We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a nature natural Lagrangian submanifold of it. A well known result is that any submanifold $X={(p,f(p)): p\in M}$, where $f$ is a closed one form is Lagrangian. Denote by $[f]$ the de Rham cohomology group class of $f$. Assume that we flow $X$ in a Hamiltonian direction to $Y$, then $Y$ will be a Lagrangian submanifold of $T^\star M$. My question is can we write it as $Y={(p,g(p)): p\in M}$ for some closed one form $g$.? If so, do we have $ [g]=[f]?$ Thanks in advance!

We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a nature Lagrangian submanifold of it. A well known result is that any submanifold $X={(p,f(p)): p\in M}$, where $f$ is a closed one form is Lagrangian. Denote by $[f]$ the Hodge homology de Rham cohomology group of $f$. Assume that we flow $X$ in a Hamiltonian direction to $Y$, then $Y$ will be a Lagrangian submanifold of $T^\star M$. My question is can we write it as $Y={(p,g(p)): p\in M}$ for some closed one form $g$.? If so, do we have $ [g]=[f]?$ Thanks in advance!Edit: I edit the question according to Serge's comments.

We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a nature Lagrangian submanifold of it. A well known result is that any Lagrangian submanifold $X$ in $T^\star M$ can be written as $X={(p,f(p)): p\in M}$, where $f$ is a closed one form is Lagrangian. Denote by $[f]$ the Hodge homology group of $f$. Assume that we flow $X$ in a Hamiltonian direction to $Y$, then $Y$ will be a Lagrangian submanifold of $T^\star M$and we . My question is can we write it as $Y={(p,g(p)): p\in M}$ for some closed one form $g$. My question is: g$.? If so, do we have $ [g]=[f]?$ Thanks in advance! Edit: I edit the question according to Serge's comments.

We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a nature Lagrangian submanifold of it. A well known result is that any Lagrangian submanifold $X$ in $T^\star M$ can be written as $X={(p,f(p)): p\in M}$, where $f$ is a closed one form. Denote by $[f]$ the Hodge homology group of $f$. Assume that we flow $X$ in a Hamiltonian direction to $Y$, then Y $Y$ will be a Lagrangian submanifold of $T^\star M$ and we can write it as $Y={(p,g(p)): p\in M}$ for some closed one form $g$. My question is: $ [g]=[f]?$ Thanks in advance!

We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a nature Lagrangian submanifold of it. A well known result is that any Lagrangian submanifold $X$ in $T^\star M$ can be written as $X={(p,f(p)): p\in M}$, where $f$ is a closed one form. Denote by $[f]$ the Hodge homology group of $f$. Assume that we flow $X$ in a Hamiltonian direction to $Y$, then Y will be a Lagrangian submanifold of $T^\star M$ and we can write it as $Y={(p,g(p)): p\in M}$ for some closed one form $g$. My question is: $ [g]=[f]?$ Thanks in advance!