- #1
Regel
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- 0
I don't understand the physical meaning of one of my dimensions/variables.
Let there be a gravitational potential [tex]\phi (x_a), a=1,2,3.[/tex]
Equation of motions of a freely falling particle is:
[tex] \frac{d^2 x_a}{d t} = - \frac{\partial \phi}{\partial x_a}.[/tex]
If there are 2 particles falling, family of paths
[tex]x_a = x_a(u,v)[/tex] (u can be considered the Newtonian absolute time). u is the curve parameter and "v labels the paths".
There are tangent vector fields:
Single particle motion: [tex]U_a = \frac{\partial x_a}{\partial u}[/tex]
and relative motion of the two particles [tex]V_a=\frac{\partial x_a}{\partial v}[/tex].
After this we came up with a couple of equations and the class ended.
I understand u, which could be understood as time. But what is v? What does [tex]\frac{\partial x_a}{\partial v}[/tex] mean? I need some real physical explanation instead of "it's a parameter" or something similar. Just like with u, instead of "curve" I now have time.
And what is this uv-space? How is v related to [tex]x_a[/tex]s.
Let there be a gravitational potential [tex]\phi (x_a), a=1,2,3.[/tex]
Equation of motions of a freely falling particle is:
[tex] \frac{d^2 x_a}{d t} = - \frac{\partial \phi}{\partial x_a}.[/tex]
If there are 2 particles falling, family of paths
[tex]x_a = x_a(u,v)[/tex] (u can be considered the Newtonian absolute time). u is the curve parameter and "v labels the paths".
There are tangent vector fields:
Single particle motion: [tex]U_a = \frac{\partial x_a}{\partial u}[/tex]
and relative motion of the two particles [tex]V_a=\frac{\partial x_a}{\partial v}[/tex].
After this we came up with a couple of equations and the class ended.
I understand u, which could be understood as time. But what is v? What does [tex]\frac{\partial x_a}{\partial v}[/tex] mean? I need some real physical explanation instead of "it's a parameter" or something similar. Just like with u, instead of "curve" I now have time.
And what is this uv-space? How is v related to [tex]x_a[/tex]s.
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