After reducing a system of higher-order regular Lagrangian into first-order singular Lagrangian using constrained auxiliary description, the Hamilton-Jacobi function is constructed. Besides, the quantization of the system is investigated using the canonical path integral approximation.

The efforts to quantize systems with constraints started with the work of Dirac [

A new formalism for investigating first-order singular systems-, the canonical-, was developed by Rabei and Guler [

Moreover, the quantization of constrained systems has been studied for first-order singular Lagrangians using the WKB approximation [

The Hamiltonian formulation for systems with higher-order regular Lagrangians was first developed by Ostrogradski [

In Ostrogradski's construction, this problem can be resolved within the well-established context of constrained systems [

The purpose of the present paper is to study the canonical path integral quantization for singular systems with arbitrary higher-order Lagrangian. In fact, this work is a continuation of the previous work [

The present work is organized as follows: in Section

The starting point is a singular Lagrangian

The canonical formulation [_{0}_{a}_{i}_{μ}

Consider a higher-order Lagrangian system of

The Euler-Lagrange equations of motion are obtained as [

Theories with higher derivatives, which have been first developed by Ostrogradski [

With this procedure, the phase space, described in terms of the canonical variables

Recall the higher-order Lagrangian given in (

Upon introducing the canonical momenta:

If the coordinates

In this section, the procedure described throughout this paper will be illustrated by the following two examples.

As a first example, let us consider a one-dimensional second-order regular lagrangian of the form:

Using (

As a second example, consider the three-dimensional second-order regular lagrangian:

In this work, we have investigated the canonical path integral quantization of higher-order regular Lagrangians. Where the higher-order regular Lagrangians are first treated as first-order singular Lagrangians, this means that each velocity term

Once the extended Lagrangian is obtained, it is treated using the well-known Hamilton-Jacobi method which enables us to obtain the equations of motion. Besides, the action integral can be derived and the quantization of the system may be investigated using the canonical path integral approximation.

In this treatment, we believe that the local structure of phase space and its local simplistic geometry is more transparent than in Ostrogradski's approach. In Ostrogradski's approach, the structure of phase space leads to confusion when considering canonical path integral quantization.