[Paper Review] Third-order relativistic dynamics: classical spinning particle travelling in a plane

Roman Matsyuk|arXiv (Cornell University)|Apr 28, 2013

Quantum chaos and dynamical systems5 references4 citations

TL;DR

This paper derives the relativistic dynamics of a classical spinning particle in planar motion from third-order variational principles, showing that Mathisson's 'new mechanics' for spinning particles in special relativity emerges naturally from relativistic invariance, third-order derivatives, and variationality. The key result is that the Hamiltonian formulation recovers the Dixon equations under the Pirani supplementary condition, establishing a consistent Lagrangian and Hamiltonian framework for planar spinning particle motion.

ABSTRACT

Mathisson's 'new mechanics' of a relativistic spinning particle is shown to follow, in the case of planar motion, from only general requirements of relativistic invariance and of the dependence on third order derivatives along with the 'variationality' feature. The Hamiltonian counterpart ultimately recovers the Dixon system of equations for this case with the Mathisson-Pirani supplementary condition.

Motivation & Objective

• To establish a variational foundation for third-order relativistic dynamics of a spinning particle in a plane.
• To demonstrate that Mathisson's equations for a spinning particle arise from general principles: relativistic invariance, third-order derivatives, and variationality.
• To construct a Hamiltonian formulation equivalent to the Dixon equations under the Pirani supplementary condition.
• To resolve the inverse variational problem for third-order relativistic equations in two spatial dimensions.
• To provide a comprehensive Lagrangian and Hamiltonian description of planar motion for a relativistic spinning particle in flat spacetime.

Proposed method

• Applies the formal theory of variational calculus to third-order differential equations, using necessary and sufficient conditions for the existence of a Lagrangian.
• Imposes Poincaré invariance and relativistic invariance as fundamental constraints on the dynamical system.
• Derives the third-order equation of motion from the Euler-Poisson equation of a variational principle with third-order derivatives.
• Uses the 'hamiltonization' prescription from [6] to transform the third-order Lagrangian into a Hamiltonian system.
• Applies the Pirani supplementary condition $\frak{u}_q \frak{S}^{pq} = 0$ to reduce the system to planar motion and recover the Dixon equations.
• Constructs the Hamiltonian function explicitly via canonical momenta and solves the inverse problem using differential geometric techniques on jet bundles.

Experimental results

Research questions

• RQ1 Can the relativistic dynamics of a spinning particle in planar motion be derived from a variational principle involving third-order derivatives?
• RQ2 Does the requirement of relativistic invariance and variationality uniquely determine the third-order equation of motion for a spinning particle?
• RQ3 How does the Hamiltonian formulation of the third-order system relate to the well-known Dixon equations for spinning particles?
• RQ4 What is the role of the Pirani supplementary condition in connecting the third-order Lagrangian dynamics to the standard relativistic spinning particle equations?
• RQ5 Can the inverse variational problem for third-order relativistic equations be solved in two spatial dimensions with Poincaré symmetry?

Key findings

• The third-order equation of motion (2) is the unique relativistic equation with third-order derivatives that admits a Lagrangian description under Poincaré invariance.
• The Hamiltonian formulation derived from the Lagrangian yields a system equivalent to the Dixon equations under the Pirani condition $\frak{u}_q \frak{S}^{pq} = 0$.
• The canonical momentum $\mathbf{p}$ is shown to coincide with $\frak{s}^3$ times the Hamiltonian momentum, confirming consistency with the relativistic spinning particle model.
• The system's equations of motion are shown to describe the kernel of two differential forms, confirming the geometric structure of the Hamiltonian system.
• The determinant $\Delta$ in the Hamiltonian (35) is explicitly computed as $\Delta = (1 + \mathbf{v} \cdot \mathbf{v})^3$, which enters the momentum transformation and ensures relativistic invariance.
• The equation $\frac{d\mathbf{p}}{dt} = \mathbf{0}$ confirms momentum conservation in the Hamiltonian system, with the remaining equations selecting holonomic solutions.