Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.

For an H-J eqn, I mean a first order pde $H\circ dS=0$ in an unknown scalar function $S$ defined on a smooth manifold $M$, where $H\in C^\infty (T^\ast M,\mathbb{R})$.

If $S$ is a solution then the image $\Lambda$ of its differential $dS$ is included in $H^{-1}(0)$ and has the following properties:

1. $\Lambda$ is a lagrangian submanifold of $(T^\ast M,d\theta_M)$,
2. $\Lambda$ is transversal to the fibers of $\tau_M^{\ast}:T^\ast M\to M$,
3. the restriction of $\tau_M^{\ast}$ to $\Lambda$ is injective.

Conversely, if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies the properties 1, 2, and 3, then it is equal to the image of the differential of a solution, unique up to a constant.

But if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies only the conditions 1 and 2, then, around each of its points, it is again equal to the image of the differential of a solution, but this can fail to holds globally.

The idea of Sophus Lie was to give up both conditions 2 and 3.

Adopting this point of view, we define a generalized (or geometric) solution is defined of $H\cic dS=0$ to be any lagrangian submanifold $\Lambda$ of $(T^\ast M,d\theta_M)$ whic which is included in $H^{-1}(0)$.

I don't think that this generalization is only due to the sake of abstractness. Infact, considering generalized solutions, it is possible, arguing with tecniques from symplectic geometry, to prove the local existence and uniqueness theorem, at the same time, for generalized and usual solutions.

But I am hoping to find "more" practical applications which illustrate the meaningfulness of geometric solutions. I would like to learn if ther is some physical or geometrical problem involving an H.-J. eqn, whose comprehension is sensibly augmented by the consideration of generalized solutions. So my question is:

What are the practical possible arguments and applications that motivate and help to interpret the notion of geometric solutions for an Hamilton-Jacobi equation?

Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.

For an H-J eqn, I mean a first order pde $H\circ dS=0$ in an unknown scalar function $S$ defined on a smooth manifold $M$, where $H\in C^\infty (T^\ast M,\mathbb{R})$.

If $S$ is a solution then the image $\Lambda$ of its differential $dS$ is included in $H^{-1}(0)$ and has the following properties:

1. $\Lambda$ is a lagrangian submanifold of $(T^\ast M,d\theta_M)$,
2. $\Lambda$ is transversal to the fibers of $\tau_M^{\ast}:T^\ast M\to M$,
3. the restriction of $\tau_M^{\ast}$ to $\Lambda$ is injective.

Conversely, if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies the properties 1, 2, and 3, then it is equal to the image of the differential of a solution, unique up to a constant.

But if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies only the conditions 1 and 2, then, around each of its points, it is again equal to the image of the differential of a solution, but this can fail to holds globally.

The idea of Sophus Lie was to give up both conditions 2 and 3.
Adopting this point of view, a generalized (or geometric) solution is defined to be any lagrangian submanifold $Lambda$ \Lambda$ of $(T^\ast M,d\theta_M)$ whic is included in $H^{-1}(0)$.

I don't think that this generalization is only due to the sake of abstractness. Infact, considering generalized solutions, it is possible, arguing with tecniques from symplectic geometry, to prove the local existence and uniqueness theorem, at the same time, for generalized and usual solutions.

For this reason,

But I would know if the admission of geometric solutions is meaningful even for am hoping to find "more" practical applications which illustrate the meaningfulness of geometric solutions. So my question is:

What are the practical applications that motivate and help to understand interpret the notion of geometric solutions for H.-J. eqnan Hamilton-Jacobi equation?

Studying the Hamilton-Jacobi equation, I meet a generalization of the notion of its solutions, which is found already in the work of Sophus Lie.

For an H-J eqn, I mean a first order pde $H\circ dS=0$ in an unknown scalar function $S$ defined on a smooth manifold $M$, where $H\in C^\infty (T^\ast M,\mathbb{R})$.

If $S$ is a solution then the image $\Lambda$ of its differential $dS$ is included in $H^{-1}(0)$ and has the following properties:

1. $\Lambda$ is a lagrangian submanifold of $(T^\ast M,d\theta_M)$,
2. $\Lambda$ is transversal to the fibers of $\tau_M^{\ast}:T^\ast M\to M$,
3. the restriction of $\tau_M^{\ast}$ to $\Lambda$ is injective.

Conversely, if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies the properties 1, 2, and 3, then it is equal to the image of the differential of a solution, unique up to a constant.

But if a submanifold $\Lambda$ of $T^\ast M$, included in $H^{-1}(0)$, satisfies only the conditions 1 and 2, then, around each of its points, it is again equal to the image of the differential of a solution, but this can fail to holds globally.

The idea of Sophus Lie was to give up both conditions 2 and 3.
Adopting this point of view, a generalized (or geometric) solution is defined to be any lagrangian submanifold $Lambda$ of $(T^\ast M,d\theta_M)$ whic is included in $H^{-1}(0)$.

I don't think that this generalization is only due to the sake of abstractness. Infact, considering generalized solutions, it is possible, arguing with tecniques from symplectic geometry, to prove the local existence and uniqueness theorem, at the same time, for generalized and usual solutions.

For this reason, I would know if the admission of geometric solutions is meaningful even for "more" practical applications. So my question is:

What are the practical applications that motivate and help to understand the notion of geometric solutions for H.-J. eqn?