- Source: Relativistic system (mathematics)
- Howard Percy Robertson
- Laju cahaya
- Relativistic system (mathematics)
- Relativistic mechanics
- Non-autonomous system (mathematics)
- Mass in special relativity
- Relativistic beaming
- Relativistic Doppler effect
- Mathematical formulation of quantum mechanics
- Coordinate system
- Frame of reference
- Relativistic Lagrangian mechanics
In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle
Q
→
R
{\displaystyle Q\to \mathbb {R} }
over
R
{\displaystyle \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
{\displaystyle Q}
whose fibration over
R
{\displaystyle \mathbb {R} }
is not fixed. Such a system admits transformations of a coordinate
t
{\displaystyle t}
on
R
{\displaystyle \mathbb {R} }
depending on other coordinates on
Q
{\displaystyle Q}
. Therefore, it is called the relativistic system. In particular, Special Relativity on the
Minkowski space
Q
=
R
4
{\displaystyle Q=\mathbb {R} ^{4}}
is of this type.
Since a configuration space
Q
{\displaystyle Q}
of a relativistic system has no
preferable fibration over
R
{\displaystyle \mathbb {R} }
, a
velocity space of relativistic system is a first order jet
manifold
J
1
1
Q
{\displaystyle J_{1}^{1}Q}
of one-dimensional submanifolds of
Q
{\displaystyle 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
1
Q
→
Q
{\displaystyle J_{1}^{1}Q\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
)
{\displaystyle (q^{0},q^{i})}
on
Q
{\displaystyle Q}
, a first order jet manifold
J
1
1
Q
{\displaystyle J_{1}^{1}Q}
is provided with the adapted coordinates
(
q
0
,
q
i
,
q
0
i
)
{\displaystyle (q^{0},q^{i},q_{0}^{i})}
possessing transition functions
q
′
0
=
q
′
0
(
q
0
,
q
k
)
,
q
′
i
=
q
′
i
(
q
0
,
q
k
)
,
q
′
0
i
=
(
∂
q
′
i
∂
q
j
q
0
j
+
∂
q
′
i
∂
q
0
)
(
∂
q
′
0
∂
q
j
q
0
j
+
∂
q
′
0
∂
q
0
)
−
1
.
{\displaystyle q'^{0}=q'^{0}(q^{0},q^{k}),\quad q'^{i}=q'^{i}(q^{0},q^{k}),\quad {q'}_{0}^{i}=\left({\frac {\partial q'^{i}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{i}}{\partial q^{0}}}\right)\left({\frac {\partial q'^{0}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{0}}{\partial q^{0}}}\right)^{-1}.}
The relativistic velocities of a relativistic system are represented by
elements of a fibre bundle
R
×
T
Q
{\displaystyle \mathbb {R} \times TQ}
, coordinated by
(
τ
,
q
λ
,
a
τ
λ
)
{\displaystyle (\tau ,q^{\lambda },a_{\tau }^{\lambda })}
, where
T
Q
{\displaystyle TQ}
is the tangent bundle of
Q
{\displaystyle Q}
. Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads
(
∂
λ
G
μ
α
2
…
α
2
N
2
N
−
∂
μ
G
λ
α
2
…
α
2
N
)
q
τ
μ
q
τ
α
2
⋯
q
τ
α
2
N
−
(
2
N
−
1
)
G
λ
μ
α
3
…
α
2
N
q
τ
τ
μ
q
τ
α
3
⋯
q
τ
α
2
N
+
F
λ
μ
q
τ
μ
=
0
,
{\displaystyle \left({\frac {\partial _{\lambda }G_{\mu \alpha _{2}\ldots \alpha _{2N}}}{2N}}-\partial _{\mu }G_{\lambda \alpha _{2}\ldots \alpha _{2N}}\right)q_{\tau }^{\mu }q_{\tau }^{\alpha _{2}}\cdots q_{\tau }^{\alpha _{2N}}-(2N-1)G_{\lambda \mu \alpha _{3}\ldots \alpha _{2N}}q_{\tau \tau }^{\mu }q_{\tau }^{\alpha _{3}}\cdots q_{\tau }^{\alpha _{2N}}+F_{\lambda \mu }q_{\tau }^{\mu }=0,}
G
α
1
…
α
2
N
q
τ
α
1
⋯
q
τ
α
2
N
=
1.
{\displaystyle G_{\alpha _{1}\ldots \alpha _{2N}}q_{\tau }^{\alpha _{1}}\cdots q_{\tau }^{\alpha _{2N}}=1.}
For instance, if
Q
{\displaystyle Q}
is the Minkowski space with a Minkowski metric
G
μ
ν
{\displaystyle G_{\mu \nu }}
, 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).