On a Time-Dependent Nonholonomic Oscillator
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c Pleiades Publishing, Ltd., 2020.
On a Time-Dependent Nonholonomic Oscillator A. V. Tsiganov∗,1 ∗
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia, E-mail: 1 [email protected] Received May 7, 2020; Revised May 14, 2020; Accepted May 28, 2020
Abstract. In this note, we compare the first integrals and exact solutions of equations of motion for scleronomic and rheonomic, and holonomic and nonholonomic oscillators. DOI 10.1134/S1061920820030115
1. INTRODUCTION Let us consider a general rheonomic Lagrangian system with nonintegrable constraints characterized by a Lagrange function L(q, q, ˙ t) and ideal independent constraints that are linear at velocities, fk (q, q, ˙ t) = 0 ,
k = 1, . . . , m .
Here q = (q1 , . . . , qn ) are the independent Lagrangian coordinates, t is time, and q˙i = dqi /dt are the generalized velocities. The well-known equations of motion d dt
∂L ∂ q˙i
∂fk ∂L = λk , ∂qi ∂qi m
−
and
fk = 0 ,
k = 1, . . . , m ,
(1.1)
k=0
are equivalent to a variational equation expressing the form of the Lagrange-d’Alembert principle or the Hamilton principle in the H¨older form; for details, see [3, 22, 29]. The greatest advantage of Lagrangian formulation is that it brings out the connection between conservation laws and important symmetry properties of dynamical systems. The knowledge of conservation laws is of great importance for the analysis of dynamical systems since they enable one to find exact solutions of dynamical systems. Explicit solutions enable us to test analytical and numerical schemes applied to the given mathematical model and to choose a reasonable approximation to solutions of the model. If time t does not explicitly enter equations (1.1), then these systems are the so-called scleronomic systems. In other cases, for example, when involving a dependence on time of the constraints fk , time t does explicitly enter, and such systems are called rheonomic Lagrangian systems. Rheonomic nonholonomic systems are divided into two families 1. systems with time-independent Lagrangian ∂L(q, q, ˙ t) = 0; ∂t 2. systems with time-dependent original Lagrangian ∂L(q, q, ˙ t) = 0 . ∂t In the first case, the infinitesimal work of the constraint forces vanishes for any admissible infinitesimal virtual displacement according to Chetaev’s rule [12]. As a result, the Jacobi integral H=
n
q˙i
i=1
∂L −L ∂qi
remains the first integral of equations (1.1) which can be used for the explicit solution of equations of motion (1.1). In the other case, we have dH ∂L =− , dt ∂t 399
400
TSIGANOV
and, instead of the Jacobi integral, we can use the Ermakov–Lewis type invariants to solve the equations of motion (1.1). For instance, consider a Lagrangian describing the time-dependent harmonic oscillator L(q, q, ˙ t) =
1 2 q˙ − ω 2 (t)q 2 . 2m(t)
Introduce a new dynamical variable Q and a new time τ using the transformation dt = m(t) f 2 (t) dτ,
Q = qf (t) ,
where the function f (t) satisfies the Ermakov equation [16] d Ω2 mf˙ + mω 2 f = 3 , dt f
Ω ∈ R,
depend
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