formulate and prove the following proposition:

***Lemma:*** *The Euler-Lagrange equations for two conformal Lagrangians transform covariantly under the conformal transformation relating the Lagrangians iff the Hamiltonian vanishes.*

***Proof:*** Consider the Lagrangian $L=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}-V\left(  x^{k}\right)  $ whose Euler-Lagrange equations are: $$\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{i}\dot{x}^{j}+V^{,i}=0 \label{CLN1.01}$$ where $\Gamma_{jk}^{i}$ are the Christofell symbols. The corresponding Hamiltonian is given by $$E=\frac{1}{2}g_{ij}\dot{x}^{i}\dot{x}^{j}+V\left(  x^{k}\right)  \;.
\label{CLN1.02}$$ For the conformally related Lagrangian $\bar{L}\left(  x^{k},x^{\prime
k}\right)  =\left(  \frac{1}{2}N^{2}\left(  x^{k}\right)  g_{ij}x^{\prime
i}x^{\prime j}-\frac{V\left(  x^{k}\right)  }{N^{2}\left(  x^{k}\right)
}\right)  $ where $N_{,j}\neq0$ the resulting Euler Lagrange equations are$$x^{\prime\prime i}+\hat{\Gamma}_{jk}^{i}x^{\prime j}x^{\prime k}+\frac
{1}{N^{4}}V^{,i}-\frac{2V}{N^{5}}N^{,i}=0 \label{CLN1.03}$$ where $$\hat{\Gamma}_{jk}^{i}=\Gamma_{jk}^{i}+(\ln N)_{,k}\delta_{j}^{i}+(\ln
N)_{,j}\delta_{k}^{i}-(\ln N)^{,i}g_{jk} \label{CLN1.04}$$ and the corresponding Hamiltonian is$$\bar{E}=\frac{1}{2}N^{2}\left(  x^{k}\right)  g_{ij}\dot{x}^{i}\dot{x}^{j}+\frac{V\left(  x^{k}\right)  }{N^{2}\left(  x^{k}\right)  }. \label{CLN1.05}$$ In order to show that the two equations of motion are conformally related we start from Eq.(\[CLN1.03\]) and apply the conformal transformation $$\begin{aligned}
x^{\prime i}  &  =\frac{dx^{i}}{d\tau}=\frac{dx^{i}}{dt}\frac{dt}{d\tau}=\dot{x}^{i}\frac{1}{N^{2}}\\
x^{^{\prime\prime}i}  &  =\ddot{x}^{i}\frac{1}{N^{4}}-2\dot{x}^{i}\dot{x}^{j}\left(  \ln N\right)  _{,j}\frac{1}{N^{4}}.\end{aligned}$$ Replacing in Eq.(\[CLN1.03\]) we find:$$\ddot{x}^{i}\frac{1}{N^{4}}-2\dot{x}^{i}\dot{x}^{j}\left(  \ln N\right)
_{,j}\frac{1}{N^{4}}+\frac{1}{N^{4}}\hat{\Gamma}_{jk}^{i}\dot{x}^{j}\dot
{x}^{k}+\frac{1}{N^{4}}V^{,i}-\frac{2V}{N^{5}}N^{,i}=0\,.$$ Replacing $\hat{\Gamma}_{jk}^{i}$ from Eq.(\[CLN1.04\]) we have $$\begin{aligned}
\ddot{x}^{i}-2\dot{x}^{i}\dot{x}^{j}\left(  \ln N\right)  _{,j}+\Gamma
_{jk}^{i}\dot{x}^{j}\dot{x}^{k}+2(\ln N)_{,j}\dot{x}^{j}\dot{x}^{i}\\
-(\ln N)^{,i}g_{jk}\dot{x}^{j}\dot{x}^{k}+V^{,i}-2V(\ln N)^{,i}=0\end{aligned}$$ from which follows$$\ddot{x}^{i}+\Gamma_{jk}^{i}\dot{x}^{j}\dot{x}^{k}+V^{,i}-(\ln N)^{,i}\left(
g_{jk}\dot{x}^{j}\dot{x}^{k}+2V\right)  =0.$$ Obviously, the above Euler-Lagrange equations coincide with Eqs.(\[CLN1.01\]) if and only if $\left(  g_{jk}\dot{x}^{j}\dot{x}^{k}+2V\right)  =0$, which implies that the Hamiltonian of Eq.(\[CLN1.02\]) vanishes. The steps are reversible hence the inverse 