A particle moving in a plane has potential energy (in joules

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Partial differentiation, chain rules and Jacobians

Problem

A particle moving in a plane has potential energy

U(x,y) = x^4 - 2x^2 + 4y^2

(in joules, with

x

and

y

in metres). ((a)) Show that the origin

(0,0)

is a stationary point of

U

. Along the line

y=0

, use the one-variable second-derivative test to decide whether

U

has a local maximum or a local minimum there; do the same along the line

x=0

. ((b)) Use the discriminant

\Delta_2 = U_{xx}U_{yy} - U_{xy}^2

to classify the origin as a whole, and reconcile the result with part (a). ((c)) At the point

(0.50\,\mathrm{m}, 0.20\,\mathrm{m})

, use the total differential to estimate the change in

U

produced by a displacement

\Delta x = +0.05\,\mathrm{m}

\Delta y = +0.03\,\mathrm{m}

. Compute the exact change as well, and state the order in

\Delta x, \Delta y

of the term the total differential leaves out.

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