# Distance not defined in phase spaces?

1. Jul 31, 2011

### wdlang

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. Aug 1, 2011

### khemist

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. Aug 1, 2011

### vanhees71

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.