Fundamentals of Time and Relativity

Equation of Motion

To derive the equation of motion one parameterizes the path $x = x (σ)$ of the particle in Minkowski space with respect to a parameter $σ$ . As a function of the proper time $τ$ , this parameter increases monotonically with $τ$ , but is otherwise arbitrary. Doing so we replace (10.1) by:

10.3

S_{free} [x (σ)] : = - m \int_{1}^{2} d σ \sqrt{\dot{x} \cdot \dot{x}}

Here the symbol $\dot{x}$ denotes the derivative $\dot{x} (σ) : = d x / d σ$ . The integrand $L = - m \sqrt{\dot{x} \cdot \dot{x}}$ of the action integral (10.3) is the invariant Lagrangian of the system, different from the non-covariant Lagrangian $L$ ; see (10.2).

We now vary the action $δ S : = S [x (σ) + δ x (σ)] - S [x (σ)]$ with respect to the path. The quantities to be varied are the coordinate $x (σ)$ and its derivative. Because the Lagrangian of a free particle only depends on $\dot{x} (σ)$ , this gives

10.4

δ S_{free} = \int_{1}^{2} d σ \frac{\partial L}{\partial \dot{x}} \cdot δ \dot{x} = - m \int_{1}^{2} d σ \frac{\dot{x}}{\sqrt{\dot{x} \cdot \dot{x}}} \cdot δ \dot{x}

Then setting $δ \dot{x} = d (δ x) / d σ, \sqrt{\dot{x} \cdot \dot{x}} = d τ / d σ$ , we obtain by an integration by parts the variational equation

10.5

δ S_{free} [x (τ)] = - m \frac{d x (τ)}{d τ} \cdot {δ x (τ) |}_{1}^{2} + m \int_{1}^{2} d τ [\frac{d}{d τ} \frac{d x (τ)}{d τ}] \cdot δ x (τ)

Given the values of $x (τ)$ at the initial proper time $τ = τ_{1}$ and final proper time $τ = τ_{2}$ , the actual path $\bar{x} (τ)$ of the particle is such that the action is extremal, $δ S = 0$ , for small variations $δ x (τ)$ that vanish at the two endpoints 1 and 2. Since the variations are otherwise arbitrary, the integrand between brackets at the right-hand side of (10.5) must vanish and we obtain the covariant equation of motion:

10.6

m \frac{d^{2} x (τ)}{d τ^{2}} = \frac{d p (τ)}{d τ} = 0

In this way the equation of motion of a relativistic free particle is rederived from Hamilton’s action principle.

The action (10.3) takes the same form regardless of the choice of parameterization. This is sometimes called a 'symmetry', but is more of a redundancy similar to the gauge freedom of the Maxwell theory.
Equations of motion derived from an action principle, are usually referred to as Euler-Lagrange equations.