second shifting theorem
Real systems do not always start at t = 0; a voltage might switch on after a delay, or a force might begin only at t = 2 seconds. The second shifting theorem is the rule for transforming such a delayed signal — a function that has been picked up and slid later in time and switched on only from that moment.
The delayed signal is written using the Heaviside step u(t - a), which is 0 before t = a and 1 after. The theorem states that L{f(t - a) u(t - a)} = e^{-a s} F(s), where F(s) = L{f(t)} and a > 0. In words: delaying a function by a in time multiplies its transform by e^{-a s}. The factor e^{-a s} is the signature of a pure time delay, and conversely, whenever you see e^{-a s} multiplying a transform, you read it backward as a shifted, switched-on copy of the inverse of the rest.
This theorem is what lets the Laplace method handle piecewise inputs cleanly — square pulses, staircases, signals that turn on and off — without splitting the problem into separate intervals. The careful point that trips everyone is the argument: the function must be f(t - a), genuinely shifted, not f(t) merely multiplied by the step. If your function is g(t) u(t - a) with g not already shifted, you first rewrite g(t) in terms of (t - a) before applying the rule.
A force that is 0 until t = 2 then equals (t - 2)^2 is (t - 2)^2 u(t - 2); since L{t^2} = 2/s^3, its transform is e^{-2 s} (2/s^3).
Because the force is already written as a function of t - 2, the rule applies directly and stamps on the delay factor e^{-2 s}.
The most common error is treating g(t) u(t - a) as if it were already shifted; you must re-express g in terms of (t - a) first, otherwise the e^{-a s} formula gives the wrong answer.