Jacobi integral in generalized mechanics

Abstract

It is shown that even for systems whose Lagrangians depend on higher derivatives of the generalized coordinates, a Jacobi integral (or Hamiltonian function) can still be found in the usual sense that it becomes a conserved quantity whenever the Lagrangian is not explicitly dependent on time. The term 'generalized mechanics' refers to study of mechanical systems whose Lagrangians depend on higher derivatives of the generalized coordinates q_j. The conventional form of the Euler-Lagrange equation (∂L/∂q_j) − d[(∂L/∂q̇_j)]/dt = 0, no longer holds in such cases since this was derived for the usual monogenic Lagrangian — one that depends only on the generalized coordinates q_j, the generalized velocities q̇_j = dq_j/dt, and time t. If for instance the system's Lagrangian depends on up to the jerk q_j⁽³⁾ = d³q_j /dt³, then the appropriate equation of motion is found to be
(∂L/∂q_j) − d[(∂L/∂q̇_j)]/dt + d²[(∂L/∂q̈_j)]/dt² − d³[(∂L/∂q_j⁽³⁾)]/dt³ = 0, 
provided the variations at the endpoints are assumed to vanish as in the conventional case:
δq_j(t₁) = δq_j(t₂) = δq̇_j(t₁) = δq̇_j(t₂) = δq̈_j(t₁) = δq̈_j(t₂) = 0.
The extended Euler-Lagrange equation is the appropriate equation of motion for systems that are subject to a generalized potential that depends on up to the third derivative of the generalized coordinates. Such jerky potentials are rarely given non-perturbative analyses not only because of the threatening mathematical complexities that it portends but also because of the lack of decent non-perturbative formalisms for the subject in the literature. More often than not, because these systems are immediately branded as nonconservative, dissipative, and highly nonlinear, one resigns in defeat and resorts to oftentimes unreasonable assumptions or approximations. It is then not surprising that people have failed to realize that even for such highly nonlinear dissipative systems, a Jacobi function can still be defined. Although it will in general be dependent on higher derivatives of the generalized coordinates, it nevertheless possesses the important property that it remains constant in time even when the Lagrangian of the system does not explicitly depend on time. This is the thesis of this paper: to show that the concept of the Jacobi integral need not be confined to ordinary monogenic systems and that no matter how many derivatives of the generalized coordinates are involved in the Lagrangian, an appropriately defined Jacobi integral can be found with the usual property of being conserved whenever the Lagrangian is not explicitly dependent on time.