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Given any smooth function $F$, we can define a Hamiltonian vector field $X_F$, and a corresponding flow $\varphi_{X_F}$. Given some parameter value $t$, $\varphi_{X_F}(\Omega, t)$ constitutes a canonical transformation (i.e. a change of coordinates). We would then call $F$ the generator of the transformation $\varphi_{X_F}$.

 

Note the similarity to quantum mechanics: Given any self-adjoint operator $F$, we can define a family of unitary operators $\exp[iFt]$. Given some parameter value $t$, $\exp[iFt]$ constitutes a unitary transformation (i.e. a change of basis). We would then call $F$ the generator of the transformation $\exp[iFt]$.

 

You ask specifically about exponentiation - the set of flows generated by Hamiltonian vector fields is called the Hamiltonian Symplectomorphism Group, whose Lie algebra is given by the Hamiltonian vector fields. In the sense of Lie theory, one obtains the flow by exponentiating the corresponding Hamiltonian vector field.

Given any smooth function $F$, we can define a Hamiltonian vector field $X_F$, and a corresponding flow $\varphi_{X_F}$. Given some parameter value $t$, $\varphi_{X_F}(\Omega, t)$ constitutes a canonical transformation (i.e. a change of coordinates). We would then call $F$ the generator of the transformation $\varphi_{X_F}$.

Note the similarity to quantum mechanics: Given any self-adjoint operator $F$, we can define a family of unitary operators $\exp[iFt]$. Given some parameter value $t$, $\exp[iFt]$ constitutes a unitary transformation (i.e. a change of basis). We would then call $F$ the generator of the transformation $\exp[iFt]$.

You ask specifically about exponentiation - the set of flows generated by Hamiltonian vector fields is called the Hamiltonian Symplectomorphism Group, whose Lie algebra is given by the Hamiltonian vector fields. In the sense of Lie theory, one obtains the flow by exponentiating the corresponding Hamiltonian vector field.

Note the similarity to quantum mechanics: Given any self-adjoint operator $F$, we can define a family of unitary operators $\exp[iFt]$. Given some parameter value $t$, $\exp[iFt]$ constitutes a unitary transformation (i.e. a change of basis). We would then call $F$ the generator of the transformation $\exp[iFt]$.

You ask specifically about exponentiation - the set of flows generated by Hamiltonian vector fields is called the Hamiltonian Symplectomorphism Group, whose Lie algebra is given by the Hamiltonian vector fields. In the sense of Lie theory, one obtains the flow by exponentiating the corresponding Hamiltonian vector field.

Note the similarity to quantum mechanics: Given any self-adjoint operator $F$, we can define a family of unitary operators $\exp[iFt]$. Given some parameter value $t$, $\exp[iFt]$ constitutes a unitary transformation (i.e. a change of basis). We would then call $F$ the generator of the transformation $\exp[iFt]$.

You ask specifically about exponentiation - the set of flows generated by Hamiltonian vector fields is called the Hamiltonian Symplectomorphism Group, whose Lie algebra is given by the Hamiltonian vector fields. In the sense of Lie theory, one obtains the flow by exponentiating the corresponding Hamiltonian vector field.

Let's apply this to the momentum observable. $\mathcal P(q,p,t)=p$, so $$X_{\mathcal P} = \frac{\partial}{\partial q}$$ The integral curves can be found via $$\frac{d}{ds} \Gamma(s) = X_{\mathcal P}\big|_{\Gamma(s)}$$ which, in component form where $\Gamma(s) = (\Gamma^1(s),\Gamma^2(s))$, reads $$\frac{d\Gamma^1}{ds} \frac{\partial}{\partial q} = \frac{\partial}{\partial q}$$ $$\frac{d\Gamma^2}{ds} \frac{\partial}{\partial p} = 0$$ which implies that, starting from the initial condition that $\Gamma(0)=(q_0,p_0)$, $$\Gamma^1(s) = q_0 + s$$ $$\Gamma^2(s) = p_0$$ From there, the corresponding flow is $$\varphi_{X_\mathcal P}(q,p,s) = (q+s,p)$$

Let's apply this to the momentum observable. $\mathcal P(q,p,t)=p$, so $$X_{\mathcal P} = \frac{\partial}{\partial q}$$ The integral curves can be found via $$\frac{d}{ds} \Gamma(s) = X_{\mathcal P}\big|_{\Gamma(s)}$$ which, in component form where $\Gamma(s) = (\Gamma^1(s),\Gamma^2(s))$, reads $$\frac{d\Gamma^1}{ds} \frac{\partial}{\partial q} = \frac{\partial}{\partial q} \implies \frac{d\Gamma^1}{ds} = 1$$ $$\frac{d\Gamma^2}{ds} \frac{\partial}{\partial p} = 0 \implies \frac{d\Gamma^2}{ds} = 0$$ which implies that, starting from the initial condition that $\Gamma(0)=(q_0,p_0)$, $$\Gamma^1(s) = q_0 + s$$ $$\Gamma^2(s) = p_0$$ From there, the corresponding flow is $$\varphi_{X_\mathcal P}(q,p,s) = (q+s,p)$$