Thorton - Marion Prob 2.52

u[x_] := u0 ( 2 (x/a)^2 - (x/a)^4)

Part (a)

(* force *) - D[u[x], x]
% // Simplify

Part (b)

Plot[u[x]/. u0 -> 1 /. a -> 1, {x, -2,2}, PlotLabel -> "Normalized Potential Energy u[x/a]/u0"]

(* equilibrium means "no force" *) - D[u[x], x] == 0

Solve[%, x]

(* stable when D[u[x], {x, 2}] > 0 *) D[u[x], {x, 2}]
% /. %%

So, only x = 0 is a “stable” equilibrium.

Part (c)

m D[x[t], {t, 2}] == - D[u[x[t]], x[t]]

Series[D[u[x[t]], x[t]], {x[t], 0, 2}]

m D[x[t], {t, 2}] == - Normal[Series[D[u[x[t]], x[t]], {x[t], 0, 2}]]

DSolve[%, x[t], t]

Part (d)

Conservation of energy…

escapeEnergy = m v^2/2;

escapeEnergy == (u[x]/. x -> a) - (u[x] /. x -> 0)

Solve[%, v]

Part (e)

m v[t]^2/2 == u0 - u[x[t]]

Solve[%, v[t]]

sol = DSolve[{x'[t] == Sqrt[2 u0/m] (1 - (x[t]/a)^2), x[0] == 0}, x[t], t] // First

Limit[x[t] /. sol, t -> 0]

Limit[x[t] /. sol, t -> ∞, Assumptions -> {a > 0, u0 > 0, m > 0}]