Английская Википедия:Acceleration (differential geometry)

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In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend".

Formal definition

Consider a differentiable manifold <math>M</math> with a given connection <math>\Gamma</math>. Let <math>\gamma \colon\R \to M</math> be a curve in <math>M</math> with tangent vector, i.e. velocity, <math>{\dot\gamma}(\tau)</math>, with parameter <math>\tau</math>.

The acceleration vector of <math>\gamma</math> is defined by <math>\nabla_{\dot\gamma}{\dot\gamma} </math>, where <math>\nabla </math> denotes the covariant derivative associated to <math>\Gamma</math>.

It is a covariant derivative along <math>\gamma</math>, and it is often denoted by

<math>\nabla_{\dot\gamma}{\dot\gamma} =\frac{\nabla\dot\gamma}{d\tau}.</math>

With respect to an arbitrary coordinate system <math>(x^{\mu})</math>, and with <math>(\Gamma^{\lambda}{}_{\mu\nu})</math> being the components of the connection (i.e., covariant derivative <math>\nabla_{\mu}:=\nabla_{\partial/\partial x^\mu}</math>) relative to this coordinate system, defined by

<math>\nabla_{\partial/\partial x^\mu}\frac{\partial}{\partial x^{\nu}}= \Gamma^{\lambda}{}_{\mu\nu}\frac{\partial}{\partial x^{\lambda}},</math>

for the acceleration vector field <math>a^{\mu}:=(\nabla_{\dot\gamma}{\dot\gamma})^{\mu}</math> one gets:

<math>a^{\mu}=v^{\rho}\nabla_{\rho}v^{\mu} =\frac{dv^{\mu}}{d\tau}+ \Gamma^{\mu}{}_{\nu\lambda}v^{\nu}v^{\lambda}= \frac{d^2x^{\mu}}{d\tau^2}+ \Gamma^{\mu}{}_{\nu\lambda}\frac{dx^{\nu}}{d\tau}\frac{dx^{\lambda}}{d\tau},</math>

where <math>x^{\mu}(\tau):= \gamma^{\mu}(\tau)</math> is the local expression for the path <math>\gamma</math>, and <math>v^{\rho}:=({\dot\gamma})^{\rho}</math>.

The concept of acceleration is a covariant derivative concept. In other words, in order to define acceleration an additional structure on <math>M</math> must be given.

Using abstract index notation, the acceleration of a given curve with unit tangent vector <math>\xi^a</math> is given by <math>\xi^{b}\nabla_{b}\xi^{a}</math>.

See also

• Acceleration
• Covariant derivative

References

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book