Complex formulation of classical mechanics

In summary, the conversation discusses the possibility of a complex analogue for the Lagrangian in Lagrangian mechanics, where the state of a system is represented by a single complex number. This would result in two independent variables, similar to the vector (x(t),v(t)). However, it is not clear if this formulation would provide any new insights. A potential example is the Complex Elliptic Pendulum, proposed by Carl M. Bender, Daniel W. Hook, and Karta Kooner.
  • #1
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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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  • #2
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.
 
  • #3
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
 

1. What is classical mechanics?

Classical mechanics is a branch of physics that deals with the motion and behavior of macroscopic objects, such as planets, cars, and baseballs. It explains how these objects move under the influence of forces, and is based on three laws of motion proposed by Sir Isaac Newton.

2. What is a complex formulation in classical mechanics?

A complex formulation in classical mechanics refers to the use of complex numbers and mathematical techniques, such as differential equations and complex analysis, to describe and analyze the behavior of physical systems. It is often used to solve problems related to quantum mechanics and relativity.

3. How is the complex formulation different from the traditional formulation of classical mechanics?

The traditional formulation of classical mechanics uses real numbers and simpler mathematical concepts, such as vectors and calculus, to describe the behavior of physical systems. The complex formulation, on the other hand, allows for a more precise and detailed analysis of systems that cannot be described by traditional methods.

4. What are some examples of problems that can be solved using the complex formulation of classical mechanics?

The complex formulation of classical mechanics can be used to solve a variety of problems, such as the behavior of particles in a magnetic field, the motion of a rigid body in three dimensions, and the dynamics of a simple pendulum. It is also used in advanced topics like quantum field theory and string theory.

5. How important is the complex formulation in understanding and advancing our understanding of classical mechanics?

The complex formulation is crucial in understanding and advancing our understanding of classical mechanics. It allows for a more comprehensive and accurate analysis of physical systems, and has played a significant role in the development of modern physics. Without the complex formulation, many of the theories and concepts in classical mechanics would not have been possible.

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