_1,q_2,q_3,q_4\in\mathbb{R}
\end{align*}
The equations of motion are determined by the Lagrangian
\begin{equation}\label{eq:Lagrangian}
\mathcal{L} = \frac{1}{2}(\dot{q_1}^2 + \dot{q_2}^2 + \dot{q_3}^2 + \dot{q_4}^2) – U(q_1,q_2,q_3,q_4)
\end{equation}
where $U$ is the potential energy.
The equations of motion are determined by the Euler-Lagrange equations
\begin{equation}\label{eq:EL}
\ddot{q_i} = -\frac{\partial U}{\partial q_i}, \quad i = 1,\dots,4
\end{equation}
The Hamiltonian of the system is given by
\begin{equation}\label{eq:Hamiltonian}
\mathcal{H} = \frac{1}{2}(p_1^2 + p_2^2 + p_3^2 + p_4^2) + U(q_1,q_2,q_3,q_4)
\end{equation}
where $p_i$ is the canonical momentum conjugate to $q_i$.
The equations of motion derived from the Hamiltonian are given by Hamilton’s equations
\begin{align}\label{eq:Hammot}
\dot{q_i} &= \frac{\partial \mathcal{H}}{\partial p_i}, \quad i = 1,\dots,4 \notag \\
\dot{p_i} &= -\frac{\partial \mathcal{H}}{\partial q_i}, \quad i = 1,\dots,4
\end{align}
The equations of motion can also be derived from the Poisson brackets
\begin{align}\label{eq:PB}
\dot{q_i} &= \{q_i, \mathcal{H}\}, \quad i = 1,\dots,4 \notag \\
\dot{p_i} &= \{p_i, \mathcal{H}\}, \quad i = 1,\dots,4
\end{align}
The equations of motion can also be derived from the Lagrangian by using the Lagrange equations
\begin{align}\label{eq:Lagmot}
\ddot{q_i} &= \frac{\partial \mathcal{L}}{\partial q_i} – \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q_i}}, \quad i = 1,\dots,4
\end{align}
The equations of motion can also be derived by using the Hamilton-Jacobi equation
\begin{equation}\label{eq:HJ}
\frac{\partial S}{\partial t} + \frac{1}{2}\left(\frac{\partial S}{\partial q_1} \right)^2 + \frac{1}{2}\left(\frac{\partial S}{\partial q_2} \right)^2 + \frac{1}{2}\left(\frac{\partial S}{\partial q_3} \right)^2 + \frac{1}{2}\left(\frac{\partial S}{\partial q_4} \right)^2 + U(q_1,q_2,q_3,q_4) = 0
\end{equation}
where $S$ is the Hamilton-Jacobi function.
The equations of motion can also be derived by using the Liouville equation
\begin{equation}\label{eq:Liouville}
\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial q_1} \left(\rho \frac{\partial \mathcal{H}}{\partial p_1}\right) + \frac{\partial}{\partial q_2} \left(\rho \frac{\partial \mathcal{H}}{\partial p_2}\right) + \frac{\partial}{\partial q_3} \left(\rho \frac{\partial \mathcal{H}}{\partial p_3}\right) + \frac{\partial}{\partial q_4} \left(\rho \frac{\partial \mathcal{H}}{\partial p_4}\right) = 0
\end{equation}
where $\rho$ is the probability density function.
The equations of motion can also be derived from the quantum mechanical Schrödinger equation
\begin{equation}\label{eq:Schrodinger}
i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi
\end{equation}
where $\hat{H}$ is the Hamiltonian operator and $\psi$ is the wavefunction.
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