We see that, if all higher coefficients       β    4    ,      β    6    ,  ⋯ are not zero, there will be other solutions. Thus, the only possible form is  F  =      β    2                      d        2            x              d              t        2              .

Why is Newton's second law only second-order in position? For instance, could there exist higher-order masses m_i with F(x)=mx¨+sum_(i=3)^∞m_ix(i)?

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