|Jul31-11, 09:59 PM||#1|
distance not defined in phase spaces?
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.
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|Aug1-11, 12:32 AM||#2|
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.
|Aug1-11, 03:04 AM||#3|
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 form-invariant under canonical transformations, which are mathematically speaking symplectomorphism, i.e., differentiable one-to-one 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 quantum-theoretical models, which provides an important quantization method.
For more information, see the nice wikipedia page
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