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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

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?

*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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