Introduction

Since Bateman proposed the time-dependent Hamiltonian in a classical context ¹ for the illustration of dissipative systems, there has been much attention paid to quantum-mechanical treatments of nonlinear and nonconservative systems. In studying nonlinear systems, it is essential to introduce a time-dependent Hamiltonian which describes the frictional cases. This was discovered first by Caldirola ², and rederived independently by Kanai ³ via Bateman's dual Hamiltonian, and afterward by several others ⁴.

Hamilton Jacobi equations (HJE) are nonlinear first order equations which have been first introduced in classical mechanics, but find applications in many other fields of mathematics. Our interest in these equations lies mainly in the connection with calculus of variations and optimal control.

However, Hamilton-Jacobi method has been studied for a wide range of systems with time-independent Hamiltonians. For systems with time-dependent Hamiltonians, however, due to the complexity of dynamics, little has been known about quantum of action variables.

However, Hamilton-Jacobi theory builds a bridge between classical mechanics and other branches of physics. Mainly, the Hamilton–Jacobi equation can be viewed as a precursor to the Schrödinger equation ^5-11.

Our primary goals will be to extend the HJ formulation for time-dependent systems, building on the previous work by Rabei et al. (2002), the idea is to construct the Hamiltonian function and the corresponding equation of motion for dissipative systems. The methodology for that, the principal function is determined using the method of separation of variables.  The equation of motion can then be readily obtained.

Hamilton-Jacobi Formalism

We start with the Lagrangian



Here L₀(q,q) stands for the usual Lagrangian and λ is the dissipation factor. The generalized momentum is defined by ¹².



The corresponding Hamiltonian is



Hamilton's Jacobi equation is differential equation of the form:



It is a partial differential of (n+1) variables, q₁….q_n; t.

The complete solution of Eq. (4) can be written in the form ⁶



Eq. (5) presents S as a function of n coordinates, the time t, and n independent quantities α_i. We can take the n constants of integration to be constants of momenta:





The relationship between p and q then describes the orbit in phase space in terms of these constants of motion, furthermore the quantities



Are the equations also constants of motion, and these equations can be inverted to find q as a function of all α and β constants and time.

Thus, the Hamilton-Jacobi function is given by



The resulting action S is



or



We must write S in the separable form


 

The time-independent function W(q, α) is sometimes called Hamilton characteristic function.

Differentiating Eq. (13) with respect to t, we find that



From Eq. (10), it follows that



Therefore, the time derivative ∂S/∂t in HJE must be a constant, usually denoted by(-α).



It follows that



The equations of transformation are





While these equations resemble Eq. (7) and (8)respectively for Hamilton’s principal function S, the condition now determining W is that it is the new canonical momentum αi


 

Examples

Friction Linear in the Velocity

The Lagrangian depending on time is ^13-14.



The linear momentum is given by



This equation can readily be solving to give



The canonical Hamiltonian has the standard form



Now, substituting Eqs. (21) and (23) into (24), we get



The momentum can be computed from



Substituting Eq. (26) into Eq. (25), we find that



The Hamilton-Jacobi equation is



Substituting Eq. (27) into Eq. (28), we get



With the change of variable,



we can eliminate the factor τ in Eq. (29) which is transformed in



Now it is possible to propose



That is





Substituting Eqs. (33) and (34) into Eq. (31), we get



So that



Taking the square root of each side of Eq. (36), we have



Integrating Eq. (37), we have



Substituting this value of W into Eq. (32), we get



Returning to the variable t results in



Then we can obtain



and



We impose the initial condition x₀ = 0 for t = 0, which gives



Substituting Eq. (42) into Eq. (43) we can get the expression for x,



Then



The value for α is set from equation (41), taking p=p₀ for t = 0, that is



Finally, we have



In fact, this result is in agreement with that obtained by Euler's equation.

Friction Quadratic in the Velocity

It is also known that the equation of motion for a particle with Newtonian friction f=-mλv² can be derived from the Lagrangian ^14,15.



The linear momentum is given by



This equation can readily be solved to give



The canonical Hamiltonian has the standard from



Equation (51) becomes



The momentum can be computed from



Substituting Eq. (53) into Eq. (52)



With this Hamiltonian we can write the Hamilton-Jacobi equation



It is possible to propose



That is



And



Substituting Eqs. (57) and (58) into (55), we get



so that



Taking the square root of each side Eq. (60), we have



Integrating Eq. (61), we get



Substituting Eq. (62) into Eq. (56), we have



Applying the usual method, we obtain



And



Since v=pe^-2λx, if we take  x(t=0)=0 , then p₀ =v₀

the Eq.(64) is



Taking the square of each sides and substituting x = 0

 


Substituting Eq. (68) into Eq. (65) when x(t=0) =0, we get



Finally, we can obtain x from the Eqs. (65), (68) and (69)




Taking the logarithm of each side



In fact, this result is in agreement with that obtained by Euler's equation

Linearly Damped Particle with Constant Force

A suitable Lagrangian for the linearly damped particle moving in one dimension under a constant force is ^16.



The linear momentum is given by



This equation can readily be solved to give



The Hamiltonian is



Substituting Eqs. (73) and (75) into Eq. (76)



The conjugate momentum is then



Eq. (77) becomes



The HJE is



Where S = S (x, α, t),α is a parameter, the x and t variables are separated by the assumption



So,

 


The Eq. (80) becomes




Divide Eq. (84) to T²W  and multiply by e^γt



Let



So,



Substituting Eqs. (86) and (87) into Eq. (85), we get



Eq. (88) becomes



We can write Eq. (89)  as below



Differentiation Eq. (90) and WËŠ  replaced by y Eq. (90) becomes




Divide Eq. (91) on 2



Which is separable



Integration Eq. (94), we get



Let



So.



Differentiate Eq. (97)



Substituting Eqs. (97) and (98) into Eq. (95), we get




Eq. (100) becomes



Instead of its value u in Eq. (101)



Differentiating Eq. (102) with respect to c



Unification of the denominators yield in Eq. (103), we have




Take out (γyËŠ) a common factor in Eq. (105), we have



Eq. (106) becomes



So,



Eq. (109) becomes



From Eq. (110)



Replace WËŠby y in Eq. (111), we obtain



Note that when derived Eq. (112) the limit (-2gx) equal zero



Multiply Eq. (113) by (-1/2), we get



From Eqs. (110) and (114), we have



From Eqs. (106) and (115), we get



Substituting Eq. (86) into Eq. (81)



Then, if the parameter α is identified as c,



So,



From Eq. (116),we obtain



Substituting Eq. (119) into Eq. (120)



Conclusion

In this paper, we have identified explicit time-ependent first integrals for the damped systems valid in different parameter regimes using the modified Hamilton-Jacobi approach. We have constructed the appropriate Hamiltonians from the time-dependent first integrals and transformed the corresponding Hamiltonian forms to standard Hamiltonian forms using suitable canonical transformations.

In addition, the solution of the Hamilton-Jacobi equations for such dissipative Hamiltonians have been constructed. We have derived an expression for the Hamilton-Jacobi equation and have applied our results for a number of time-dependent models including dissipation terms. Among them are: friction linear in the velocity; friction quadratic in the velocity; friction quadratic in the velocity in a constant gravitational field; the linearly damped particle with constant force.

Acknowledgment

The author thanks Mutah University for its continuous support for scientific research and for providing various research resources.

Funding Sources

There are no funding sources.

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