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.:

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

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

[tex] H\left(q,\frac{\partial S}{\partial q},t\right)+\frac{\partial S}{\partial t}=0 [/tex]

we find they are compatible provided we let

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

Since [tex] p_i=\frac{\partial S}{\partial q_i} [/tex], we can rewrite it as

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

or

[tex] H=T+W [/tex]

where

[tex] W=\sum_i p_i+V(q_i) [/tex].

Here we see that [tex] W=f(p_i,q_i) [/tex], 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?

 

[GSXR750]

The Strange Hamilton Jacobi equation is an interesting and complex equation that describes the dynamics of a particle under a potential. It is given by (dS/dt)+(gra(S))^2/2m+(LS)+V(x), where S is the action, L is the Laplacian Operator, and V is the potential. It can also be written as (dS/dt)+(H)+(V(x)), where H is the Hamiltonian operator.

This equation can be considered as the Hamilton Jacobi equation for a particle under a potential Vtotal=V(x)+(LS). The addition of the term (LS) in the potential reflects the presence of a non-conservative force, which can be thought of as a sort of friction or resistance acting on the particle. This term also makes the equation more complex and "strange" compared to the standard Hamilton Jacobi equation.

The presence of the Laplacian operator in the equation also adds a spatial component, making it more relevant for describing the motion of a particle in a physical space. This equation is often used in quantum mechanics to describe the behavior of particles in a potential field, and has many applications in various areas of physics and mathematics.

Overall, the Strange Hamilton Jacobi equation is a powerful tool for understanding the dynamics of particles under a potential and can provide valuable insights into the behavior of complex systems. Its complexity and versatility make it an important equation in the study of physics and its applications.