Hamilton-Jacobi Formalism of Singular Lagrangians with Linear Accelerations

This paper examined a new model for solving mechanical problems of second-order linear Lagrangian systems, using the Hamilton-Jacobi formalism. Lagrangians linear in accelerations with coefficients given by functions of coordinates alone yield primary constraints. It is shown that the equations of motion can be obtained from the action integral and these equations are equivalent to the canonical method.


Introduction
The canonical formalism for investigating singular systems has been developed by (Rabei & Guler, 1992;Pimentel & Teixeiria, 1996, 1998).A set of Hamilton-Jacobi partial differential equations was obtained and the equations of motion were written as total differential equations.
The Hamilton-Jacobi treatment has been studied for singular Lagrangians (Rabei et al., 2004).The Hamilton-Jacobi functions in configuration space have been obtained by solving the HJPDEs.This has led to another approach for solving mechanical problems for these singular systems.
Singular Lagrangians with linear velocities have been studied (Rabei et al., 2003) by using the canonical method.In this method, the integrable action was obtained directly without considering the total variation of constraints.In this paper, we wish to extend the model for second-order linear Lagrangian.
More recently, the path integral quantization of Lagrangians with linear accelerations has been investigated (Hasan, 2014) by using the canonical method.It is shown that by calculating the integrable action and constructing the wave function, the quantization has been carried out.This paper is organized as follow.In Section 2, a new model of singular Lagrangian with linear acceleration is proposed.In Section 3, several illustrative examples are examined.The work closes with some concluding remarks in Section 4.

The Model of Hamilton-Jacobi Formalism for Lagrangian with Linear Acceleration
The general form of a second-order linear Lagrangian is ( , , ) ( , ) ( , ) The associated Euler-Lagrange equations Have at most order three.Lagrangians linear in accelerations with coefficients given by functions of coordinates alone yield primary constraints.If , then the general form of a second-order linear Lagrangian becomes ( , , ) ( ) ( ) The generalized momenta i p , i  conjugate to the generalized coordinates i q , i q  , respectively: Equations (2.4) and (2.5) become
The canonical Hamiltonian 0 H is given by: The corresponding HJPDEs The equations of motion are obtained as total differential equations follows: The set of Equations (2.10) are integrable (Muslih & Guler, 1998), the total variation of Equation (2.6) and Equation (2.7) can be written as: which is equivalent to ( ) ( ) If the inverse of the matrix ij f exist, then we can solve all the dynamics i q , while if the rank of the matrix ij f is n-R, then we can solve the dynamics a q in terms of independent parameters ( , , .
The total derivative of the Hamilton-Jacobi function can be obtained as: Using the HJPDEs Equations (2.9), we get (2.17) One can integrate the above Equation (2.17) to give We can use the fact that ( ) (2.20) And using the fact that Assuming that the function ( ) i a q and ( ) V q satisfy the following conditions The action S to be an integrable function, the terms in the brackets must be zero, i.e.


(2.23) Equation (2.23) gives the equation of motion for the coordinates j q .

The First Example
Consider the following singular Lagrangian: The potential of this Lagrangian is given by and the coefficients 1 a and 2 a are The generalized momenta by using Equation (2.4) and Equation (2.5) are: From Equation (2.6) and Equation (2.7) the primary constraints are given as (3.4) Making use of (2.23), we can obtain the equation of motion for 1 q and 2 q These equations are given by 0 2 (3.7b)

The Second Example
Let consider the singular Lagrangian: The potential of this Lagrangian is given by and the coefficients 1 a , 2 a and 3 a are The generalized momenta by using Equation (2.4) and Equation (2.5) are: By Equation (2.6) and Equation (2.7) the primary constraints are given as (3.12) Making use of (2.23), we can obtain the equation of motion for 3 q

Conclusion
This paper investigated the Hamilton-Jacobi formalism for singular Lagrangian with linear acceleration.Lagrangians linear in accelerations with coefficients given by functions of coordinates alone yield primary constraints.It is proven that the total derivative of the Hamilton-Jacobi function has been constructed using the HJPDEs and Hamilton-Jacobi function is integrable.It is shown that both the equations of motion and the integrable action are obtained from the integrability conditions and the number of independent parameters are determined from the rank of matrix ij f .