2D Relativistic Dynamics

Let:

$\vec{V} = v_x \ \hat{i} + v_y \ \hat{j}$

$\vec{p} = p_x \ \hat{i} + p_y \ \hat{j}$

$p_x = \frac{v_x}{\sqrt{1-v_x^2-v_y^2}}$ $p_y = \frac{v_y}{\sqrt{1-v_x^2-v_y^2}}$

Applying Newton's second law gives: $F_x = \frac{dp_x}{dt} = \frac{\partial p_x}{\partial v_x} \dot{v}_x + \frac{\partial p_x}{\partial v_y} \dot{v}_y = m_{11}\dot{v}_x + m_{12}\dot{v}_y$ $F_y = \frac{dp_y}{dt} = \frac{\partial p_y}{\partial v_x} \dot{v}_x + \frac{\partial p_y}{\partial v_y} \dot{v}_y= m_{21}\dot{v}_x + m_{22}\dot{v}_y$

The partial derivative computations have not been shown here. This implies:

$\left[\begin{matrix} F_x\\F_y\end{matrix}\right] = \left[\begin{matrix} m_{11}&m_{12}\\m_{21} &m_{22} \end{matrix}\right] \left[\begin{matrix} \dot{v}_x\\\dot{v}_y\end{matrix}\right]$

This implies:

$\boxed{\left[\begin{matrix} \dot{v}_x\\\dot{v}_y\end{matrix}\right] = \left[\begin{matrix} m_{11}&m_{12}\\m_{21} &m_{22} \end{matrix}\right] ^{-1}\left[\begin{matrix} F_x\\F_y\end{matrix}\right]}$

Beyond this point, numerical integration does the rest. The Lagrangian formulation gives the same. The motion of the system can be summarised in the following plots:

One can see that in the non-relativistic case, the speed of the particle eventually exceeds the speed of light, which is an impossibility.

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Just as general information, the 'kinetic energy' term can be derived by realising that in classical mechanics, the expression for generalised momentum (for 1D motion) is:

$\frac{\partial \mathcal{T}}{\partial \dot{x}} = p$ For the relativistic case:

$\frac{\partial \mathcal{T}}{\partial \dot{x}} = p = \frac{mc\dot{x}}{\sqrt{c^2-\dot{x}^2}}$

Integrating the above gives the required expression.