Adiabatic Invariant in Variable Mass Oscillator

The equation of motion for " ice ball slowly melting connected to a spring"

you start with the kinetic energy $T_c$

$$T_c=frac 12 m(tau),dot x^2$$

and the potential energy of the spring $U$

$$U=frac 12 k,x^2$$

with Euler- Lagrange you obtain the EOM.

$$ begin {array}{c} m left( tau right) {ddot x}+kx+ left( { frac {d}{dtau}}m left( tau right) right) {dot x}end {array} =0 $$

you can make this Ansatz for $m(tau)$

$$m(tau)=m_{{0}},{{e}^{-{frac {tau}{T}}}}$$

where $T$ is the time constant $~[s]$ for the ice melting process and $m_0$ is the initial ice mass

the EOM is now

$$ddot x+underbrace{frac{e^{(tau/T)},k}{m_0}},x-frac{1}{T},dot x$$

Edit

The total energy is:

$$E=frac 12,m_{{0}}{{rm e}^{-{frac {tau}{T}}}}{{dot x}}^{2}+frac 12,k{x}^{2}$$

with :

$$T=frac {2pi}{omega}=frac{2,pi}{sqrt{frac{k}{m_0}}}~,k=m_0,omega^2$$

$$E=frac 12,m_{{0}}{{rm e}^{-frac{1}{2,pi},sqrt {{frac {k}{m_{{0}}}}}tau}}{{dot x}}^{2}+frac 12,m_{{0}}{omega}^{2}{x}^{2} $$

for $tau mapsto infty$

$$E=frac 12,m_{{0}}{omega}^{2}{x}^{2} ~,frac{E}{omega}=frac 12,m_0,omega,x^2=text{const.}$$