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I am trying to understand the definition of a Hamiltonian vector field. I take as my symplectic manifold the sphere $$p_1^2+p_2^2+p_3^2=1$$ with symplectic form $$\omega=p_1 dp_2 \wedge dp_3+ p_2 dp_3 \wedge dp_1 + p_3 dp_1 \wedge dp_2.$$ I take as the Hamiltonian $H$ something very simple, namely, $$H(p_1,p_2,p_3)=p_1.$$ Then $$dH=dp_1.$$ By definition, the Hamiltonian vector field $X_H$ is the unique vector field such that $\omega(X_H,\cdot)=dH$. I compute $$\omega(X_H, \cdot)=(p_2 X_3-p_3 X_2) dp_1+(p_3 X_1 -p_1 X_3)dp_2+(p_1 X_2-p_2 X_1)dp_3,$$ so that equating coefficients of $dp_i$, I get $$1=p_2X_3-p_3 X_2$$ $$0=p_3X_1-p_1 X_3$$ $$0=p_1 X_2-p_2 X_1,$$ which is inconsistent. What went wrong?

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You have to solve the equation $dH=\omega(X_H,.)$ on the sphere, not on $R^3$. The form $\omega_H$ has max rank 2 on $R^3$, so in $R^3$ the equation does not has always a solution, you have to write the equation of the tangent of a point of $S^2$ and the equation of a vector field of $S^2$ tangent to each of its point

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