The Hamiltonian dynamics of non-holonomic systems

1951 ◽  
Vol 205 (1083) ◽  
pp. 564-583 ◽  
Keyword(s):  
Nonholonomic System ◽  
Integrable Function ◽  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Holonomic System ◽  
Principal Function ◽  
Holonomic Constraints ◽  
Second Derivatives ◽  
General Dynamical System ◽  
Hamilton Jacobi Equation

A formalism is developed which makes it possible to express the equations of motion of a nonholonomic system in Poisson bracket form. The main difficulty which has to be overcome arises from the fact that the Lagrangian co-ordinates and their corresponding momenta do not form a canonical set. However, at each instant of time, these variables can be expressed in a unique way as functions of a canonical set called the locally free co-ordinates and momenta. Poisson brackets can be formed with respect to the locally free variables, and it is shown that these lead to the correct equations of motion for a general dynamical system subject to a given set of non-holonomic constraints. Hamilton’s principle applies to a non-holonomic system , so a principal function can be formed, and its properties are studied in the second part of this paper. In addition to the usual Hamilton-Jacobi equation, the principal function satisfies a set of equations corresponding to the set of constraints. It is shown that these equations imply an indefinite or non-integrable principal function. A non-integrable function is one for which the order of double differentiation is not reversible. A precise method is given for defining the principal function for a non-holonomic system, and it is shown how this leads to indefiniteness in its second derivatives.

Lie Symmetry and Approximate Hojman Conserved Quantity of Lagrange Equations for a Weakly Nonholonomic System

2013 ◽  
Vol 30 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Y.-L. Han ◽  
X.-X. Wang ◽  
M.-L. Zhang ◽  
L.-Q. Jia
Keyword(s):  
Nonholonomic System ◽  
Equations Of Motion ◽  
Lie Symmetry ◽  
Conserved Quantity ◽  
Holonomic System ◽  
Lagrange Equations ◽  
Definition Of ◽  
Weakly Nonholonomic System ◽  
Infinitesimal Transformations ◽  
Hojman Conserved Quantity

ABSTRACTThe Lie symmetry and Hojman conserved quantity of Lagrange equations for a weakly nonholonomic system and its first-degree approximate holonomic system are studied. The differential equations of motion for the system are established. Under the special infinitesimal transformations of group in which the time is invariable, the definition of the Lie symmetry for the weakly nonholonomic system and its first-degree approximate holonomic system are given, and the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are obtained. Finally, an example is given to study the exact and approximate Hojman conserved quantity for the system.

The Maggi or Canonical Form of Lagrange’s Equations of Motion of Holonomic Mechanical Systems

1990 ◽  
Vol 57 (4) ◽  
pp. 1004-1010 ◽  
Author(s):  
John G. Papastavridis
Keyword(s):  
Linear Algebra ◽  
Nonlinear Equations ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Holonomic System ◽  
Holonomic Constraints ◽  
Constrained Mechanical Systems ◽  
New Variables

This paper formulates the simplest possible, or canonical, form of the Lagrangean-type of equations of motion of holonomically constrained mechanical systems. This is achieved by introducing a new special set of n holonomic (system) coordinates in terms of which the m ( < n) holonomic constraints are expressed in their simplest, or uncoupled, form: the first m of these new coordinates vanish; the remaining (n-m) (nonvanishing) new coordinates of the (n-m) degree-of-freedom system are then independent. From the resulting equations of motion: (a) The last (n-m) are reactionless canonical equations (the holonomic counterpart of the linear or nonlinear equations, either of Maggi (in the old variables), or of Boltzmann/Hamel (in the new variables)) whose solution yields the motion, while (b) the first m supply the system reactions, in the old or new coordinates, once the motion is known. Special forms of these equations and a simple example are also given. The geometrical interpretation of the above, in modern vector/linear algebra language is summarized in the Appendix.

The Hamiltonian Form of Field Dynamics

1951 ◽  
Vol 3 ◽  
pp. 1-23 ◽  
Author(s):  
P. A. M. Dirac
Keyword(s):  
Equations Of Motion ◽  
Dynamical Theory ◽  
Point Of View ◽  
Newtonian Mechanics ◽  
Classical Dynamics ◽  
Principal Function ◽  
The Family ◽  
Deep Significance ◽  
Hamilton Jacobi Equation ◽  
Insight Into

In classical dynamics one has usually supposed that when one has solved the equations of motion one has done everything worth doing. However, with the further insight into general dynamical theory which has been provided by the discovery of quantum mechanics, one is lead to believe that this is not the case. It seems that there is some further work to be done, namely to group the solutions into families (each family corresponding to one principal function satisfying the Hamilton-Jacobi equation). The family does not have any importance from the point of view of Newtonian mechanics; but it is a family which corresponds to one state of motion in the quantum theory, so presumably the family has some deep significance in nature, not yet properly understood.

scholarly journals The correct and unusual coordinate transformation rules for electromagnetic quadrupoles

2018 ◽  
Vol 474 (2213) ◽  
pp. 20170652 ◽  
Author(s):  
J. Gratus ◽  
T. Banaszek
Keyword(s):  
Coordinate Transformation ◽  
Equations Of Motion ◽  
Flat Space ◽  
Polar Coordinates ◽  
Cartesian Coordinates ◽  
Transformation Rules ◽  
Integral Transformations ◽  
Second Derivatives ◽  
Electric Dipoles ◽  
Derivatives Of

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space–time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates-free manner. This is particularly useful given their complicated coordinate transformation.

scholarly journals GRAVITATIONAL FIELD OF A ROTATING GRAVITATIONAL DYON

2002 ◽  
Vol 17 (15n17) ◽  
pp. 1091-1096 ◽  
Author(s):  
N. DADHICH ◽  
Z. YA. TURAKULOV
Keyword(s):  
Gravitational Field ◽  
Gordon Equation ◽  
Equations Of Motion ◽  
Polar Angle ◽  
Vacuum Equation ◽  
Axially Symmetric ◽  
Einstein Vacuum Equation ◽  
Einstein Vacuum ◽  
Klein Gordon ◽  
Hamilton Jacobi Equation

We have obtained the general solution of the Einstein vacuum equation for the axially symmetric stationary metric in which both the Hamilton-Jacobi equation for particle motion and the Klein - Gordon equation are separable. It can be interpreted to describe the gravitational field of a rotating dyon, a particle endowed with both gravoelectric (mass) and gravomagnetic (NUT parameter) charges. Further, there also exists a duality relation between the two charges and the radial and the polar angle coordinates which keeps the solution invariant. The solution can however be transformed into the known Kerr - NUT solution indicating its uniqueness under the separability of equations of motion.

The quantum mechanics of non-holonomic systems

1951 ◽  
Vol 205 (1083) ◽  
pp. 583-595 ◽  
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Classical Dynamics ◽  
Holonomic System ◽  
Holonomic Systems ◽  
Classical Analogue ◽  
Heisenberg Equations ◽  
Subsequent Motion ◽  
Heisenberg Equations Of Motion

Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q (0) r , then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q (0) r .

Inverse Problems of Mei Symmetry for Nonholonomic Systems with Variable Mass

2015 ◽  
Vol 31 (5) ◽  
pp. 515-523 ◽  
Author(s):  
W.-L. Huang ◽  
J.-L. Cai
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Nonholonomic System ◽  
Research Result ◽  
Variable Mass ◽  
Nonholonomic Systems ◽  
Noether Symmetry ◽  
Additional Restriction ◽  
Holonomic System ◽  
Mei Symmetry

AbstractThe inverse problem of the Mei symmetry for nonholonomic systems with variable mass is studied. Firstly, the authors discuss the Mei symmetry of the holonomic system opposite to a nonholonomic system. Secondly, weak and strong Mei symmetries of a nonholonomic system are concluded through restriction equations and additional restriction equations. Thirdly, the relevant conserved quantity is deduced by means of the structure equation for the gauge function. Fourthly, the inverse problem of the Mei symmetry is obtained by the Noether symmetry. Finally, the paper offers an example to illustrate the application of the research result.

A New Method of Stabilization for Holonomic Constraints

1983 ◽  
Vol 50 (4a) ◽  
pp. 869-870 ◽  
Author(s):  
J. W. Baumgarte
Keyword(s):  
Essential Feature ◽  
Equations Of Motion ◽  
New Method ◽  
Lagrangian Multipliers ◽  
Holonomic Constraints ◽  
Asymptotic Stabilization

A new method for the asymptotic stabilization of holonomic constraints is presented. The essential feature of this approach is the introduction of stabilizing momenta of constraint. The advantage of the method is the fact that to obtain the nonclassical Lagrangian multipliers in the equations of motion, the holonomic constraints need to be differentiated only once with respect to time.

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