Poisson Brackets Explained: Understanding the Relationship between {x,p} = 1"

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SUMMARY

The Poisson brackets for the canonical variables {x, p} equal 1, as derived from the formula (dx/dx)(dp/dp) - (dx/dp)(dp/dx). By calculating the derivatives, it is established that dx/dx = 1, dp/dp = 1, dx/dp = 0, and dp/dx = 0, leading to the conclusion that 1 - 0 = 1. The misunderstanding arose from incorrectly assuming that (dx/dp) and (dp/dx) were both equal to 1, which is not the case since x and p are independent variables.

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  • Understanding of Poisson brackets in classical mechanics
  • Familiarity with calculus, specifically derivatives
  • Knowledge of independent variables in physics
  • Basic concepts of Hamiltonian mechanics
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can anyone tell me why the poisson brackets for {x,p} = 1 ..from (dx/dx)(dp/dp) - (dx/dp)(dp/dx)... shouldn this equal 0??
 
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I don't know why you think it should be 0.

Calculate each derivative and plug the values into the formula. You should get 1. Remember that x and p are independent.
 
dx/dx = 1. dp/dp = 1. dx/dp = 0. dp/dx = 0. 1 - 0 = 1. Which of these is not clear to you?
 
oh right, my lecture notes said that (dx/dp) = 1... i then assumed that (dp/dx) = 1.. oh yes and p is velocity dependent, now i get it... thanks!
 
{x,p} = 1 therefore
(dx/dx)(dp/dp) - (dx/dp)(dp/dx)=i
 

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