My question concerns whether or not the topology of a trajectory in the configuration space of an integrable system, or rather the area of configuration space accessible by trajectories, can be determined by using the projection of a trajectory from the phase space into the configuration space?

For context, I am interested in determining the topology of trajectories in the configuration space, specifically as billiard trajectories within the triaxial ellipsoid in $\mathbb{R}^3$. Due to the billiard system within the ellipsoid being integrable it is well known that trajectories, ignoring singular cases, are geodesics on Liouville 3-tori, that foliate the phase space. Within the ellipsoid itself, every line of the trajectory is tangent to two confocal quadrics known as caustics and these caustics define a space within the ellipsoid that trajectories are then limited to and it is the topology of this space that I am interested in.

According to another result, the caustics arise as the singular components of the projection of these Liouville tori (defined by trajectories) from the phase space to the configuration space. I have seen this result mentioned a number of times but for whatever reason, I have not been able to find a direct computation or proof of it. I have also seen a few times such a projection written as $\pi \, : \, T^*Q \rightarrow Q$ where $T^*Q$ is the cotangent bundle (phase space) and $Q$ is a smooth manifold (the configuration space), but again I have not had much luck in the way of finding computations that are more specific to a concrete system.

Therefore, if a solution to my problem is already known, a point in the right direction would be greatly appreciated. Otherwise (or additionally), where might I find an explicit example/computation and/or proof of the result concerning projecting phase space Liouville tori to configuration space (either ellipsoidal billiards specific or broadly in Hamiltonian dynamics)?

Thank you!

My question concerns whether or not the topology of a trajectory in the configuration space of an integrable system, or rather the area of configuration space accessible by trajectories can be determined by using the projection of a trajectory from the phase space into the configuration space?

For context, I am interested in determining the topology of trajectories in the configuration space, specifically as billiard trajectories within the triaxial ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$, in $\mathbb{R}^3$. Due to the billiard system within the ellipsoid being integrable it is well known that trajectories, ignoring singular cases, are geodesics on Liouville 3-tori, that foliate the phase space. Within the ellipsoid itself, every line of the trajectory is tangent to two confocal quadrics known as caustics and these caustics define a space within the ellipsoid that trajectories are then limited to and it is the topology of this space that I am interested in.

According to another result, the caustics arise as the singular components of the projection of these Liouville tori (defined by trajectories) from the phase space to the configuration space. I have seen this result mentioned a number of times but for whatever reason, I have not been able to find a direct computation or proof of it. I have also seen a few times such a projection written as $\pi \, : \, T^*Q \rightarrow Q$ where $T^*Q$ is the cotangent bundle (phase space) and $Q$ is a smooth manifold (the configuration space), but again I have not had much luck in the way of finding computations that are more specific to a concrete system.

Therefore, if a solution to my problem is already known, a point in the right direction would be greatly appreciated. Otherwise (or additionally), where might I find an explicit example/computation and/or proof of the result concerning projecting phase space Liouville tori to configuration space (either ellipsoidal billiards specific or broadly in Hamiltonian dynamics)?

Thank you!

My question concerns whether or not the topology of a trajectory in the configuration space of an integrable system, or rather the area of configuration space accessible by trajectories, can be determined by using the projection of a trajectory from the phase space into the configuration space?

For context, I am interested in determining the topology of trajectories in the configuration space, specifically as billiard trajectories within the triaxial ellipsoid in $\mathbb{R}^3$. Due to the billiard system within the ellipsoid being integrable it is well known that trajectories, ignoring singular cases, are geodesics on Liouville 3-tori, that foliate the phase space. Within the ellipsoid itself, every line of the trajectory is tangent to two confocal quadrics known as caustics and these caustics define a space within the ellipsoid that trajectories are then limited to and it is the topology of this space that I am interested in.

According to another result, the caustics arise as the singular components of the projection of these Liouville tori (defined by trajectories) from the phase space to the configuration space. I have seen this result mentioned a number of times but for whatever reason, I have not been able to find a direct computation or proof of it. I have also seen a few times such a projection written as $\pi \, : \, T^*Q \rightarrow Q$ where $T^*Q$ is the cotangent bundle (phase space) and $Q$ is a smooth manifold (the configuration space), but again I have not had much luck in the way of finding computations that are more specific to a concrete system.

Therefore, if a solution to my problem is already known a point in the right direction would be greatly appreciated. Otherwise (or additionally), where might I find an explicit example/computation and/or proof of the result concerning projecting phase space Liouville tori to configuration space (either ellipsoidal billiards specific or broadly in Hamiltonian dynamics)?

Thank you!

My question concerns whether or not the topology of a trajectory in the configuration space of an integrable system, or rather the area of configuration space accessible by trajectories, can be determined by using the projection of a trajectory from the phase space into the configuration space?

For context, I am interested in determining the topology of trajectories in the configuration space, specifically as billiard trajectories within the triaxial ellipsoid in $\mathbb{R}^3$. Due to the billiard system within the ellipsoid being integrable it is well known that trajectories, ignoring singular cases, are geodesics on Liouville 3-tori, that foliate the phase space. Within the ellipsoid itself, every line of the trajectory is tangent to two confocal quadrics known as caustics and these caustics define a space within the ellipsoid that trajectories are then limited to and it is the topology of this space that I am interested in.

According to another result, the caustics arise as the singular components of the projection of these Liouville tori (defined by trajectories) from the phase space to the configuration space. I have seen this result mentioned a number of times but for whatever reason, I have not been able to find a direct computation or proof of it. I have also seen a few times such a projection written as $\pi \, : \, T^*Q \rightarrow Q$ where $T^*Q$ is the cotangent bundle (phase space) and $Q$ is a smooth manifold (the configuration space), but again I have not had much luck in the way of finding computations that are more specific to a concrete system.

Therefore, if a solution to my problem is already known, a point in the right direction would be greatly appreciated. Otherwise (or additionally), where might I find an explicit example/computation and/or proof of the result concerning projecting phase space Liouville tori to configuration space (either ellipsoidal billiards specific or broadly in Hamiltonian dynamics)?

Thank you!

My question concerns whether or not the topology of a trajectory in configuration space, or rather the area of configuration space accessible by trajectories, can be determined by using the projection of a trajectory from the phase space into the configuration space?

For context, I am interested in determining the topology of trajectories in the configuration space, specifically as billiard trajectories within the triaxial ellipsoid in $\mathbb{R}^3$. Due to the billiard system within the ellipsoid being integrable it is well known that trajectories, ignoring singular cases, are geodesics on Liouville 3-tori, that foliate the phase space. Within the ellipsoid itself, every line of the trajectory is tangent to two confocal quadrics known as caustics and these caustics define a space within the ellipsoid that trajectories are then limited to and it is the topology of this space that I am interested in.

According to another result, the caustics arise as the singular components of the projection of these Liouville tori (defined by trajectories) from the phase space to the configuration space. I have seen this result mentioned a number of times but for whatever reason, I have not been able to find a direct computation or proof of it. I have also seen a few times such a projection written as $\pi \, : \, T^*Q \rightarrow Q$ where $T^*Q$ is the cotangent bundle (phase space) and $Q$ is a smooth manifold (the configuration space), but again I have not had much luck in the way of finding computations that are more specific to a concrete system.

Therefore, if a solution to my problem is already known a point in the right direction would be greatly appreciated. Otherwise (or additionally), where might I find an explicit example/computation and/or proof of the result concerning projecting phase space Liouville tori to configuration space (either ellipsoidal billiards specific or broadly in Hamiltonian dynamics)?

Thank you!

My question concerns whether or not the topology of a trajectory in the configuration space of an integrable system, or rather the area of configuration space accessible by trajectories, can be determined by using the projection of a trajectory from the phase space into the configuration space?

For context, I am interested in determining the topology of trajectories in the configuration space, specifically as billiard trajectories within the triaxial ellipsoid in $\mathbb{R}^3$. Due to the billiard system within the ellipsoid being integrable it is well known that trajectories, ignoring singular cases, are geodesics on Liouville 3-tori, that foliate the phase space. Within the ellipsoid itself, every line of the trajectory is tangent to two confocal quadrics known as caustics and these caustics define a space within the ellipsoid that trajectories are then limited to and it is the topology of this space that I am interested in.

According to another result, the caustics arise as the singular components of the projection of these Liouville tori (defined by trajectories) from the phase space to the configuration space. I have seen this result mentioned a number of times but for whatever reason, I have not been able to find a direct computation or proof of it. I have also seen a few times such a projection written as $\pi \, : \, T^*Q \rightarrow Q$ where $T^*Q$ is the cotangent bundle (phase space) and $Q$ is a smooth manifold (the configuration space), but again I have not had much luck in the way of finding computations that are more specific to a concrete system.

Therefore, if a solution to my problem is already known a point in the right direction would be greatly appreciated. Otherwise (or additionally), where might I find an explicit example/computation and/or proof of the result concerning projecting phase space Liouville tori to configuration space (either ellipsoidal billiards specific or broadly in Hamiltonian dynamics)?

Thank you!