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Distance not defined in phase spaces? 
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#1
Jul3111, 09:59 PM

P: 307

is it meaningful to define a distance between two points in a phase space?
it is interesting that we can define volume in a phase space but not distance it seems that it is useless to define the distance between two points as the euclidean distance. 


#2
Aug111, 12:32 AM

P: 267

My teacher last semester claimed that a phase space is a convenient way to represent things which are periodic. He emphasized that it is nothing more than a way to visualize this that are happening, such as the lag/ lead of current and voltage.
Now that I think about it, the phase space could be some sort of complex plane, but I am not quite sure. I gather that they deal with rotations more than they deal with length or distance. 


#3
Aug111, 03:04 AM

Sci Advisor
Thanks
P: 2,548

The phase space in Hamiltonian mechanics lives on the cotangent bundle of the configuration space. The additional mathematical feature, very important for physics, is not a metric but a symplectic form in terms of Poisson brackets on the space of differentiable functions on phase space. Hamiltonian mechanics is forminvariant under canonical transformations, which are mathematically speaking symplectomorphism, i.e., differentiable onetoone mappings (diffeomorphisms) which leave the Poisson brackets invariant.
Another important feature is that this structure builds a Lie algebra which at the same time is a derivation algebra and thus gives rise to representations of symmetries, which can be mapped easily to quantumtheoretical models, which provides an important quantization method. For more information, see the nice wikipedia page http://en.wikipedia.org/wiki/Hamilto...tonian_systems 


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