Strange Hamilton Jacobi equation

[eljose]

let be (dS/dt)+(gra(S))^2/2m+(LS)+V(x) where L is the Laplacian Operator and V is the potential...could it be considered as the Hamiltan Jacobi equation for a particle under a potential Vtotal=V(x)+(LS) where S is the action

 

[Coelum]

let be (dS/dt)+(gra(S))^2/2m+(LS)+V(x) where L is the Laplacian Operator and V is the potential...could it be considered as the Hamiltan Jacobi equation for a particle under a potential Vtotal=V(x)+(LS) where S is the action

I assume you mean to equate to 0, i.e.:

$$\frac{\partial S}{\partial t}\right)+\frac{(\vec\nabla S)^2}{2m}+\vec\nabla S+V(q)=0$$

If we compare it to the Hamillton-Jacobi equation for the generating function S (a concept more general than the "action")

$$H\left(q,\frac{\partial S}{\partial q},t\right)+\frac{\partial S}{\partial t}=0$$

we find they are compatible provided we let

$$H\left(q,\frac{\partial S}{\partial q},t\right)=\frac{(\vec\nabla S)^2}{2m}+\vec\nabla S+V(q)$$

Since $$p_i=\frac{\partial S}{\partial q_i}$$, we can rewrite it as

$$H\left(q_i,p_i,t\right)=\frac{(\sum_i p_i)^2}{2m}+\vec\nabla S+V(q)$$

or

$$H=T+W$$

where

$$W=\sum_i p_i+V(q_i)$$.

Here we see that $$W=f(p_i,q_i)$$, in other words the "potential" W is not conservative and the meaning of W is that of "virtual work". Is that the source of your doubts?