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## Main Question or Discussion Point

I have an odd, yet intriguing question. I want to describe a particle that is constrained to move along a straight line in the x direction. The location of the particle can be described with theoretically infinite parametric parameters, d, along the line. For example:

all describe the location of a particle in the x direction. These parameters are dependent on time and are a parametric generalized coordinate of units distance, distance,

[tex]\dot{\bf{R}}= [\dot{d},0,0]^T[/tex]

[tex]\ddot{\bf{R}}= [\ddot{d},0,0]^T[/tex]

and newton's law gives that:

[tex]\dot{\bf{L}} = m*\ddot{\bf{R}}= m\ddot{d} = {F}[/tex]

Simple, easy peasy right? But the other parametric descriptions are more complicated, and don't appear to yield the correct results.

[tex]{\bf{R}}= [{d^2},0,0]^T[/tex]

[tex]\dot{\bf{R}}= [2\dot{d}d,0,0]^T[/tex]

[tex]\ddot{\bf{R}}= [2\ddot{d}d+2\dot{d}^2,0,0]^T[/tex]

[tex]\dot{\bf{L}} = m*\ddot{\bf{R}}= m(2\ddot{d}d+2\dot{d}^2) = {F}[/tex]

which gives the final second order differential equation:

[tex]\ddot{d}= (F-2\dot{d}^2)/(2md)[/tex]

Of interesting note is that this equation can not be solved for zero initial conditions. Also of interest is that the behavior of the system is dependent upon d, which is not the case with the first model of the system. To get the coordinates of the particle, one would of course have to square the numerical solution to the differential equation for d because

[tex]{\bf{R}}= [{d^2},0,0]^T[/tex]

I have also done this derivation utilizing Lagrange's equations and I obtained the exact same differential equation. What am I doing wrong?

**R**= [d 0 0]^{T}**R**= [d^{2}0 0]^{T}**R**= [d^{3}0 0]^{T}all describe the location of a particle in the x direction. These parameters are dependent on time and are a parametric generalized coordinate of units distance, distance,

^{1/2}and distance^{1/3}. I am interested in describing the motion of the particle to a simple input force along the x direction. The first case is very simple:[tex]\dot{\bf{R}}= [\dot{d},0,0]^T[/tex]

[tex]\ddot{\bf{R}}= [\ddot{d},0,0]^T[/tex]

and newton's law gives that:

[tex]\dot{\bf{L}} = m*\ddot{\bf{R}}= m\ddot{d} = {F}[/tex]

Simple, easy peasy right? But the other parametric descriptions are more complicated, and don't appear to yield the correct results.

[tex]{\bf{R}}= [{d^2},0,0]^T[/tex]

[tex]\dot{\bf{R}}= [2\dot{d}d,0,0]^T[/tex]

[tex]\ddot{\bf{R}}= [2\ddot{d}d+2\dot{d}^2,0,0]^T[/tex]

[tex]\dot{\bf{L}} = m*\ddot{\bf{R}}= m(2\ddot{d}d+2\dot{d}^2) = {F}[/tex]

which gives the final second order differential equation:

[tex]\ddot{d}= (F-2\dot{d}^2)/(2md)[/tex]

Of interesting note is that this equation can not be solved for zero initial conditions. Also of interest is that the behavior of the system is dependent upon d, which is not the case with the first model of the system. To get the coordinates of the particle, one would of course have to square the numerical solution to the differential equation for d because

[tex]{\bf{R}}= [{d^2},0,0]^T[/tex]

I have also done this derivation utilizing Lagrange's equations and I obtained the exact same differential equation. What am I doing wrong?