Hamiltonian Mechanics for Functionals Involving Second-Order Derivatives

2002 ◽  
Vol 69 (6) ◽  
pp. 749-754 ◽  
Author(s):  
B. Tabarrok ◽  
C. M. Leech
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Jacobi Equation ◽  
Equations Of Motion ◽  
Second Order ◽  
Hamiltonian Mechanics ◽  
Energy Functionals ◽  
First Order ◽  
Independent Variable ◽  
Hamilton Jacobi Equation

Hamilton’s principle was developed for the modeling of dynamic systems in which time is the principal independent variable and the resulting equations of motion are second-order differential equations. This principle uses kinetic energy which is functionally dependent on first-order time derivatives, and potential energy, and has been extended to include virtual work. In this paper, a variant of Hamiltonian mechanics for systems whose motion is governed by fourth-order differential equations is developed and is illustrated by an example invoking the flexural analysis of beams. The variational formulations previously associated with Newton’s second-order equations of motion have been generalized to encompass problems governed by energy functionals involving second-order derivatives. The canonical equations associated with functionals with second order derivatives emerge as four first-order equations in each variable. The transformations of these equations to a new system wherein the generalized variables and momenta appear as constants, can be obtained through several different forms of generating functions. The generating functions are obtained as solutions of the Hamilton-Jacobi equation. This theory is illustrated by application to an example from beam theory the solution recovered using a technique for solving nonseparable forms of the Hamilton-Jacobi equation. Finally whereas classical variational mechanics uses time as the primary independent variable, here the theory is extended to include static mechanics problems in which the primary independent variable is spatial.

scholarly journals I. Memoir on the integration of partial differential equations second order in three independent variables when an intermediary integral does not exist in general

1898 ◽  
Vol 191 ◽  
pp. 1-86 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Second Order ◽  
Original Equation ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Independent Variable ◽  
Two Independent Variables

The general feature of most methods for the integration of partial differential equations in two independent variables is, in some form or other, the construction of a set of subsidiary equations in only a single independent variable; and this applies to all orders. In particular, for the first order in any number of variables (not merely in two), the subsidiary system is a set of ordinary equations in a single independent variable, containing as many equations as dependent variables to be determined by that subsidiary system. For equations of the second order which possess an intermediary integral, the best methods (that is, the most effective as giving tests of existence) are those of Boole, modified and developed by Imschenetsky, and that of Goursat, initially based upon the theory of characteristics, but subsequently brought into the form of Jacobian systems of simultaneous partial equations of the first order. These methods are exceptions to the foregoing general statement. But for equations of the second order or of higher orders, which involve two independent variables and in no case possess an intermediary integral, the most general methods are that of Ampere and that of Darboux, with such modifications and reconstruction as have been introduced by other writers; and though in these developments partial differential equations of the first order are introduced, still initially the subsidiary system is in effect a system with one independent variable expressed and the other, suppressed during the integration, playing a parametric part. In oilier words, the subsidiary system practically has one independent variable fewer than the original equation. In another paper I have given a method for dealing with partial differential equations of the second order in three variables when they possess an intermediary integral; and references will there be found to other writers upon the subject. My aim in the present paper has been to obtain a method for partial differential equations of the second order in three variables when, in general, they possess no intermediary integral. The natural generalisation of the idea in Darboux’s method has been adopted, viz., the construction of subsidiary equations in which the number of expressed independent variables is less by unity than the number in the original equation; consequently the number is two. The subsidiary equations thus are a set of simultaneous partial differential equations in two independent variables and a number of dependent variables.

Hamiltonian mechanics

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Configuration Space ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Hamiltonian Mechanics ◽  
Lagrange Equations ◽  
First Order ◽  
Point Transformations ◽  
First Order Differential Equations

This chapter gives a brief overview of Hamiltonian mechanics. The complexity of the Newtonian equations of motion for N interacting bodies led to the development in the late 18th and early 19th centuries of a formalism that reduces these equations to first-order differential equations. This formalism is known as Hamiltonian mechanics. This chapter shows how, given a Lagrangian and having constructed the corresponding Hamiltonian, Hamilton’s equations amount to simply a rewriting of the Euler–Lagrange equations. The feature that makes the Hamiltonian formulation superior is that the dimension of the phase space is double that of the configuration space, so that in addition to point transformations, it is possible to perform more general transformations in order to simplify solving the equations of motion.

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

Coherent states, Schrödinger equation, and the Hamilton–Jacobi equation

1995 ◽  
Vol 73 (7-8) ◽  
pp. 478-483
Author(s):  
Rachad M. Shoucri
Keyword(s):  
Schrödinger Equation ◽  
Jacobi Equation ◽  
Coherent States ◽  
Schrodinger Equation ◽  
Hidden Variables ◽  
Classical Equation ◽  
Equation Of Motion ◽  
Phase Variable ◽  
Independent Variable ◽  
Hamilton Jacobi Equation

The self-adjoint form of the classical equation of motion of the harmonic oscillator is used to derive a Hamiltonian-like equation and the Schrödinger equation in quantum mechanics. A phase variable ϕ(t) instead of time t is used as an independent variable. It is shown that the Hamilton–Jacobi solution in this case is identical with the solution obtained from the Schrödinger equation without the need to introduce the idea of hidden variables or quantum potential.

DIFFERENTIAL EQUATIONS IN A HYPERCOMPLEX SYSTEM

2020 ◽  
Vol 69 (1) ◽  
pp. 7-11
Author(s):  
A.K. Abirov ◽  
◽  
N.K. Shazhdekeeva ◽  
T.N. Akhmurzina ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Inhomogeneous System ◽  
Homogeneous Solutions ◽  
First Order ◽  
Zero Divisors ◽  
Hypercomplex System ◽  
Independent Variable ◽  
The Right

The article considers the problem of solving an inhomogeneous first-order differential equation with a variable with a constant coefficient in a hypercomplex system. The structure of the solution in different cases of the right-hand side of the differential equation is determined. The structure of solving the equation in the case of the appearance of zero divisors is shown. It turns out that when the component of a hypercomplex function is a polynomial of an independent variable, the differential equation turns into an inhomogeneous system of real variables from n equations and its solution is determined by certain methods of the theory of differential equations. Thus, obtaining analytically homogeneous solutions of inhomogeneous differential equations in a hypercomplex system leads to an increase in the efficiency of modeling processes in various fields of science and technology.

On the Number of Limit Cycles of Discontinuous Liénard Polynomial Differential Systems

2018 ◽  
Vol 28 (14) ◽  
pp. 1850175
Author(s):  
Fangfang Jiang ◽  
Zhicheng Ji ◽  
Yan Wang
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Second Order ◽  
Discontinuous Differential Equations ◽  
Differential Systems ◽  
Lower Upper ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Averaging Theorem

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one.

A universal Hamilton-Jacobi equation for second-order ODEs

1999 ◽  
Vol 32 (5) ◽  
pp. 827-844 ◽  
Author(s):  
G E Prince ◽  
J E Aldridge ◽  
G B Byrnes
Keyword(s):  
Jacobi Equation ◽  
Second Order ◽  
Hamilton Jacobi Equation

Convergent Liouville–Green expansions for second-order linear differential equations, with an application to Bessel functions

1993 ◽  
Vol 440 (1908) ◽  
pp. 37-54 ◽  
Keyword(s):  
Differential Equations ◽  
Bessel Functions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Large Parameter ◽  
Order Linear ◽  
Factorial Series ◽  
Independent Variable ◽  
Linear Differential ◽  
Asymptotic Validity

A class of second-order linear differential equations with a large parameter u is considered. It is shown that Liouville–Green type expansions for solutions can be expressed using factorial series in the parameter, and that such expansions converge for Re ( u ) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity. The theory is then applied to obtain convergent expansions for modified Bessel functions of large order.

Sign in / Sign up

Export Citation Format

Share Document