Hi, I have some questions concerning Newtonian physics which confuse me, especially the Galilean boosts. First of all, the Galilei group is(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

x^i \rightarrow R^i_{\ j}x^j + v^i t + d^i, \ \ \ \ \ \ \ \ x^0 \rightarrow x^0 + \xi^0

[/tex]

where R is an element of SO(3), v is a constant velocity, d is a constant vector and xi^0 is a constant shift in time. The boost is explicitly (R=I, d=0)

[tex]

x^i \rightarrow x^{'i} = x^i + v^i t

[/tex]

We know that in Newtonian physics we have a spatial metric which is Euclidean (space is flat!). So I have the line element

[tex]

ds^2 = \delta_{ij}dx^i dx^j

[/tex]

Under a boost however, I get via

[tex] dx^{'i} = \frac{\partial x^{'i}}{\partial x^j}dx^j + \frac{\partial x^{'i}}{\partial t}dt = dx^i + v^i dt

[/tex]

that

[tex]

ds^2 \rightarrow \delta_{ij} [dx^i + v^i dt][dx^j + v^j dt] = \delta_{ij}dx^idx^j + 2 \delta_{ij} dx^i v^j dt + \delta_{ij}v^i v^j dtdt

[/tex]

What does it mean for the Euclidean line element NOT to be invariant under Galilei boosts? A same situation I ofcourse have if I calculate the variation of the Lagrangian

[tex]

L = m\delta_{ij}\dot{x}^i \dot{x}^j

[/tex]

under boosts; that becomes a total derivative, but the action is invariant, so that's OK (the Noether charge belonging to boosts has to be adjusted) and I do understand that.

Second, how would I proof that the Newtonian EOM

[tex]

\ddot{x}^i + \frac{\partial \phi}{\partial x^i} = 0

[/tex]

is invariant under boostsinfinitesimally? Physically I understand that it should be invariant, but mathematically I have some problems showing this explicitly. For finite transformations it's quite obvious,

[tex]

\ddot{x}^i \rightarrow \ddot{x}^i, \ \ \ \ \frac{\partial \phi}{\partial x^i} \rightarrow \frac{\partial x^j}{\partial x^i}\frac{\partial \phi}{\partial x^j} = \frac{\partial \phi}{\partial x^i}

[/tex]

but how would one show this infinitesimally?

These may be confusing questions with simple answers, but I don't see it :)

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# Newton and Galilei boosts

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