Relativistic system (mathematics)

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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.

References[edit]

  • 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).

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