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Abstract： In this paper，we present a class of exactly solvable higher derivative models with and without constraints based on the Pais Uhlenbeck Oscillator，through which we show explicitly how the constraints remove the Ostrogradski instability brought by higher derivatives.We find the sufficient and necessary condition to establish a stable system with constraints，in which the Lagrangian is involved with 2nd time derivative.We also generalize the investigations of linear systems to a higher derivative system with an arbitrary potential term.We propose a criterion of choosing constraints，which is to eliminate the harmful and retain the favorable in physics from the original dynamic behavior.In the linear case，this criterion would enable us to eliminate the unstable QNMs and retain the stable ones.All these discussions would help us to find a reasonable field theory with higher derivatives.
Key words： Pais Uhlenbeck Oscillator； Ostrogradski instability； higher derivative models； constraints
CLC number： O 412.1Document code： AArticle ID： 1000 5137（2014）06-0618-11
1Introduction
Recently，Chen et al.[1] proved that the Ostrogradski instability can be removed by the addition of constraints if the original theory′s phase space is reduced.Based on their study，we find the sufficient and necessary condition to establish a stable higher derivative models with constraints，which would help us to find a reasonable field theory with higher derivatives.There has been a reviviscence of interest in higher derivative theories，especially within attempts to modify Einstein′s gravity theory[2-3].Modified gravity theories can predict observed cosmic phenomena without the need for dark energy[4-5] and maybe even dark matter[6-7].The string and brane theories both show that the higher derivative terms could have interesting cosmological implications in high energy realm[8].The higher derviative terms may be interpreted as the quantum corrections of the matter fields[9].The flat spacetime Galileon is a scalar field theory whose Lagrangian involves the higher derivative terms[10].It seems that we should study higher derivative theory if we try to explain some current riddles of physics with deviation from Einstein′s general relativity.However，Ostrogradski[11] proved in the middle of the 19th century that there exists a linear instability in any nondegenerate theory whose fundamental dynamical variable is higher than 2nd order in time derivative.In various modified gravity models，the higher order gravity theories employing terms such as RμνRμν，RμνρδRμνρδ or CμνρδCμνρδ generally suffer the sickness of Ostrogradski instability.It is often possible to accommodate higher derivatives in the Lagrangian while retaining 2nd order field equations.The usual practices are that the original Lagrangian is added by a judicious term including first or higher derivatives，which is basic ingredient for the modified gravity.So far，one does not know if there is a universal Dirac′s method[12] of Hamiltonian analysis to curve Ostragradski′s instability by the addition of constraints. We refer to a system，L（q，q·，…，q（N）），whose L/q（N） depends on q（N） ，as a non degenerate system.We can use this system as a toy model of Hamiltonian analysis，where the Lagrangian involves higher derivative terms.Euler Lagrangian equation is 2N order in time derivative，i.e.，
and the canonical phase space should have 2N coordinates.Following Ostrogradski′s choices，the canonical coordinates are
Nondegeneracy means that there exists a function a（Q1， …，QN，PN），which can be expressed in terms of canonical coordinates by Eq.（3）
It is easy to find that the Hamiltonian is linear in P1，…， PN-1，so it is not bounded from below （or above）.It will become problematic when the interactions with other degrees of freedom are introduced，whose Hamiltonians are bounded from below （or above）[5].
Imposing constraints is a desirable way out to cure Ostrogradski instability for higher derivative theories[1]，which makes the Hamiltonian bounded.In other words，the Ostrogradski instability can be removed by the addition of constraints，which reduce the dimensionality of the phase space of original system.One can introduce auxiliary variables to realize the reduction.In fact， the total phase space is enlarged after introducing auxiliary variables，but the dimensionality of the reduced phase space keeps the same or even gets smaller，because trajectories are constrained.It has been proven that the addition of Lagrange multipliers can not reduce the original phase space，so the constraints must contain second or higher derivative terms of auxiliary variables.
The Pais Uhlenbeck （PU） oscillator[13] is a fascinating toy model for the investigation of higher derivative theories[14-18].It is a quantum mechanical analog to a field theory containing both the second and the fourth order derivative terms.On the classical level， the simplest interaction is an external dissipative force for a PU oscillator[15].On the quantum level，Bender and Mannheim showed that PT symmetric Hamiltonians are ghost free up to fourth order[16].A stable and unitary quantum system of the PU oscillator is discussed in Refs.[14].
In this paper，we find the sufficient and necessary condition to establish a stable linear system with constraints，in which the Lagrangian involved with 2nd time derivatives.We also move on and generalize the discussions of linear system to a second time derivative system with arbitrary potential V（q）.We present a class of exactly solvable higher derivative models with and without constraints.We show explicitly that the Ostrogradski instability can be removed by imposing constraints in this model.In the case without constraints，the exact solution of classical dynamics includes four quasi normal modes （QNMs），among which two modes increase exponentially.That indicates an exponential instability of the system.While in the case with constraints，the two divergent modes are eliminated and the good ones are retained.So the exact solution includes only two stable QNMs.All these discussions and conclusions would pave the way to establish a reasonable field theory with higher order derivative. This paper is organized as follows.In section 2 we investigate the PU oscillator with an external dissipative force to illustrate the instability of a higher derivative system.In section 3，using Dirac′s Hamlitonian method，we obtain the sufficient and necessary condition to establish a stable linear system with constraints，in which the Lagrangina invloved with 2nd time derivative.We also discuss the generalized system with an arbitray potential term.In section 4，we reexamine the PU model with constraints suggested by Chen et al.[1]，and find that model is still not stable within some range of the parameters ω1 and ω2，although it exorcises Ostrogradski instability.In section 5，using the sufficient and necessary condition obtained in Sec.3，we find a stable PU model with constraints for all ω1 and ω2.And this model maintains the characteristics of the PU oscillator.In the last section，we summarize all the investigations，and discuss what we can learn from them. 2A Pais Uhlenbeck oscillator with an external dissipative interaction
Woodard has shown that the physical disaster of Ostragradski′s instability manifested by an interaction of the higher derivative system and other system[5].On the classical level，the simplest interaction is an external dissipative force.We consider a PU oscillator with an external dissipative force，± 2 ω1 ω2 ω21+ω22 q·.The PU action is given by
where ω1 and ω2 are positive constants，and the Euler Lagrange equation is
The Lagrangian contains the whole information about the system dynamics for the fundamental physics，in which possible constraints of dynamical variables can be included.The existence of constraints means that the total phase space is reduced to a submanifold in physics.In order to amend the Ostrogradski instability，the dimensionality of original system′s phase space must be reduced by imposing constraints.Constraints can be classified into two classes.The first class constraints are those associated with gauge freedoms in the theory，and the second class constraints are physical，which means the solutions of motion equations are different with or without constraints.In this paper，we can check all the constraints are connected with second class.
The term C（q，q·）q·· in the second time derivative Lagrangian can be absorbed into other terms by using the mathematical identity，
Therefore，the general second time derivative Lagrangian is given by
Furthermore，the most general Lagrangian of second time derivative linear system with constraints can be written as where α，β，a，b，c and e are real parameters.Following Dirac′s analysis of constraints systems[18] and Ostrogradski′s choice[17]，we define canonical variables as From Eq.（20），we have q··=P2-eQ3 as a function of canonical variables P2 and Q3.Through the Legendre transform，the total Hamiltonian is
HT= P1Q2+12P22-αQ22-βQ21-bQ1Q3-cQ2Q3-eP2Q3-aQ32+12e2Q32+u1Φ1，（22）
where the primary constraint Φ1 is a functional given by P3=0，which can be denoted Φ1：P3=0.u1 is a function of canonical variables which can be found by the so-called consistency relations.In fact，we must ensure all constraints are preserved in order that dynamical system can be consistent in the evolution of time.Since P3=0，consistency suggests that its equation of motion P·3={P3，HT} must also vanish on the submanifold of phase space where all the constraints are satisfied.In other words，the Poisson bracket {P3，HT}～0，where the sign "～" means equal after all constraints have been enforced due to Dirac[18].The consistency relations lead up to the fact that there exists new constraints called secondary constraints.So，the following secondary constraint is expected
That is to say，further consistency relations only give rise to the form of the arbitrary function u1，and there′s no further constraints.The source of Ostrogradski instability is the vexatious linear term P1Q2.To exorcise this instability we have to find a constraint where Q2 is some function of P1.Therefore，we have
between the parameters a and e，if we want to exorcise Ostrogradski instability.
In the 2a=e2 case，repeating the procedures above，one can find the further consistency relations
It is easy to check all the constraints are second class constraints.By using these constraints，the reduced Hamiltonian can be written as
HR=1e2b22-e2βQ21+（αe2+b e-c22）（bcQ1+e2P1）2e2（2αe2+be -c2）2.（28）
Obviously，the effective dimensionality of phase space is reduced from four （Q1，Q2，P1，P2） to two （Q1，P1），and the Ostrogradski instability is exorcised.Furthermore，the reduced Hamiltonian will be positive or negative definite when suitable relations are satisfied among the parameters α，β，b，c and e.
From the discussions above，we obtain Theorem 1 stated as follows，
Theorem 1
The most general Lagrangian （18） of second time derivative linear system with constraints is stable and nontrivial iff 2a=e2，e≠0 and
Also，we can move on and generalize the discussions of the system Eq.（18） to a second time derivative system with arbitrary potentials，whose Lagrangian is as follows，
and Q2 can be inverted as a function of Q1 and P1 for a fixed c（Q1，Q2） in Eq.（31）.So，the dimensionality of phase space can be reduced to two （Q1，P1） from four （Q1，P1，Q2，P2）.From Eq.（31） and （32），we obtain Theorem 2，stated as follows， Theorem 2
For a second time derivative dynamical system with constraints，Eq.（30），
（i） By choosing some appropriate function c（Q1，Q2），one can always change the original system into a stable one，and the dimensionality of the phase space is reduced to two （Q1，P1） from four （Q1，P1，Q2，P2） ，but the characteristics of the constrained system might change essentially.
（ii） If one choose c（q，q·） to be a constant c（q，q·）=c0，then the reduced system will retain the complete characteristics of the original dynamical system with the higher derivative terms left out.
（iii） If one choose c=c0+δ（q，q·） and δ（q，q·）C0，then the reduced system roughly keeps the characteristics of the original dynamical system with the higher derivative terms left out，with some corresponding corrections against the original one，quantitatively.
Since Ω1 is positive definite，the solutions are unstable in the case of ω1ω2>2+1 or ω1ω2<2-1.This conclusion is completely consistent with the above mentioned result by the Hamiltonian method.Meanwhile，we find that the characteristics of the PU oscillator is radically changed in this system with constraints.Especially，the oscillating behavior becomes monotonically decreasing or increasing.It is obvious that the choice of constraints is crucial.Inappropriate choosing of constraints will completely change dynamic behavior of the original system，which is certainly not what we expect.
which is negative definite for all values of ω1>0 and ω2>0 （See Fig.1）.The reduced system is not only free from Ostrogradski′s ghost，but also stable.
Next，we consider this sytem with the addition of external dissipative force 2ω1ω2（ω21+ω22）12 q·.The equations of motion are
Finally，we compare the PU oscillators with and without constraints，both encountering the friction force 2ω1ω2（ω21+ω22）12q·.We find from the comparison of Eq.（15） with Eq.（61） that the effective dimensionality of phase space is reduced from four （Q1，Q2，P1，P2） to two （Q1，P1）.There exists the term P1 Q2 in the Hamiltonian of the system without constraints，which gives rise to Ostrogradski′s ghost.On the contrary，the system with constraints is ghost free and its reduced Hamiltonian is negative definite， therefore the system with the constraints is stable.The solvability of both systems allows us to compare further their analytical solutions.For the PU oscillator system without constraint，there are four QNMs.Two of them oscillate and decrease exponentially，but the other two oscillate and increase exponentially which lead to the instability of the system.On the contrary，there only exist two stable QNMs for the PU oscillator with constraints，which is in accordance to the facts that the dimensionality of phase space from four to two and HR is negative definite.It is easy to find from the discussion above that the choice of the constraints is crucial.The criterion of choosing constraints is to eliminate the harmful and retain the favorable from the original dynamic behavior，and meanwhile to maintain the stability of the reduced system.In the linear case，the criterion would enable us to eliminate the unstable QNMs and retain the stable ones. 6Conclusion and discussion
As is known to all，there exists linear instabilities （Ostrogradski instability） in any nondegenerate theory whose fundamental dynamical variable is higher than 2nd order in time derivative.But these instabilities can be removed by imposing constraints which reduce the dimensionality of original system′s phase space.In this paper，we have investigated the most general system of second time derivative linear system with constraints，and showed the sufficient and necessary condition in Theorem 1.Furthermore，we also generalized the investigations of linear system to a higher derivative system with an arbitrary potential term V（q）.
In this paper，we investigated three generalized PU oscillator models with external dissipative forces and found the exact analytic solutions.The solutions of the first model include four QNMs.Two oscillate and decrease exponentially and the other two oscillate and increase exponentially.Therefore，the PU oscillator system without constraints is an unstable system，which is a typical example of systems with time derivative higher than 2nd order.The second model has been discussed by the authors of Ref.[1] using Hamiltonian method.But there is some minor error in their computation，which leads to a conclusion not quite correct actually.We reexamined this model using Hamiltonian method and found although this model can avoid the Ostrogradski′s ghost，it is still unstable when the parameter ratio ω1ω2>2+1.Furthermore，the solutions of this model are monotonous，which have no longer the oscillating characteristics of the original PU oscillator.The third model satisfies the stability conditions given in Theorem 1.The phase space （Q1，Q2，P1，P2） of the model is reduced to （Q1，P1） and the reduced Hamiltonian HR is negative definite，which leads to a stable system.The solutions of the third model indicate that the constraints exclude two unstable QNMs and retain two stable ones，although the frequency and the amplitude of the modes need some appropriate corrections.From the discussion above of the three solvable models，we could learn the following helpful lessons：
Lesson 1： In order to obtain a stable system，we have to choose the Lagrangian of the auxiliary field seriously.In the restricted case of the linear system，Theorem 1 gives an explicit answer，but we should explore further for more general cases.Theorem 2 is just a preliminary study of nonlinear cases.
Lesson 2： The solutions of motion equations are different between the two models with and without second class constraints.One significant reason for the revival of interest in higher derivative theories is related to modified gravity.A reasonable modified gravity model with nondegenerate higher derivatives ought to be free from Ostrogradski instability brought by the higher derivatives，maintain the virtue of the general relativity，and also be competitive in solving the problems such as the accelerating expansion of the universe.When we study modified gravity with nondegnerate higher derivatives，the models are sure to meet these requirements. Lesson 3： The consistency relation，which introduces a complete set of reasonable constraints，is the key point to remove the factor causing instability of the system.For example，in a linear case，good constraints could eliminate the divergent QNMs，but retain the stable ones.
Finally，for general nonlinear higher derivative theories with constraints，we have found a class of system with the following Lagrangian
which is free from Ostrogradski′s ghost.We will investigate the stability conditions of this system in the forthcoming work.In any case，the higher derivative theory mentioned above is only a class of toy models for the fundamental physics.It seems that we have to go beyond simple models，if we attempt to explain some current riddles of physics with a deviation from general relativity.However，our experience from the higher derivative theory discussed above would help us to find a reasonable field theory with higher derivatives.
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带有约束的稳定高导数模型 翟向华 敖犀晨
摘要： 给出了一类精确可解的高导数模型，表明了约束如何去除高导数带来的Ostrogradski不稳定性，对于作用量含有二阶时间导数的系统，找到了建立稳定约束系统的充分必要条件.
关键词： Pais Uhlenbeck振子； Ostrogradski不稳定性； 高导数模型； 约束系统
（责任编辑：顾浩然）
Key words： Pais Uhlenbeck Oscillator； Ostrogradski instability； higher derivative models； constraints
CLC number： O 412.1Document code： AArticle ID： 1000 5137（2014）06-0618-11
1Introduction
Recently，Chen et al.[1] proved that the Ostrogradski instability can be removed by the addition of constraints if the original theory′s phase space is reduced.Based on their study，we find the sufficient and necessary condition to establish a stable higher derivative models with constraints，which would help us to find a reasonable field theory with higher derivatives.There has been a reviviscence of interest in higher derivative theories，especially within attempts to modify Einstein′s gravity theory[2-3].Modified gravity theories can predict observed cosmic phenomena without the need for dark energy[4-5] and maybe even dark matter[6-7].The string and brane theories both show that the higher derivative terms could have interesting cosmological implications in high energy realm[8].The higher derviative terms may be interpreted as the quantum corrections of the matter fields[9].The flat spacetime Galileon is a scalar field theory whose Lagrangian involves the higher derivative terms[10].It seems that we should study higher derivative theory if we try to explain some current riddles of physics with deviation from Einstein′s general relativity.However，Ostrogradski[11] proved in the middle of the 19th century that there exists a linear instability in any nondegenerate theory whose fundamental dynamical variable is higher than 2nd order in time derivative.In various modified gravity models，the higher order gravity theories employing terms such as RμνRμν，RμνρδRμνρδ or CμνρδCμνρδ generally suffer the sickness of Ostrogradski instability.It is often possible to accommodate higher derivatives in the Lagrangian while retaining 2nd order field equations.The usual practices are that the original Lagrangian is added by a judicious term including first or higher derivatives，which is basic ingredient for the modified gravity.So far，one does not know if there is a universal Dirac′s method[12] of Hamiltonian analysis to curve Ostragradski′s instability by the addition of constraints. We refer to a system，L（q，q·，…，q（N）），whose L/q（N） depends on q（N） ，as a non degenerate system.We can use this system as a toy model of Hamiltonian analysis，where the Lagrangian involves higher derivative terms.Euler Lagrangian equation is 2N order in time derivative，i.e.，
and the canonical phase space should have 2N coordinates.Following Ostrogradski′s choices，the canonical coordinates are
Nondegeneracy means that there exists a function a（Q1， …，QN，PN），which can be expressed in terms of canonical coordinates by Eq.（3）
It is easy to find that the Hamiltonian is linear in P1，…， PN-1，so it is not bounded from below （or above）.It will become problematic when the interactions with other degrees of freedom are introduced，whose Hamiltonians are bounded from below （or above）[5].
Imposing constraints is a desirable way out to cure Ostrogradski instability for higher derivative theories[1]，which makes the Hamiltonian bounded.In other words，the Ostrogradski instability can be removed by the addition of constraints，which reduce the dimensionality of the phase space of original system.One can introduce auxiliary variables to realize the reduction.In fact， the total phase space is enlarged after introducing auxiliary variables，but the dimensionality of the reduced phase space keeps the same or even gets smaller，because trajectories are constrained.It has been proven that the addition of Lagrange multipliers can not reduce the original phase space，so the constraints must contain second or higher derivative terms of auxiliary variables.
The Pais Uhlenbeck （PU） oscillator[13] is a fascinating toy model for the investigation of higher derivative theories[14-18].It is a quantum mechanical analog to a field theory containing both the second and the fourth order derivative terms.On the classical level， the simplest interaction is an external dissipative force for a PU oscillator[15].On the quantum level，Bender and Mannheim showed that PT symmetric Hamiltonians are ghost free up to fourth order[16].A stable and unitary quantum system of the PU oscillator is discussed in Refs.[14].
In this paper，we find the sufficient and necessary condition to establish a stable linear system with constraints，in which the Lagrangian involved with 2nd time derivatives.We also move on and generalize the discussions of linear system to a second time derivative system with arbitrary potential V（q）.We present a class of exactly solvable higher derivative models with and without constraints.We show explicitly that the Ostrogradski instability can be removed by imposing constraints in this model.In the case without constraints，the exact solution of classical dynamics includes four quasi normal modes （QNMs），among which two modes increase exponentially.That indicates an exponential instability of the system.While in the case with constraints，the two divergent modes are eliminated and the good ones are retained.So the exact solution includes only two stable QNMs.All these discussions and conclusions would pave the way to establish a reasonable field theory with higher order derivative. This paper is organized as follows.In section 2 we investigate the PU oscillator with an external dissipative force to illustrate the instability of a higher derivative system.In section 3，using Dirac′s Hamlitonian method，we obtain the sufficient and necessary condition to establish a stable linear system with constraints，in which the Lagrangina invloved with 2nd time derivative.We also discuss the generalized system with an arbitray potential term.In section 4，we reexamine the PU model with constraints suggested by Chen et al.[1]，and find that model is still not stable within some range of the parameters ω1 and ω2，although it exorcises Ostrogradski instability.In section 5，using the sufficient and necessary condition obtained in Sec.3，we find a stable PU model with constraints for all ω1 and ω2.And this model maintains the characteristics of the PU oscillator.In the last section，we summarize all the investigations，and discuss what we can learn from them. 2A Pais Uhlenbeck oscillator with an external dissipative interaction
Woodard has shown that the physical disaster of Ostragradski′s instability manifested by an interaction of the higher derivative system and other system[5].On the classical level，the simplest interaction is an external dissipative force.We consider a PU oscillator with an external dissipative force，± 2 ω1 ω2 ω21+ω22 q·.The PU action is given by
where ω1 and ω2 are positive constants，and the Euler Lagrange equation is
The Lagrangian contains the whole information about the system dynamics for the fundamental physics，in which possible constraints of dynamical variables can be included.The existence of constraints means that the total phase space is reduced to a submanifold in physics.In order to amend the Ostrogradski instability，the dimensionality of original system′s phase space must be reduced by imposing constraints.Constraints can be classified into two classes.The first class constraints are those associated with gauge freedoms in the theory，and the second class constraints are physical，which means the solutions of motion equations are different with or without constraints.In this paper，we can check all the constraints are connected with second class.
The term C（q，q·）q·· in the second time derivative Lagrangian can be absorbed into other terms by using the mathematical identity，
Therefore，the general second time derivative Lagrangian is given by
Furthermore，the most general Lagrangian of second time derivative linear system with constraints can be written as where α，β，a，b，c and e are real parameters.Following Dirac′s analysis of constraints systems[18] and Ostrogradski′s choice[17]，we define canonical variables as From Eq.（20），we have q··=P2-eQ3 as a function of canonical variables P2 and Q3.Through the Legendre transform，the total Hamiltonian is
HT= P1Q2+12P22-αQ22-βQ21-bQ1Q3-cQ2Q3-eP2Q3-aQ32+12e2Q32+u1Φ1，（22）
where the primary constraint Φ1 is a functional given by P3=0，which can be denoted Φ1：P3=0.u1 is a function of canonical variables which can be found by the so-called consistency relations.In fact，we must ensure all constraints are preserved in order that dynamical system can be consistent in the evolution of time.Since P3=0，consistency suggests that its equation of motion P·3={P3，HT} must also vanish on the submanifold of phase space where all the constraints are satisfied.In other words，the Poisson bracket {P3，HT}～0，where the sign "～" means equal after all constraints have been enforced due to Dirac[18].The consistency relations lead up to the fact that there exists new constraints called secondary constraints.So，the following secondary constraint is expected
That is to say，further consistency relations only give rise to the form of the arbitrary function u1，and there′s no further constraints.The source of Ostrogradski instability is the vexatious linear term P1Q2.To exorcise this instability we have to find a constraint where Q2 is some function of P1.Therefore，we have
between the parameters a and e，if we want to exorcise Ostrogradski instability.
In the 2a=e2 case，repeating the procedures above，one can find the further consistency relations
It is easy to check all the constraints are second class constraints.By using these constraints，the reduced Hamiltonian can be written as
HR=1e2b22-e2βQ21+（αe2+b e-c22）（bcQ1+e2P1）2e2（2αe2+be -c2）2.（28）
Obviously，the effective dimensionality of phase space is reduced from four （Q1，Q2，P1，P2） to two （Q1，P1），and the Ostrogradski instability is exorcised.Furthermore，the reduced Hamiltonian will be positive or negative definite when suitable relations are satisfied among the parameters α，β，b，c and e.
From the discussions above，we obtain Theorem 1 stated as follows，
Theorem 1
The most general Lagrangian （18） of second time derivative linear system with constraints is stable and nontrivial iff 2a=e2，e≠0 and
Also，we can move on and generalize the discussions of the system Eq.（18） to a second time derivative system with arbitrary potentials，whose Lagrangian is as follows，
and Q2 can be inverted as a function of Q1 and P1 for a fixed c（Q1，Q2） in Eq.（31）.So，the dimensionality of phase space can be reduced to two （Q1，P1） from four （Q1，P1，Q2，P2）.From Eq.（31） and （32），we obtain Theorem 2，stated as follows， Theorem 2
For a second time derivative dynamical system with constraints，Eq.（30），
（i） By choosing some appropriate function c（Q1，Q2），one can always change the original system into a stable one，and the dimensionality of the phase space is reduced to two （Q1，P1） from four （Q1，P1，Q2，P2） ，but the characteristics of the constrained system might change essentially.
（ii） If one choose c（q，q·） to be a constant c（q，q·）=c0，then the reduced system will retain the complete characteristics of the original dynamical system with the higher derivative terms left out.
（iii） If one choose c=c0+δ（q，q·） and δ（q，q·）C0，then the reduced system roughly keeps the characteristics of the original dynamical system with the higher derivative terms left out，with some corresponding corrections against the original one，quantitatively.
Since Ω1 is positive definite，the solutions are unstable in the case of ω1ω2>2+1 or ω1ω2<2-1.This conclusion is completely consistent with the above mentioned result by the Hamiltonian method.Meanwhile，we find that the characteristics of the PU oscillator is radically changed in this system with constraints.Especially，the oscillating behavior becomes monotonically decreasing or increasing.It is obvious that the choice of constraints is crucial.Inappropriate choosing of constraints will completely change dynamic behavior of the original system，which is certainly not what we expect.
which is negative definite for all values of ω1>0 and ω2>0 （See Fig.1）.The reduced system is not only free from Ostrogradski′s ghost，but also stable.
Next，we consider this sytem with the addition of external dissipative force 2ω1ω2（ω21+ω22）12 q·.The equations of motion are
Finally，we compare the PU oscillators with and without constraints，both encountering the friction force 2ω1ω2（ω21+ω22）12q·.We find from the comparison of Eq.（15） with Eq.（61） that the effective dimensionality of phase space is reduced from four （Q1，Q2，P1，P2） to two （Q1，P1）.There exists the term P1 Q2 in the Hamiltonian of the system without constraints，which gives rise to Ostrogradski′s ghost.On the contrary，the system with constraints is ghost free and its reduced Hamiltonian is negative definite， therefore the system with the constraints is stable.The solvability of both systems allows us to compare further their analytical solutions.For the PU oscillator system without constraint，there are four QNMs.Two of them oscillate and decrease exponentially，but the other two oscillate and increase exponentially which lead to the instability of the system.On the contrary，there only exist two stable QNMs for the PU oscillator with constraints，which is in accordance to the facts that the dimensionality of phase space from four to two and HR is negative definite.It is easy to find from the discussion above that the choice of the constraints is crucial.The criterion of choosing constraints is to eliminate the harmful and retain the favorable from the original dynamic behavior，and meanwhile to maintain the stability of the reduced system.In the linear case，the criterion would enable us to eliminate the unstable QNMs and retain the stable ones. 6Conclusion and discussion
As is known to all，there exists linear instabilities （Ostrogradski instability） in any nondegenerate theory whose fundamental dynamical variable is higher than 2nd order in time derivative.But these instabilities can be removed by imposing constraints which reduce the dimensionality of original system′s phase space.In this paper，we have investigated the most general system of second time derivative linear system with constraints，and showed the sufficient and necessary condition in Theorem 1.Furthermore，we also generalized the investigations of linear system to a higher derivative system with an arbitrary potential term V（q）.
In this paper，we investigated three generalized PU oscillator models with external dissipative forces and found the exact analytic solutions.The solutions of the first model include four QNMs.Two oscillate and decrease exponentially and the other two oscillate and increase exponentially.Therefore，the PU oscillator system without constraints is an unstable system，which is a typical example of systems with time derivative higher than 2nd order.The second model has been discussed by the authors of Ref.[1] using Hamiltonian method.But there is some minor error in their computation，which leads to a conclusion not quite correct actually.We reexamined this model using Hamiltonian method and found although this model can avoid the Ostrogradski′s ghost，it is still unstable when the parameter ratio ω1ω2>2+1.Furthermore，the solutions of this model are monotonous，which have no longer the oscillating characteristics of the original PU oscillator.The third model satisfies the stability conditions given in Theorem 1.The phase space （Q1，Q2，P1，P2） of the model is reduced to （Q1，P1） and the reduced Hamiltonian HR is negative definite，which leads to a stable system.The solutions of the third model indicate that the constraints exclude two unstable QNMs and retain two stable ones，although the frequency and the amplitude of the modes need some appropriate corrections.From the discussion above of the three solvable models，we could learn the following helpful lessons：
Lesson 1： In order to obtain a stable system，we have to choose the Lagrangian of the auxiliary field seriously.In the restricted case of the linear system，Theorem 1 gives an explicit answer，but we should explore further for more general cases.Theorem 2 is just a preliminary study of nonlinear cases.
Lesson 2： The solutions of motion equations are different between the two models with and without second class constraints.One significant reason for the revival of interest in higher derivative theories is related to modified gravity.A reasonable modified gravity model with nondegenerate higher derivatives ought to be free from Ostrogradski instability brought by the higher derivatives，maintain the virtue of the general relativity，and also be competitive in solving the problems such as the accelerating expansion of the universe.When we study modified gravity with nondegnerate higher derivatives，the models are sure to meet these requirements. Lesson 3： The consistency relation，which introduces a complete set of reasonable constraints，is the key point to remove the factor causing instability of the system.For example，in a linear case，good constraints could eliminate the divergent QNMs，but retain the stable ones.
Finally，for general nonlinear higher derivative theories with constraints，we have found a class of system with the following Lagrangian
which is free from Ostrogradski′s ghost.We will investigate the stability conditions of this system in the forthcoming work.In any case，the higher derivative theory mentioned above is only a class of toy models for the fundamental physics.It seems that we have to go beyond simple models，if we attempt to explain some current riddles of physics with a deviation from general relativity.However，our experience from the higher derivative theory discussed above would help us to find a reasonable field theory with higher derivatives.
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带有约束的稳定高导数模型 翟向华 敖犀晨
摘要： 给出了一类精确可解的高导数模型，表明了约束如何去除高导数带来的Ostrogradski不稳定性，对于作用量含有二阶时间导数的系统，找到了建立稳定约束系统的充分必要条件.
关键词： Pais Uhlenbeck振子； Ostrogradski不稳定性； 高导数模型； 约束系统
（责任编辑：顾浩然）