Labware - MA35 Multivariable Calculus - Single Variable Calculus

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Decomposition of Acceleration (Page: 1 | 2 )

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Suppose some object is traveling along a curve. Geodesic curvature describes quantitatively the change in the direction of the object's travel at some point on the curve.

Consider a parameterized curve X(t) = (x(t),y(t)) with velocity vector X'(t) = (x'(t),y'(t)). The unit tangent vector is defined as T(t) = (x'(t),y'(t))/s'(t) and the unit normal is defined as U(t) = (-y'(t),x'(t))/s'(t). As long as T(t) and U(t) exist and are non-degenerate, they form a basis for R2. Therefore, we can write the accleration vector as a linear combination of T(t) and U(t):

X''(t) = a(t)T(t) + b(t)U(t)

By differentiating the expression for the velocity vector, X'(t)=s'(t)T(t), we obtain

X''(t) = s''(t)T(t) + s'(t)T'(t)

Observe that T(t) is a unit vector, so that T(t)ĚT(t) = 1 and taking the derivative of both sides gives 2T(t)ĚT'(t) = 0. So T'(t) is perpendicular and T(t) and it follows that T'(t) must be a multiple of U(t). We define T'(t) = κg(t)s'(t)U(t) and write the acceleration vector as

X''(t) = s''(t)T(t) + κg(t)s'(t)2U(t)

The function κg(t) is the geodesic curvature of the curve at the point X(t).

Demos

Exercises

  • 1. Examine the acceleration vectors and curvature of the ellipse X(t) = (c*cos(t), sin(t)).
  • 2. Examine the acceleration vectors and curvature of the cardioid X(t) = (1 + cos(t))*(cos(t), sin(t)).
  • 3. Examine the acceleration vectors and curvature of the exponential spiral: X(t) = e^t*(cos(t),sin(t)).