Dynamics of Nonholonomic Systems

1961 ◽  
Vol 28 (4) ◽  
pp. 574-578 ◽  
Author(s):  
T. R. Kane
Keyword(s):  
Differential Equations ◽  
Nonholonomic Systems ◽  
General Method

A general method for obtaining the differential equations governing motions of a class of nonholonomic systems is presented. Several supplementary theorems are stated, and the use of the method is illustrated by means of two examples.

scholarly journals A Continuous Analogue of Lattice Path Enumeration

10.37236/8788 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Quang-Nhat Le ◽  
Sinai Robins ◽  
Christophe Vignat ◽  
Tanay Wakhare
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lattice Paths ◽  
Lattice Path ◽  
Continuous Version ◽  
Continuous Analog ◽  
Continuous Analogue ◽  
Lattice Path Enumeration ◽  
Multinomial Coefficients ◽  
General Method

Following the work of Cano and Díaz, we consider a continuous analog of lattice path enumeration. This process allows us to define a continuous version of many discrete objects that count certain types of lattice paths. As an example of this process, we define continuous versions of binomial and multinomial coefficients, and describe some identities and partial differential equations that they satisfy. Finally, as an important byproduct of these continuous analogs, we illustrate a general method to recover discrete combinatorial quantities from their continuous analogs, via an application of the Khovanski-Puklikov discretizing Todd operators.  

Another Look at Nonholonomic Systems

1973 ◽  
Vol 40 (1) ◽  
pp. 101-104 ◽  
Author(s):  
C. E. Passerello ◽  
R. L. Huston
Keyword(s):  
Analytical Methods ◽  
Classical Problem ◽  
Nonholonomic Systems ◽  
Constraint Forces ◽  
Advantages And Disadvantages ◽  
Arbitrary Choice ◽  
Special Cases ◽  
Dependent Variables ◽  
Automatic Elimination ◽  
General Method

The relative advantages and disadvantages of various analytical methods for nonholonomic systems are briefly presented and discussed. The techniques of Kane’s method are then used to develop a derivation of a general method which consolidates and employs the advantages of the various classical methods. These advantages include the automatic elimination of nonworking constraint forces while avoiding the computation of vector components of acceleration. The method also provides for the arbitrary choice of dependent variables so that it may be adapted to a variety of nonholonomic systems. Two special cases are considered and the method is then illustrated in the classical problem of the rolling coin.

scholarly journals Direct construction method for conservation laws of partial differential equations Part II: General treatment

2002 ◽  
Vol 13 (5) ◽  
pp. 567-585 ◽  
Author(s):  
STEPHEN C. ANCO ◽  
GEORGE BLUMAN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
General Treatment ◽  
Local Conservation ◽  
Partial Differential ◽  
Direct Construction ◽  
Computational Implementation ◽  
Direct Construction Method ◽  
General Method

This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.

scholarly journals Using Ordinary Differential Equations to Explore Cancer-Immune Dynamics and Tumor Dormancy

2016 ◽  
Author(s):  
Kathleen P. Wilkie ◽  
Philip Hahnfeldt ◽  
Lynn Hlatky
Keyword(s):  
Mathematical Modeling ◽  
Immune Response ◽  
Differential Equations ◽  
Tumor Growth ◽  
Ordinary Differential Equations ◽  
Systemic Disease ◽  
Growth Dynamics ◽  
Tumor Dormancy ◽  
Cytotoxic Action ◽  
General Method

AbstractCancer is not solely a disease of the genome, but is a systemic disease that affects the host on many functional levels, including, and perhaps most notably, the function of the immune response, resulting in both tumor-promoting inflammation and tumor-inhibiting cytotoxic action. The dichotomous actions of the immune response induce significant variations in tumor growth dynamics that mathematical modeling can help to understand. Here we present a general method using ordinary differential equations (ODEs) to model and analyze cancer-immune interactions, and in particular, immune-induced tumor dormancy.

scholarly journals Composition sum identities related to the distribution of coordinate values in a discrete simplex

10.37236/1498 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
R. Milson
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Algebraic Methods ◽  
Algebraic Proof ◽  
Geometric Properties ◽  
Linear Differential ◽  
Exponential Formula ◽  
Ordinary Linear Differential Equations ◽  
General Method

Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first and second order, ordinary, linear differential equations. Regarding the first class, the corresponding identities amount to a proof of the exponential formula of labelled counting. The identities in the second class can be used to establish certain geometric properties of the simplex of bounded, ordered, integer tuples. We present three theorems that support the conclusion that the inner dimensions of such an order simplex are, in a certain sense, more ample than the outer dimensions. As well, we give an algebraic proof of a bijection between two families of subsets in the order simplex, and inquire as to the possibility of establishing this bijection by combinatorial, rather than by algebraic methods.

scholarly journals The Laplace transform as an alternative general method for solving linear ordinary differential equations

2021 ◽  
Vol 1 (4) ◽  
pp. 309
Author(s):  
William Guo
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Ordinary Differential Equations ◽  
Case Studies ◽  
Initial Conditions ◽  
Linear Ordinary Differential Equations ◽  
The Laplace Transform ◽  
Linear Odes ◽  
General Method

<p style='text-indent:20px;'>The Laplace transform is a popular approach in solving ordinary differential equations (ODEs), particularly solving initial value problems (IVPs) of ODEs. Such stereotype may confuse students when they face a task of solving ODEs without explicit initial condition(s). In this paper, four case studies of solving ODEs by the Laplace transform are used to demonstrate that, firstly, how much influence of the stereotype of the Laplace transform was on student's perception of utilizing this method to solve ODEs under different initial conditions; secondly, how the generalization of the Laplace transform for solving linear ODEs with generic initial conditions can not only break down the stereotype but also broaden the applicability of the Laplace transform for solving constant-coefficient linear ODEs. These case studies also show that the Laplace transform is even more robust for obtaining the specific solutions directly from the general solution once the initial values are assigned later. This implies that the generic initial conditions in the general solution obtained by the Laplace transform could be used as a point of control for some dynamic systems.</p>

SOLITON EIGENFUNCTION EQUATIONS: THE IST INTEGRABILITY AND SOME PROPERTIES

1990 ◽  
Vol 02 (04) ◽  
pp. 399-440 ◽  
Author(s):  
B.G. KONOPELCHENKO
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Wide Class ◽  
Eigenvalue Problems ◽  
Nonlinear Differential Equations ◽  
Soliton Equations ◽  
Spectral Transform ◽  
Dimensional Soliton ◽  
Operator Representations ◽  
General Method

Eigenfunctions of the linear eigenvalue problems for the soliton equations obey nonlinear differential equations. It is shown that these eigenfunction equations are integrable by the inverse spectral transform (IST) method. They have triad operator representations. Eigenfunction equations are the generating equations and possess other interesting properties. Eigenfunction equations form a new wide class of nonlinear integrable equations. Eigenfunction equations for several typical, well-known (1+1)-, (2+1)- and multi-dimensional soliton equations are considered. A general method for constructing the auxiliary linear systems for the eigenfunction equations is proposed. It is shown that the vertical hierarchies of the eigenfunction equations contain only finite numbers of different members in the cases considered. The properties of such hierarchies for soliton equations are closely connected with their Painleve properties. Some “linear” properties of the eigenfunction equations are also discussed.

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