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You can add the bouncing behavior using a mirroring method, where we relax the system and instead of bouncing the ball one allows it to cross to y<0 and changes the sign of the potential accordingly. It can be written as

$H=\frac{p^2}{2m}+mg|y|$

where $|y|=y$ if $y>=0$ and $|y|=-y$ if $y<0$. Potential then looks like (for $m,g=1$)

Equation of motion becomes

$\ddot y(t)=-mg\,sgn(y)$,

where $sgn(y)$ is the sign function of y (a Heaviside-theta-like function). To allow equlilibrium at $y=0$ to exist, we can define $sgn(0)\equiv0$.

Writing this in Wolfram Alpha (with $m=g=1$ for simplicity) gives the following graphs related to explicit solution $y(t)$ and phase diagram trajectories (similar to a spring but instead of $sin$/$cos$ functions, its smoothly connected parabolas):

Inserting the Hamiltonian gives a better look at phase space appearance

As for solutions, here's one:

For given initial conditions $y_0>0$, $v_0$, the (downward-facing) initial parabolic trajectory is given by

$y_{init}(t)=y_0+v_0t-\frac g2t^2$.$\,\,\,\,\,\,\,\,\,\,$(1)

It will bounce for the first time when $y_{init}(t)=0$ with $t>0$ and the "last time" it "has bounced" is the other root. The difference between the two roots is the period of bouncing $T$, given by the square root of the Bhaskara formula's $\Delta$:

$T=\sqrt{v_0^2+2g\,y_0}$.

Then, by using a sawtooth function of period $T$

$\tau(t)=t-T \lfloor t/T \rfloor$,

the bouncing trajectory is

$y(t)=y_{init}(\tau(t))$,

with $y_{init}$ given by (1).

For $C^2$ trajectories (and actual solutions to the above hamiltonian) the sign of the function should change every period.

You can add the bouncing behavior using a mirroring method, where we relax the system and instead of bouncing the ball one allows it to cross to y<0 and changes the sign of the potential accordingly. The dynamic is exactly the same with a gain: trajectories become $C^2$. It can be written as

$H=\frac{p^2}{2m}+mg|y|$

where $|y|=y$ if $y>=0$ and $|y|=-y$ if $y<0$. Potential then looks like (for $m,g=1$)

Equation of motion becomes

$\ddot y(t)=-mg\,sgn(y)$,

where $sgn(y)$ is the sign function of y (a Heaviside-theta-like function). To allow equlilibrium at $y=0$ to exist, we can define $sgn(0)\equiv0$.

Writing this in Wolfram Alpha (with $m=g=1$ for simplicity) gives the following graphs related to explicit solution $y(t)$ and phase diagram trajectories (similar to a spring but instead of $sin$/$cos$ functions, its smoothly connected parabolas):

Inserting the Hamiltonian gives a better look at phase space appearance

As for explicit solutions, here's one:

For given initial conditions $y_0>0$, $v_0$, the (downward-facing) initial parabolic trajectory is given by

$y_{init}(t)=y_0+v_0t-\frac g2t^2$.$\,\,\,\,\,\,\,\,\,\,$(1)

It will bounce for the first time when $y_{init}(t)=0$ with $t>0$ and the "last time" it "has bounced" is the other root. The difference between the two roots is the period of bouncing $T$, given by the square root of the Bhaskara formula's $\Delta$:

$T=\sqrt{v_0^2+2g\,y_0}$.

Then, by using a sawtooth function of period $T$

$\tau(t)=t-T \lfloor t/T \rfloor$,

the bouncing trajectory is

$y(t)=y_{init}(\tau(t))$,

with $y_{init}$ given by (1).

For $C^2$ trajectories (and actual solutions to the above hamiltonian) the sign of the function should change every period.

You can add the bouncing behavior using a mirroring method, where we relax the system and instead of bouncing the ball one allows it to cross to y<0 and changes the sign of the potential accordingly. It can be written as

$H=\frac{p^2}{2m}+mg|y|$

where $|y|=y$ if $y>=0$ and $|y|=-y$ if $y<0$. Potential then looks like (for $m,g=1$)

Equation of motion becomes

$\ddot y(t)=-mg\,sgn(y)$,

where $sgn(y)$ is the sign function of y (a Heaviside-theta-like function). To allow equlilibrium at $y=0$ to exist, we can define $sgn(0)\equiv0$.

Writing this in Wolfram Alpha (with $m=g=1$ for simplicity) gives the following graphs related to explicit solution $y(t)$ and phase diagram trajectories (similar to a spring but instead of $sin$/$cos$ functions, its smoothly connected parabolas):

Inserting the Hamiltonian gives a better look at phase space appearance

As for solutions, here's one:

For given initial conditions $y_0>0$, $v_0$, the (downward-facing) initial parabolic trajectory is given by

$y_{init}(t)=y_0+v_0t-\frac g2t^2$.$\,\,\,\,\,\,\,\,\,\,$(1)

It will bounce for the first time when $y_{init}(t)=0$ with $t>0$ and the "last time" it "has bounced" is the other root. The difference between the two roots is the period of bouncing $T$, given by

$T=\sqrt{v_0^2+2g\,y_0}$.

Then, by using a sawtooth function of period $T$

$\tau(t)=t-T \lfloor t/T \rfloor$,

the bouncing trajectory is

$y(t)=y_{init}(\tau(t))$,

with $y_{init}$ given by (1).

For $C^2$ trajectories (and actual solutions to the above hamiltonian) the sign of the function should change every period.

You can add the bouncing behavior using a mirroring method, where we relax the system and instead of bouncing the ball one allows it to cross to y<0 and changes the sign of the potential accordingly. It can be written as

$H=\frac{p^2}{2m}+mg|y|$

where $|y|=y$ if $y>=0$ and $|y|=-y$ if $y<0$. Potential then looks like (for $m,g=1$)

Equation of motion becomes

$\ddot y(t)=-mg\,sgn(y)$,

where $sgn(y)$ is the sign function of y (a Heaviside-theta-like function). To allow equlilibrium at $y=0$ to exist, we can define $sgn(0)\equiv0$.

Writing this in Wolfram Alpha (with $m=g=1$ for simplicity) gives the following graphs related to explicit solution $y(t)$ and phase diagram trajectories (similar to a spring but instead of $sin$/$cos$ functions, its smoothly connected parabolas):

Inserting the Hamiltonian gives a better look at phase space appearance

As for solutions, here's one:

For given initial conditions $y_0>0$, $v_0$, the (downward-facing) initial parabolic trajectory is given by

$y_{init}(t)=y_0+v_0t-\frac g2t^2$.$\,\,\,\,\,\,\,\,\,\,$(1)

It will bounce for the first time when $y_{init}(t)=0$ with $t>0$ and the "last time" it "has bounced" is the other root. The difference between the two roots is the period of bouncing $T$, given by the square root of the Bhaskara formula's $\Delta$:

$T=\sqrt{v_0^2+2g\,y_0}$.

Then, by using a sawtooth function of period $T$

$\tau(t)=t-T \lfloor t/T \rfloor$,

the bouncing trajectory is

$y(t)=y_{init}(\tau(t))$,

with $y_{init}$ given by (1).

For $C^2$ trajectories (and actual solutions to the above hamiltonian) the sign of the function should change every period.

You can add the bouncing behavior using a mirroring method, where we relax the system and instead of bouncing the ball one allows it to cross to y<0 and changes the sign of the potential accordingly. It can be written as

$H=\frac{p^2}{2m}+mg|y|$

where $|y|=y$ if $y>=0$ and $|y|=-y$ if $y<0$. Potential then looks like (for $m,g=1$)

Equation of motion becomes

$\ddot y(t)=-mg\,sgn(y)$,

where $sgn(y)$ is the sign function of y (a Heaviside-theta-like function). To allow equlilibrium at $y=0$ to exist, we can define $sgn(0)\equiv0$.

Writing this in Wolfram Alpha (with $m=g=1$ for simplicity) gives the following graphs related to explicit solution $y(t)$ and phase diagram trajectories:

Inserting the Hamiltonian gives a better look at phase space appearance

As for solutions, here's one:

For given initial conditions $y_0>0$, $v_0$, the (downward-facing) initial parabolic trajectory is given by

$y_{init}(t)=y_0+v_0t-\frac g2t^2$.

It will bounce for the first time when $y_{init}(t)=0$ with $t>0$ and the "last time" it "has bounced" is the other root. The difference between the two roots is the period of bouncing $T$, given by

$T=\sqrt{v_0^2+2g\,y_0}$.

Then, by using a sawtooth function of period $T$

$\tau(t)=t-T \lfloor t/T \rfloor$,

the bouncing trajectory is

$y(t)=y_{init}(\tau(t))$.

For $C^2$ trajectories (and actual solutions to the above hamiltonian) the sign of the function should change every period.

You can add the bouncing behavior using a mirroring method, where we relax the system and instead of bouncing the ball one allows it to cross to y<0 and changes the sign of the potential accordingly. It can be written as

$H=\frac{p^2}{2m}+mg|y|$

where $|y|=y$ if $y>=0$ and $|y|=-y$ if $y<0$. Potential then looks like (for $m,g=1$)

Equation of motion becomes

$\ddot y(t)=-mg\,sgn(y)$,

where $sgn(y)$ is the sign function of y (a Heaviside-theta-like function). To allow equlilibrium at $y=0$ to exist, we can define $sgn(0)\equiv0$.

Writing this in Wolfram Alpha (with $m=g=1$ for simplicity) gives the following graphs related to explicit solution $y(t)$ and phase diagram trajectories (similar to a spring but instead of $sin$/$cos$ functions, its smoothly connected parabolas):

Inserting the Hamiltonian gives a better look at phase space appearance

As for solutions, here's one:

For given initial conditions $y_0>0$, $v_0$, the (downward-facing) initial parabolic trajectory is given by

$y_{init}(t)=y_0+v_0t-\frac g2t^2$.$\,\,\,\,\,\,\,\,\,\,$(1)

It will bounce for the first time when $y_{init}(t)=0$ with $t>0$ and the "last time" it "has bounced" is the other root. The difference between the two roots is the period of bouncing $T$, given by

$T=\sqrt{v_0^2+2g\,y_0}$.

Then, by using a sawtooth function of period $T$

$\tau(t)=t-T \lfloor t/T \rfloor$,

the bouncing trajectory is

$y(t)=y_{init}(\tau(t))$,

with $y_{init}$ given by (1).

For $C^2$ trajectories (and actual solutions to the above hamiltonian) the sign of the function should change every period.