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Overdot — Definition, Formula & Examples

An overdot is a dot placed above a variable to indicate its derivative with respect to time. A single dot means the first time derivative, two dots mean the second time derivative, and so on.

Given a function x(t)x(t) of time tt, the overdot notation x˙\dot{x} denotes the first derivative dxdt\frac{dx}{dt}, and x¨\ddot{x} denotes the second derivative d2xdt2\frac{d^2x}{dt^2}. This convention, attributed to Newton, is standard in classical mechanics and dynamical systems.

Key Formula

x˙=dxdt,x¨=d2xdt2\dot{x} = \frac{dx}{dt}, \quad \ddot{x} = \frac{d^2x}{dt^2}
Where:
  • xx = A function of time $t$
  • x˙\dot{x} = First derivative of $x$ with respect to time (velocity if $x$ is position)
  • x¨\ddot{x} = Second derivative of $x$ with respect to time (acceleration if $x$ is position)
  • tt = Time, the independent variable

How It Works

When you see x˙\dot{x}, read it as "x-dot" and interpret it as the rate of change of xx with respect to time. If xx represents position, then x˙\dot{x} is velocity and x¨\ddot{x} is acceleration. The notation is compact and especially convenient when writing equations of motion where every quantity depends on time. Unlike Leibniz notation dxdt\frac{dx}{dt}, the overdot does not explicitly show the independent variable — it is always assumed to be time.

Worked Example

Problem: A particle's position is given by x(t)=3t2+5tx(t) = 3t^2 + 5t. Find x˙\dot{x} and x¨\ddot{x}.
Find the first time derivative: Differentiate x(t)=3t2+5tx(t) = 3t^2 + 5t with respect to tt.
x˙=dxdt=6t+5\dot{x} = \frac{dx}{dt} = 6t + 5
Find the second time derivative: Differentiate x˙\dot{x} with respect to tt.
x¨=d2xdt2=6\ddot{x} = \frac{d^2x}{dt^2} = 6
Answer: The velocity is x˙=6t+5\dot{x} = 6t + 5 and the acceleration is x¨=6\ddot{x} = 6.

Why It Matters

Overdot notation is the default in classical mechanics, robotics, and control theory. Newton's second law is often written F=mx¨F = m\ddot{x}, and Lagrangian mechanics relies heavily on expressions involving q˙\dot{q} and q¨\ddot{q}. Recognizing the overdot lets you read physics and engineering texts fluently.

Common Mistakes

Mistake: Treating the overdot as a derivative with respect to any variable, not just time.
Correction: The overdot always means differentiation with respect to time tt. For derivatives with respect to other variables, use Leibniz notation dxdy\frac{dx}{dy} or prime notation f′(x)f'(x).