Hamilton-Jacobi Equation of Time Dependent Hamiltonians

In this work, we apply the geometric Hamilton-Jacobi theory to obtain solution of Hamiltonian systems in classical mechanics that are either compatible with two structures: the first structure plays a central role in the theory of timedependent Hamiltonians, whilst the second is used to treat classical Hamiltonians including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the non stationary case can be obtained naturally in Hamilton-Jacobi formalism. CONTACT Khaled I. Nawafleh knawaflehh@yahoo.com Faculty of Science, Department of Physics Mutah University, Jordan. © 2020 The Author(s). Published by Oriental Scientific Publishing Company This is an Open Access article licensed under a Creative Commons license: Attribution 4.0 International (CC-BY). Doi: http://dx.doi.org/10.13005/OJPS05.01-02.04 Oriental Journal of Physical Sciences www.orientaljphysicalsciences.org ISSN: 2456-799X, Vol.05, No.(1-2) 2020, Pg. 09-15 Article History Received: 10 February 2020 Accepted: 27 April 2020


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 non conservative systems. In studying nonlinear systems, it is essential to introduce a timedependent 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, butfind 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][6][7][8][9][10][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. FUQARA et al., Orient. J. Phys. Sciences, Vol. 5 (1-2) 09-15 (2020)

Hamilton-Jacobi Formalism
We start with the Lagrangian Here L 0 (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: H (q 1 ……q n ; ∂S / ∂q i ,……,∂S / ∂q n ; t) + ∂S / ∂t = 0 ... (4) It is a partial differential of (n+1) variables, q 1 ..q n ; t.
The complete solution of Eq. (4) can be written in the form 6 S= S(q 1 …….q n ; α i ……α n ; t) ... (5) 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 βconstants and time.
Thus, the Hamilton-Jacobi function is given by The resulting action S is We must write S in the separable form ... (13) Thetime-independent function W(q, α) is sometimes called Hamilton characteristic function.
S (q,α,t) = W(q,α) -α(t) ... (16) 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

Examples Friction Linear in the Velocity
The Lagrangian depending on time is. [13][14] L= L= 1/2 e λt x ̇2 ... In fact, this result is in agreement with that obtained by Euler'sequation.

Friction Quadratic in the Velocity
It is also known that the equation of motion for a particle with Newtonian friction f = -mλv 2 can be derived from the Lagrangian. 14,15 .

..(48)
The linear momentum is given by This equation can readily be solved to give The canonical Hamiltonian has the standard from 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 .

Conclusion
In this paper, we have identified explicit timeependent 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, thesolution of the Hamilton-Jacobi equations forsuch 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.