Write down the kinetic energy and total energy expressions in terms of linear momentum, For one-dimensional case.

Explanation

Kinetic energy is `KE=(1)/(2)mv_(x)^(2)`
Multiply numerator and denominator by m
`KE=(1)/(2m)m^(2)v_(x)^(2)=(1)/(2m)(mv_(x))^(2)=(1)/(2m)p_(x)^(2)`
where, `p_(x)` is the linear momentum of the particle executing simple harmonic motion. Total energy can be written as sum of kinetic energy and potential energy, therefore,
`E=KE+U(x)=(1)/(2m)p_(x)^(2)+(1)/(2)momega^(2)x^(2)` = constant