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 [math]\displaystyle{ Q\to \mathbb R }[/math] over [math]\displaystyle{ \mathbb R }[/math]. 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 [math]\displaystyle{ Q }[/math] whose fibration over [math]\displaystyle{ \mathbb R }[/math] is not fixed. Such a system admits transformations of a coordinate [math]\displaystyle{ t }[/math] on [math]\displaystyle{ \mathbb R }[/math] depending on other coordinates on [math]\displaystyle{ Q }[/math]. Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space [math]\displaystyle{ Q= \mathbb R^4 }[/math] is of this type.

Since a configuration space [math]\displaystyle{ Q }[/math] of a relativistic system has no preferable fibration over [math]\displaystyle{ \mathbb R }[/math], a velocity space of relativistic system is a first order jet manifold [math]\displaystyle{ J^1_1Q }[/math] of one-dimensional submanifolds of [math]\displaystyle{ Q }[/math]. 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 [math]\displaystyle{ J^1_1Q\to Q }[/math] 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 [math]\displaystyle{ (q^0, q^i) }[/math] on [math]\displaystyle{ Q }[/math], a first order jet manifold [math]\displaystyle{ J^1_1Q }[/math] is provided with the adapted coordinates [math]\displaystyle{ (q^0,q^i,q^i_0) }[/math] possessing transition functions

[math]\displaystyle{ 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}. }[/math]

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

[math]\displaystyle{ \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, }[/math]

[math]\displaystyle{ G_{\alpha_1\ldots\alpha_{2N}}q^{\alpha_1}_\tau\cdots q^{\alpha_{2N}}_\tau=1. }[/math]

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

See also

• Non-autonomous system (mathematics)
• Non-autonomous mechanics
• Relativistic mechanics
• Special relativity

References

• Krasil'shchik, I. S., Vinogradov, A. M., [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, ISBN 0-8218-0958-X.
• Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ISBN 981-4313-72-6 (arXiv:1005.1212).