Acceleration (differential geometry) - Alchetron, the free social encyclopedia

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 M with a given connection Γ . Let γ : R → M be a curve in M with tangent vector, i.e. velocity, γ ˙ ( τ ) , with parameter τ .

The acceleration vector of γ is defined by ∇ γ ˙ γ ˙ , where ∇ denotes the covariant derivative associated to Γ .

It is a covariant derivative along γ , and it is often denoted by

∇ γ ˙ γ ˙ = ∇ γ ˙ d τ .

With respect to an arbitrary coordinate system ( x μ ) , and with ( Γ λ μ ν ) being the components of the connection (i.e., covariant derivative ∇ μ := ∇ ∂ / x μ ) relative to this coordinate system, defined by

∇ ∂ / x μ ∂ ∂ x ν = Γ λ μ ν ∂ ∂ x λ ,

for the acceleration vector field a μ := ( ∇ γ ˙ γ ˙ ) μ one gets:

a μ = v ρ ∇ ρ v μ = d v μ d τ + Γ μ ν λ v ν v λ = d 2 x μ d τ 2 + Γ μ ν λ d x ν d τ d x λ d τ ,

where x μ ( τ ) := γ μ ( τ ) is the local expression for the path γ , and v ρ := ( γ ˙ ) ρ .

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

References

Acceleration (differential geometry) Wikipedia

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