Relativistic system (mathematics)

In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle $$Q\to \mathbb R$$ over $$\mathbb R$$. For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanics. To describe relativistic mechanics, one should consider a system of ordinary differential equations on a smooth manifold $$Q$$ whose fibration over $$\mathbb R$$ is not fixed. Such a system admits transformations of a coordinate $$t$$ on $$\mathbb R$$ depending on other coordinates on $$Q$$. Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space $$Q= \mathbb R^4$$ is of this type.

Since a configuration space $$Q$$ of a relativistic system has no preferable fibration over $$\mathbb R$$, a velocity space of relativistic system is a first order jet manifold $$J^1_1Q$$ of one-dimensional submanifolds of $$Q$$. The notion of jets of submanifolds generalizes that of jets of sections of fiber bundles which are utilized in covariant classical field theory and non-autonomous mechanics. A first order jet bundle $$J^1_1Q\to Q$$ is projective and, following the terminology of Special Relativity, one can think of its fibers as being spaces of the absolute velocities of a relativistic system. Given coordinates $$(q^0, q^i)$$ on $$Q$$, a first order jet manifold $$J^1_1Q$$ is provided with the adapted coordinates $$(q^0,q^i,q^i_0)$$ possessing transition functions


 * $$q'^0=q'^0(q^0,q^k), \quad q'^i=q'^i(q^0,q^k), \quad

{q'}^i_0 = \left(\frac{\partial q'^i}{\partial q^j} q^j_0 + \frac{\partial q'^i}{\partial q^0} \right) \left(\frac{\partial q'^0}{\partial q^j} q^j_0 + \frac{\partial q'^0}{\partial q^0} \right)^{-1}.$$

The relativistic velocities of a relativistic system are represented by elements of a fibre bundle $$\mathbb R\times TQ$$, coordinated by $$(\tau,q^\lambda,a^\lambda_\tau)$$, where $$TQ$$ is the tangent bundle of $$Q$$. Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads


 * $$ \left(\frac{\partial_\lambda G_{\mu\alpha_2\ldots\alpha_{2N}}}{2N}- \partial_\mu

G_{\lambda\alpha_2\ldots\alpha_{2N}}\right) q^\mu_\tau q^{\alpha_2}_\tau\cdots q^{\alpha_{2N}}_\tau - (2N-1)G_{\lambda\mu\alpha_3\ldots\alpha_{2N}}q^\mu_{\tau\tau} q^{\alpha_3}_\tau\cdots q^{\alpha_{2N}}_\tau  + F_{\lambda\mu}q^\mu_\tau =0,$$


 * $$G_{\alpha_1\ldots\alpha_{2N}}q^{\alpha_1}_\tau\cdots q^{\alpha_{2N}}_\tau=1.$$

For instance, if $$Q$$ is the Minkowski space with a Minkowski metric $$G_{\mu\nu}$$, this is an equation of a relativistic charge in the presence of an electromagnetic field.