Complex formulation of classical mechanics

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SUMMARY

The discussion centers on the potential formulation of Lagrangian mechanics using complex numbers, specifically representing the state of a system as a complex number x + iv. Participants express skepticism regarding the utility of this approach, noting that while it mathematically transforms a vector from R to C, it does not yield a new analytic Lagrangian. The reference to the "Complex Elliptic Pendulum" by Carl M. Bender et al. suggests that while there are explorations in this area, the consensus is that such formulations do not provide significant advantages over traditional methods.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with complex numbers and their properties
  • Knowledge of vector transformations in physics
  • Basic grasp of Poisson brackets and their significance
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  • Research the implications of complex variables in classical mechanics
  • Study the "Complex Elliptic Pendulum" paper by Carl M. Bender et al.
  • Explore the relationship between Lagrangian mechanics and complex analysis
  • Investigate alternative formulations of mechanics that utilize complex numbers
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The discussion is beneficial for physicists, mathematicians, and students interested in advanced mechanics, particularly those exploring the intersection of classical mechanics and complex analysis.

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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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