How the Poisson bracket transform when we change coordinates?
Answers
Answered by
0
Hey mate ^_^
For any smooth functions F,H∈C∞(R3)F,H∈C∞(R3) of coordinates X∈R3X∈R3 with volume element d3Xd3X, the Nambu bracket {F,H}{F,H} defined by
−dC∧dF∧dH={F,H}d3X−dC∧dF∧dH={F,H}d3X
is a Poisson bracket for any choice of distinguished smooth function C:R3→RC:R3→R.
#Be Brainly⏬
For any smooth functions F,H∈C∞(R3)F,H∈C∞(R3) of coordinates X∈R3X∈R3 with volume element d3Xd3X, the Nambu bracket {F,H}{F,H} defined by
−dC∧dF∧dH={F,H}d3X−dC∧dF∧dH={F,H}d3X
is a Poisson bracket for any choice of distinguished smooth function C:R3→RC:R3→R.
#Be Brainly⏬
Answered by
3
Hello mate here is your answer.
Every Poisson bracket {⋅,⋅}{⋅,⋅} is associated with a bivector field ΛΛ(see, for instance, this book chapter 4.3):
{f,h}=Λ(df,dh).{f,h}=Λ(df,dh).
What you need to do is to compute ΛΛon the holonomic basis ∂∂Yj∂∂Yj and then apply the rule of transformation for the bivector field ΛΛ when you pass from Cartesian coordinates (Y1,Y2,Y3)(Y1,Y2,Y3) to hyperboloidal ones.
Hope it helps you.
Every Poisson bracket {⋅,⋅}{⋅,⋅} is associated with a bivector field ΛΛ(see, for instance, this book chapter 4.3):
{f,h}=Λ(df,dh).{f,h}=Λ(df,dh).
What you need to do is to compute ΛΛon the holonomic basis ∂∂Yj∂∂Yj and then apply the rule of transformation for the bivector field ΛΛ when you pass from Cartesian coordinates (Y1,Y2,Y3)(Y1,Y2,Y3) to hyperboloidal ones.
Hope it helps you.
Similar questions