Difference between identical transformation and canonical transformation
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Contact transformations were discovered by Sophus Lie in the 19th century. Within this context an infinitesimal homogeneous (time independent) contact transformation:
δqi=∂H∂piδt,δpi=−∂H∂qiδtδqi=∂H∂piδt,δpi=−∂H∂qiδt
is a coordinate transformation that leaves the system of equations:
Δ=∣∣∣∣dp1,…,dpnp1,…,pndq1,…,dqn∣∣∣∣=0,∑ipidqi=0Δ=|dp1,…,dpnp1,…,pndq1,…,dqn|=0,∑ipidqi=0
invariant [1]. In this context we can interchange contact with canonical according to Qmechanic's answer.
In the context of differential geometry, we make a distinction between symplectic transformations on dim(2n)dim(2n) symplectic manifolds and contact transformations on dim(2n+1)dim(2n+1) contact manifolds. This extends the time independent formulation into an extended phase space (time dependent). [2] We must now take care on how we use the phrase contact.
In both symplectic and contact frameworks, we can define a canonical structure,
θ=pdq,Θ=pdq−Hdtθ=pdq,Θ=pdq−Hdt
respectively, that becomes invariant under their respective transformations.
δqi=∂H∂piδt,δpi=−∂H∂qiδtδqi=∂H∂piδt,δpi=−∂H∂qiδt
is a coordinate transformation that leaves the system of equations:
Δ=∣∣∣∣dp1,…,dpnp1,…,pndq1,…,dqn∣∣∣∣=0,∑ipidqi=0Δ=|dp1,…,dpnp1,…,pndq1,…,dqn|=0,∑ipidqi=0
invariant [1]. In this context we can interchange contact with canonical according to Qmechanic's answer.
In the context of differential geometry, we make a distinction between symplectic transformations on dim(2n)dim(2n) symplectic manifolds and contact transformations on dim(2n+1)dim(2n+1) contact manifolds. This extends the time independent formulation into an extended phase space (time dependent). [2] We must now take care on how we use the phrase contact.
In both symplectic and contact frameworks, we can define a canonical structure,
θ=pdq,Θ=pdq−Hdtθ=pdq,Θ=pdq−Hdt
respectively, that becomes invariant under their respective transformations.
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