Changes

Once again, let us begin with a 1-feature alphabet <math>\mathcal{X} = \{ x_1 \} = \{ x \}.</math>  In the discussion that follows I will consider a class of trajectories that are ruled by the constraint that <math>d^k x = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m,\!</math> and I will indulge in the use of some picturesque speech to describes salient classes of such curves.  Given this finite order condition, there is a highest order non-zero difference <math>d^m x\!</math> that is exhibited at each point in the course of any determinate trajectory.  Relative to any point of the corresponding orbit or curve, let us call this highest order differential feature <math>d^m x\!</math> the ''drive'' at that point.  Curves of constant drive <math>d^m x\!</math> are then referred to as ''<math>m^\text{th}\!</math> gear curves''.

Once again, let us begin with a 1-feature alphabet <math>\mathcal{X} = \{ x_1 \} = \{ x \}.</math>  In the discussion that follows I will consider a class of trajectories that are ruled by the constraint that <math>d^k x = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m,\!</math> and I will indulge in the use of some picturesque speech to describes salient classes of such curves.  Given this finite order condition, there is a highest order non-zero difference <math>d^m x\!</math> that is exhibited at each point in the course of any determinate trajectory.  Relative to any point of the corresponding orbit or curve, let us call this highest order differential feature <math>d^m x\!</math> the ''drive'' at that point.  Curves of constant drive <math>d^m x\!</math> are then referred to as ''<math>m^\text{th}\!</math> gear curves''.

One additional piece of notation will be needed here.  Starting from the base alphabet <math>\mathcal{X} = \{ x \},</math> we define and notate <math>\operatorname{E}^j \mathcal{X} = \{ x, d^1 x, d^2 x, \ldots, d^j x \}</math> as the ''<math>j^\text{th}\!</math> order extended alphabet over <math>\mathcal{X}</math>''.

One additional piece of notation will be needed here.  Starting from the base alphabet <math>\mathcal{X} = \{ x \},</math> we define and notate <math>\operatorname{E}^j \mathcal{X} = \{ x, d^1 x, d^2 x, \ldots, d^j x \}</math> as the ''<math>j^\text{th}\!</math> order extended alphabet over <math>\mathcal{X}.</math>''

 

Let us now consider the family of <math>4^\text{th}\!</math> gear curves through the extended space <math>\operatorname{E}^4 X = \langle x, dx, d^2 x, d^3 x, d^4 x \rangle.</math>  These are the trajectories that are generated subject to the law <math>d^4 x = 1,\!</math> where it is understood in making such a statement that all higher order differences are equal to <math>0.\!</math>

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Since d^4.x and all higher order d^j.x are fixed, the entire dynamics

Since d^4.x and all higher order d^j.x are fixed, the entire dynamics

can be plotted in the extended space E^3.X = <|x, dx, d^2.x, d^3.x|>.

can be plotted in the extended space E^3.X = <|x, dx, d^2.x, d^3.x|>.