Study On Some Problems Of Lie Symmetry And Conformal Invariance And Conserved Quantities

This dissertation investigates mainly some problems on conserved quantity of me-chanical systems based on the symmetry theory. Firstly, we discuss the equivalent problem of the two different kinds of Lie symmetries for Lagrange system, Hamilton system, and the generalized Hamilton system respectively. Then, we obtain the con-served quantities deduced by the conformal invariance for the generalized Hamilton system. Secondly, we study the conformal invariance, Lie symmetry, Mei symmetry and conserved quantities of the planar Kepler equation. Thirdly, we are concerned with a new conserved quantity of Nielsen equation induced by Lie symmetry and a new conserved quantity of Appell equation induced by Mei symmetry. Finally, we consider reduction problem of the generalized homogeneous system, the condition that Lotka-Volterra system has quasi-homogeneous polynomial first integral is given, and the expression of the first integral is shown.We are concerned with the following non-autonomous systemWe denote the differential operator corresponding to system (1) byWe will take the infinitesimal generator vector of some one-parameter Lie group as followsLiterature [30] pointed out that the necessary and sufficient condition that system (1) admits one-parameter Lie group of the generator (3) is where [X*, V*]=X*V*-V*X*is the operation of Lie bracket.Introducing the first expansion of infinitesimal transformations for Lagrange sys-tem The necessary and sufficient condition that Lagrange system accepts one-parameter Lie group of the generator (5) is Literature [1] pointed out that the determining equations of Lie symmetry under the transformation (5) for Lagrange system isTheorem1That Lagrange system admits one-parameter Lie group of the gener-ator (5) is equivalent to the determining equations of Lie symmetry could be expressed as (7), namely,(6) holds if and only if (7) holds.We have a similar conclusion for Hamilton system. Suppose ζ0,ζs,ηs are in-finitesimal generators. On one hand, Literature [1] derived the determining equations of Lie symmetry for Hamilton system as follows On the other hand, if we denoted the infinitesimal generator vector of a one-parameter Lie group by we have the necessary and sufficient condition that Hamilton system accepts one-parameter Lie group of the generator (9) isTheorem2That Hamilton system admits one-parameter Lie group of the gener-ator (9) is equivalent to the determining equations of Lie symmetry could be expressed as (8), namely,(10) holds if and only if (8) holds.We can also prove a similar conclusion for the generalized Hamilton system. Sup-pose ζ0,ζi are infinitesimal generators. On one hand, Literature [1] told us that the determining equations of Lie symmetry of the generalized Hamilton system had the following expressionOn the other hand, if we write the infinitesimal generator vector of some one-parameter Lie group aswe can get the necessary and sufficient condition that the generalized Hamilton system admits one-parameter Lie group of the generator (12) isTheorem3That the generalized Hamilton system admits one-parameter Lie group of a generator (12) is equivalent to the determining equations of Lie symmetry could be expressed as (11), namely,(13) holds if and only if (11) holds.Theorem4If the infinitesimal generators ζ0,ζi satisfy (11), and there is a function λ=λ(t,x) satisfyingthen we obtain the conserved quantity resulted from the generalized Hamilton system whereTheorem5Suppose the generators under the usual infinitesimal transformation are ζ0,ζi for the generalized Hamilton system, if there exists a matrix Γik satisfyingthen the necessary and sufficient condition for the conformal invariance to be Mei symmetry for the generalized Hamilton system is Γki=lik,where lik is the conformal factor of the conformal invariance. Theorem6For the generalized Hamilton system, if the generators under the usual infinitesimal transformation ζ0,ζi satisfy then the matrix Γik in (17) will beTheorem6shows that the matrix Γik could be given by the generators, which satisfy the conformal invariance and Mei symmetry simultaneously.For the planar Kepler equation, we study Hojman conserved quantity resulted from conformal invariance by Lie symmetry. We get the following theorem.Theorem7Under the time-invariant infinitesimal transformations, If the gen-erators ζs satisfy and there exists a function λ=λ(t,q, q) so that then Hojman conserved quantity induced by the conformal invariance isWe also discuss Mei conserved quantity of Kepler equation deduced by Mei sym-metry. A conserved quantity independent with the system’s total energy and angular momentum is obtained.Theorem8For Nielsen equation, if the infinitesimal generators ζ0,ζs satisfy ζs+2αsζ0-qsζ0=X(1)(αs),(23) and there exists a function λ=λ(t,q, q) satisfying then a conserved quantity of the Nielsen equation induced by Lie symmetry iswhereTheorem9For complete system Appell equations, If the infinitesimal generators ζ0,ζs satisfy and there is a gauge function GM=GM(t,q, q) satisfyingthen a new conserved quantity of the complete system Appell equation deduced by Mei symmetry isFinally, we consider the following third-order Lotka-Volterra system,where A, B, C are nonzero real parameters. Theorem10If the systems (29) has quasi-homogeneous polynomial first integral,then the parameters A, B,C satisfy two of the following four conditions1-A+AC=0,1-C+BC=0,1-B+AB=0, ABC+1=0.(30) Theorem11When the parameters A,B,C satisfy Theorem10, system (29) has quasi-homogeneous polynomial first integral of degree1Ω1(x,y,z)=Bx-BCy-z.(31)Theorem12When the parameters A,B,C satisfy Theorem10, system (29) has quasi-homogeneous polynomial first integral of degree2Ω2(x,y,z)=A2B2x2+y2+A2z2+2ABxy-2A2Bxz-2Ayz.(32)...