Complex formulation of classical mechanics

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The discussion explores the relationship between classical mechanics and complex numbers, particularly in the context of the Lagrangian formulation. It suggests that while one can represent the state of a system as a complex number (x + iv), this does not yield a new or useful formulation since the resulting Lagrangian remains non-analytic. The participants question the meaning of transforming a vector from R to C and whether such a transformation could provide new insights into Lagrangian mechanics. Ultimately, the consensus is that this approach does not offer significant advantages, as it merely replicates the existing variables (x, v) without adding new information. The mention of a complex elliptic pendulum paper indicates some existing research in this area, though its relevance is uncertain.
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Looking at a path of system state (x(t),v(t)) as a vector, the Lagrangian strangely is a scalar function of pairs of coordinates of the vector.

If, on the other hand, the complete state of a system was captured in a single complex number x+iv, a complex analogue of the Lagrangian would simply transform a vector R->C into another vector R->C (vaguely reminiscent of the symmetry of Poisson brackets).

Is there a formulation of Lagrangian mechanics that does something like this?
 
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I'm not sure I understand what R->C means?

I'm pretty sure you can write (x,v) as an imaginary number x+iv. However, you don't really get anything out of it since your Lagrangian will not be analytic, and you'll have two independent variables x+iv and it's conjugate, which is the same as having two variables (x,v).

So I don't think such a formulation gives you anything new.
 
A quick search showed me that, no idea how serious it is:\
Complex Elliptic Pendulum
Carl M. Bender, Daniel W. Hook, Karta Kooner
http://arxiv.org/abs/1001.0131
 

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