Difference between revisions of "Relativistic dynamics"
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A branch of [[Relativity theory|relativity theory]] devoted to the study of the movement of material bodies under the action of forces applied to them. | A branch of [[Relativity theory|relativity theory]] devoted to the study of the movement of material bodies under the action of forces applied to them. | ||
In relativity theory free material points, i.e. points not subject to the action of forces, have as their world lines time-like geodesics or isotropic geodesics. This fact is an expression of the [[Law of inertia|law of inertia]] in relativity theory. | In relativity theory free material points, i.e. points not subject to the action of forces, have as their world lines time-like geodesics or isotropic geodesics. This fact is an expression of the [[Law of inertia|law of inertia]] in relativity theory. | ||
| − | If there are forces acting on a particle, then its world line does not coincide with a geodesic. The movement of a particle is described with reference to the concepts of a four-dimensional energy-momentum vector | + | If there are forces acting on a particle, then its world line does not coincide with a geodesic. The movement of a particle is described with reference to the concepts of a four-dimensional energy-momentum vector $ p ^ {i} $, |
| + | and the four-dimensional force vector $ g ^ {i} $. | ||
| + | Thus, | ||
| + | |||
| + | $$ \tag{1 } | ||
| + | p ^ {i} = \left ( | ||
| + | \frac{\mathbf E }{c} | ||
| + | ; \mathbf p \right ) , | ||
| + | $$ | ||
| − | + | where $ \mathbf E $ | |
| + | is the energy of the particle, $ m $ | ||
| + | is its rest mass and $ \mathbf p $ | ||
| + | is its three-dimensional momentum. The vector $ g ^ {i} $ | ||
| + | is defined by the relation | ||
| − | + | $$ | |
| + | g ^ {i} = \left ( {\mathbf F } \cdot | ||
| + | \frac{\mathbf V }{c ^ {2} \sqrt {1 - V ^ {2} /c ^ {2} } } | ||
| − | + | ; | |
| + | \frac{\mathbf F }{c \sqrt {1 - V ^ {2} /c ^ {2} } } | ||
| + | \right ) , | ||
| + | $$ | ||
| − | where | + | where $ \mathbf F $ |
| + | is a three-dimensional force and $ \mathbf V $ | ||
| + | is the velocity. By using these vectors, the basic equations of relativistic dynamics can be written in a form similar to that of the equations of Newton's second law: | ||
| − | + | $$ \tag{2 } | |
| + | g ^ {i} = | ||
| + | \frac{dp ^ {i} }{ds} | ||
| + | = mc | ||
| + | \frac{du ^ {i} }{ds} | ||
| + | . | ||
| + | $$ | ||
| − | The concrete form of the force | + | The concrete form of the force $ g ^ {i} $ |
| + | is determined in those branches of the theory of relativity that examine the concrete properties of various interactions. For example, the force acting on a particle in an electromagnetic field — the Lorentz force — takes the form | ||
| − | + | $$ | |
| + | g ^ {i} = | ||
| + | \frac{e}{c} | ||
| + | F ^ { ik } u _ {k} , | ||
| + | $$ | ||
| − | where | + | where $ e $ |
| + | is the charge of the particle, $ F ^ { ik } $ | ||
| + | is the tensor of the electromagnetic field and $ u _ {k} $ | ||
| + | is the four-dimensional velocity. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.D. Landau, E.M. Lifshitz, "The theory of fields" , Pergamon (1965) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.D. Landau, E.M. Lifshitz, "The theory of fields" , Pergamon (1965) (Translated from Russian)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rindler, "Essential relativity" , Springer (1977)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rindler, "Essential relativity" , Springer (1977)</TD></TR></table> | ||
Latest revision as of 08:10, 6 June 2020
A branch of relativity theory devoted to the study of the movement of material bodies under the action of forces applied to them.
In relativity theory free material points, i.e. points not subject to the action of forces, have as their world lines time-like geodesics or isotropic geodesics. This fact is an expression of the law of inertia in relativity theory.
If there are forces acting on a particle, then its world line does not coincide with a geodesic. The movement of a particle is described with reference to the concepts of a four-dimensional energy-momentum vector $ p ^ {i} $, and the four-dimensional force vector $ g ^ {i} $. Thus,
$$ \tag{1 } p ^ {i} = \left ( \frac{\mathbf E }{c} ; \mathbf p \right ) , $$
where $ \mathbf E $ is the energy of the particle, $ m $ is its rest mass and $ \mathbf p $ is its three-dimensional momentum. The vector $ g ^ {i} $ is defined by the relation
$$ g ^ {i} = \left ( {\mathbf F } \cdot \frac{\mathbf V }{c ^ {2} \sqrt {1 - V ^ {2} /c ^ {2} } } ; \frac{\mathbf F }{c \sqrt {1 - V ^ {2} /c ^ {2} } } \right ) , $$
where $ \mathbf F $ is a three-dimensional force and $ \mathbf V $ is the velocity. By using these vectors, the basic equations of relativistic dynamics can be written in a form similar to that of the equations of Newton's second law:
$$ \tag{2 } g ^ {i} = \frac{dp ^ {i} }{ds} = mc \frac{du ^ {i} }{ds} . $$
The concrete form of the force $ g ^ {i} $ is determined in those branches of the theory of relativity that examine the concrete properties of various interactions. For example, the force acting on a particle in an electromagnetic field — the Lorentz force — takes the form
$$ g ^ {i} = \frac{e}{c} F ^ { ik } u _ {k} , $$
where $ e $ is the charge of the particle, $ F ^ { ik } $ is the tensor of the electromagnetic field and $ u _ {k} $ is the four-dimensional velocity.
References
| [1] | L.D. Landau, E.M. Lifshitz, "The theory of fields" , Pergamon (1965) (Translated from Russian) |
Comments
References
| [a1] | W. Rindler, "Essential relativity" , Springer (1977) |
Relativistic dynamics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Relativistic_dynamics&oldid=48499