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

: $q'^0=q'^0(q^0,q^k), \quad q'^i=q'^i(q^0,q^k), \quad ^i_0 = \left(\frac q^j_0 + \frac \right) \left(\frac q^j_0 + \frac \right)^.$

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

: $\left(\frac- \partial_\mu G_\right) q^\mu_\tau q^_\tau\cdots q^_\tau - (2N-1)G_q^\mu_ q^_\tau\cdots q^_\tau + F_q^\mu_\tau =0,$

: $G_q^_\tau\cdots q^_\tau=1.$

For instance, if $Q$ is the Minkowski space with a Minkowski metric $G_$, 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., t al. "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, .
* Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Formulation of Classical and Quantum Mechanics (World Scientific, 2010) ({{arXiv, 1005.1212).

Differential equations
Classical mechanics
Theory of relativity