Complex formulation of classical mechanics

[2VtQCxn]

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?

 

[RedX]

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.

 

[bartschaff]

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