Correspondence between hamiltonian flows Suppose I have an hamiltonian function $H$ that is fiberwise homogenenous of degree 2 , i.e, $H(q,sp)=s^2H(q,p)$,in $(T^*M,\omega)$ the cotangent manifold with the canonical symplectic structure and a path $\gamma(t)=(q(t),p(t))$ such that $\dot \gamma=X_H(\gamma)$, I want to see that we define $\gamma_c(t)=(q(t),\frac{1}{c}p(t))$ we will have that $\dot \gamma_c=X_{cH}(\gamma_c)$. Now in my attempt to do this first we have that $X_{cH}$ is defined such that $\omega_{c\gamma}(X_{cH},.)=d_{c\gamma}cH(.)=2cH(q(t),\frac{1}{c}p(t))=\frac{2}{c}H(q(t),p(t))$ since $H$ is fiberwise homogeneous of degree $2$ and using euler's identity.
Now I tried to check that $\omega_{c\gamma}(\frac{d}{dt}( c\gamma),.)=\frac{2}{c}H(q(t),p(t))$ but I got nowhere .Maybe I am making some silly mistake but I can't seem to be able to prove this.
Any help is appreciated. Thanks in advance.
 A: I hope the following helps.

*

*Suppose you have Hamiltonians $H(q,p)$ and $\widetilde H(q,p):=c^2 H(q,p)$. Then the integral curve $(\widetilde q(t),\widetilde p(t))$ of Hamilton's equations for $\widetilde H$ with $\widetilde q(t=0)=q_0,\widetilde p(t=0)=p_0$ is $(q(c^2t),p(c^2t))$ where $(q(t),p(t))$ is the integral curve of Hamilton's equations for $H$ with initial conditions $q(t=0)=q_0,p(t=0)=p_0$.


*Suppose you have Hamiltonians $H(q,p)$ and $\widetilde H(q,p):=H(q,cp)$. Then the integral curve $(\widetilde q(t),\widetilde p(t))$ of Hamilton's equations for $\widetilde H$ with $\widetilde q(t=0)=q_0,\widetilde p(t=0)=p_0$ is $(q(ct),p(ct)/c)$ where $(q(t),p(t))$ is the integral curve of Hamilton's equations for $H$ with initial conditions $q(t=0)=q_0,p(t=0)=cp_0$.


*Combining the previous points, suppose that $H(q,cp)=c^2 H(q,p)$, for $c>0$. Then the integral curves $(q(t;q_0,p_0),p(t;q_0,p_0))$ of Hamilton's equations, now denoting also the dependence on initial data, have the following invariance
$$ (q(c^2t;q_0,p_0),p(c^2t;q_0,p_0)) =(q(ct;q_0,cp_0),p(ct;q_0,cp_0)/c) ,\qquad c>0, $$
or, just by setting $t\mapsto t/c$ in this identity,
$$
(q(ct;q_0,p_0),p(ct;q_0,p_0)) =(q(t;q_0,cp_0),p(t;q_0,cp_0)/c) ,\qquad c>0.
$$
