Complex formulation of classical mechanics

AI Thread Summary
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.
2VtQCxn
Messages
8
Reaction score
1
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?
 
Last edited:
Physics news on Phys.org
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
 

Thread 'Derivation of Hamilton's Principle: Questions'

I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
View full post »

Similar threads

A Recover Hamilton equation from 2-form defined on phase space
Replies
3
Views
1K
Why do we need the Lagrangian formulation of Mechanics?
Replies
8
Views
2K
Effective potential in a central field
Replies
3
Views
2K
B Thoughts about Galilean transformations
Replies
13
Views
2K
Difference between Hamiltonian and Lagrangian Mechanics
Replies
20
Views
10K
Motivation for Lagrangian mechanics
Replies
2
Views
2K
What Is The Meaning Of Newton's Laws
Replies
10
Views
2K
I Classical Field Theory - Something isn't clicking
Replies
12
Views
1K
I Question on derivation of a property of Poisson brackets
Replies
0
Views
1K
Can the kinetic energy be a function of the position vector?
Replies
8
Views
4K

Hot Threads

Recent Insights

Back
Top