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I was wondering if someone would be kind enough to help me understand an example in my class notes:

If we have a Lagrangian:

[tex]L=m(\dot{z}\dot{z^{*}})-V(\dot{z}\dot{z^{*}})[/tex]

where z=x+iy.

Why does it follow that

[tex]Q=X^{i}\frac{{\partial}L}{{\partial}\dot{q}^{i}}[/tex]

is equal to:

[tex]X\frac{{\partial}L}{{\partial}\dot{z}}+X^{*}\frac{{\partial}L}{{\partial}\dot{z^{*}}}[/tex]?

I mean, mathematically that seems wrong, why are we adding the second term (the one with the complex conjugate of [tex]\dot{z}[/tex])

Also, can I check if my understanding of the superscript 'i' is correct - does it correspond to the axes, for example i=1 corresponds to the x axis, i=2 corresponds to the y axis, etc. If z = x + iy, we are no longer talking about a 3 dimensional real space, so how are the superscripts relavent?

And is there a reason why the superscript 'i' has gone in the second line?

thanks