Equation of motion using Hamiltonian principle

The equation of motion in covariant form can be obtained from the Lagragian by using Hamiltonian principle

Consider the action integral

In order to write action in invariant form, we have to replace 'dt' by d𝜏 (proper time) i.e dt = 𝛾d𝜏

This is the required invariant form

Since

In this equation 𝜏 is a parameter i.e x = x(𝜏 )

Taking arc S as parameter 𝜏 -> S

We see that tilde L is function of dX_𝛼 / dS. Hence from variation principle

Interchange 𝛼 and 𝛽 in the second term,

Dividing both sides by

This is required equation for free particle.

If we consider interaction

The Lagragian equation

Using the result for free particles

These features make it a powerful and insightful tool for understanding the dynamics of relativistic systems.

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