Some Further Remarks on Hamilton's Principle

Firdaus E. Udwadia

Department of Aerospace and Mechanical Engineering, University of Southern California, 430 K Olin Hall, Los Angeles, CA 90089

George Leitmann

Department of Mechanical Engineering, University of California, Berkeley, CA 94720

Hancheol Cho

Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089

Some Further Remarks on

Hamilton’s Principle

This paper deals with the inverse problem of finding a suitable integrand so that upon the use of the calculus of variations, one obtains the equations of motion for systems in which the forces are nonpotential. New extensions and generalizations of previous results are obtained. 关DOI: 10.1115/1.4002435兴

1 Introduction

The development of the equations of motion for a mechanical system from Hamilton’s principle can be viewed as a problem in the calculus of variations when the constraints on the system are holonomic and the forces are derivable from a potential function. Rendering stationary the integral of the Lagrangian over a fixed interval of time taken between two fixed points in configuration space, then, yields the equations of motion for the system. How-ever, it is interesting to investigate the types of forces that can be engendered through the use of an appropriately chosen function 共integrand兲 whose integral when rendered stationary, yields the proper equations of motion even when the forces acting on a system do not arise from a potential, that is, when they are non-conservative.

In 1931, Bolza关1兴 gave a general procedure for finding such an integrand for a single degree-of-freedom system. This was fol-lowed by Douglas关2,3兴 who obtained the necessary and sufficient conditions for the existence of an integrand for multidegree-of-freedom systems. However, it is difficult to obtain the integrands for given, specific forces. In a 1963 note, Leitmann关4兴 provided some examples of such forces and the corresponding integrands for which a variational principle exists. A single degree-of-freedom system was considered and two examples were provided. Recently, the so-called semi-inverse method 关5兴 has attracted much attention due to its simplicity and applicability to certain cases. However, this method assumes a specific form for the func-tion f, which needs to be obtained from experience, intuifunc-tion, or both, and utilizes a Lagrange multiplier type approach.

In this paper, we extend the results in Ref. 关4兴 to some more general nonpotential systems and provide a more systematic way of handling the inverse problem of the calculus of variations. The main difficulties lie in performing the necessary integrations ex-plicitly, as will be seen. The examples given in Ref.关4兴 arise as special cases of the results provided herein. Finally, we apply the general results to some specific systems to indicate the nature of the nonpotential forces, which the results encompass.

2 The Solution to the Inverse Problem

This section deals with the so-called inverse problem of the calculus of variations. Let G关t,x共t兲,x˙共t兲兴 be a force acting on an

unconstrained single degree-of-freedom system whose mass is taken to be unity. We assume that G共·兲 is of class C1_{on R}3_{. Our} aim is to find the function f共·兲:关t1, t2兴⫻R1_⫻R1_→R1_{so that the} functional

冕

t1 t₂

f关t,x共t兲,x˙共t兲兴dt with x共t1兲 = x1 and x共t2兲 = x2 共1兲 when rendered stationary, yields the Euler–Lagrange equation of motion

x¨共t兲 = G关t,x共t兲,x˙共t兲兴 共2兲 We assume that the general solution x共t兲=g共t,a,b兲 of Eq. 共2兲 passes for some values of a and b through x共t1兲=x1 and x共t2兲 = x₂. Noting that x˙共t兲=g_t共t,a,b兲, we further assume that one can solve for a and b from the expressions for x共t兲 and x˙共t兲 so that a =␾关t,x共t兲,x˙共t兲兴 and b=␺关t,x共t兲,x˙共t兲兴, where ␾共·兲 and ␺共·兲 are the continuous functions from 关t1, t2兴⫻R2→R1. Then, x¨共t兲 = gtt共t,a,b兲 and G共·兲:关t1, t2兴⫻R2→R1takes the values G共t,x,r兲

= g_tt关t,␾共t,x,r兲,␺共t,x,r兲兴, where t, x, and r are considered to be independent variables.

Thus, we need to determine the integrand function f共·兲, i.e., 共t,x,r兲→ f共t,x,r兲, such that Eq. 共2兲 is identical with the Euler– Lagrange equation ⳵2_f ⳵t ⳵ r+ ⳵2_f ⳵x ⳵ rx˙共t兲 + ⳵2_f ⳵r2x¨共t兲 − ⳵ f ⳵x= 0 共3兲

where the partial derivatives of f共·兲 are evaluated at 关t,x共t兲,x˙共t兲兴. Substituting x¨共t兲 for Eq. 共2兲 into Eq. 共3兲, we get

⳵2_f ⳵t ⳵ r+ ⳵2_f ⳵x ⳵ rx˙共t兲 + ⳵2_f ⳵r2G − ⳵ f ⳵x= 0 共4兲

Since Eq.共2兲 gives the solution for every initial condition pair 关x共t1兲,x˙共t1兲兴, the equation ⳵2_f ⳵t ⳵ r+ ⳵2_f ⳵x ⳵ rr + ⳵2_f ⳵r2G共t,x,r兲 − ⳵ f ⳵x= 0 共5兲

must be an identity for all 共t,x,r兲苸关t1, t2兴⫻R1⫻R1. To get an explicit expression for f共t,x,r兲, we partially differentiate Eq. 共5兲 with respect to r to obtain

⳵ ⳵t

冉

⳵2_f ⳵r2

冊

+ ⳵ ⳵x

冉

⳵2_f ⳵r2

冊

r + ⳵ ⳵r

冉

⳵2_f ⳵r2

冊

G + ⳵2_f ⳵r2 ⳵G ⳵r = 0 共6兲

Contributed by the Applied Mechanics of ASME for publication in the JOURNAL OF

APPLIEDMECHANICS. Manuscript received March 8, 2010; final manuscript received July 27, 2010; accepted manuscript posted August 24, 2010; published online October 22, 2010. Assoc. Editor: Wei-Chau Xie.

Journal of Applied Mechanics Copyright © 2011 by ASME JANUARY 2011, Vol. 78 / 011014-1 Downloaded 25 Oct 2010 to 74.66.132.217. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Upon defining M共t,x,r兲
共⳵2_/⳵r2_{兲f共t,x,r兲, Eq. 共6兲 becomes the} partial differential equation

⳵M ⳵t + ⳵M ⳵xr + ⳵M ⳵rG + M ⳵G ⳵r = 0 共7兲

whose solution is obtained as关1兴

M共t,x,r兲 = ⌽关␾共t,x,r兲,␺共t,x,r兲兴⌰关t,␾共t,x,r兲,␺共t,x,r兲兴 共8兲 where⌽共·兲 is an arbitrary function of t,␾共t,x,r兲 and ␺共t,x,r兲, and

⌰共t,a,b兲
exp

冉

−

冕

⳵G

⳵r关t,g共t,a,b兲,gt共t,a,b兲兴dt

冊

共9兲

where ⳵G/⳵r关t,g共t,a,b兲,g_t共t,a,b兲兴
共⳵/⳵r兲G共t,x,r兲兩

r=gt共t,a,b兲

x=g共t,a,b兲⬎0.

Finally, the function f共·兲 is found by integrating M共t,x,r兲 twice with respect to r

f共t,x,r兲 =

冕冕

M共t,x,r˜兲dr˜dr + r␭共t,x兲 +␮共t,x兲 共10兲 where␭共t,x兲 and ␮共t,x兲 are arbitrary within the requirement of Eq.共5兲. A similar derivation for the function f共t,x,r兲 is also given in Refs.关1,4,6兴. The notation used here and our treatment closely follows Ref.关6兴.

As a simple example, let us consider the following linear oscil-lator equation:

x¨共t兲 = − cx共t兲 − 2dx˙共t兲 共11兲 where c and d are the positive constants with the initial conditions x共0兲=x0and x˙共0兲=x˙0. For simplicity, let us assume d2_{− c⬎0, the} overdamped case. Then, it is straightforward to show that

␾共t,x,r兲 = 1 ␰2−␰1e −共␰1+␰2兲t关共␰2_e␰2t₋␰1_e␰1t兲x + 共e␰1t_{− e}␰2t兲r兴 共12a兲 ␺共t,x,r兲 = 1 ␰2−␰1e −共␰1+␰2兲t关␰1␰2共e␰2t_{− e}␰1t兲x + 共␰2_e␰1t₋␰1_e␰2t兲r兴 共12b兲 where␰1
−d+

冑d

2_{− c and␰2
−d−}

冑d

2_{− c. According to Eq.共9兲,} we have

⌰共t,x,r兲 = e2dt _共13兲

Using Eq.共10兲, we have the following integrand:

f共t,x,r兲 = e2dt

冕冕

⌽关␾共t,x,r˜兲,␺共t,x,r˜兲兴dr˜dr + r␭共t,x兲 + ␮共t,x兲 共14兲 Since ⌽共·兲 is an arbitrary function of its arguments, we may use

⌽共t,x,r兲 =␾共t,x,r兲 = 1 ␰2−␰1e

−共␰1+␰2兲t关共␰2_e␰2t₋␰1_e␰1t兲x + 共e␰1t

− e␰2t兲r兴 共15兲

Substituting Eq.共15兲 into Eq. 共14兲 yields f共t,x,r兲 = 1 ␰2−␰1e 4dt

冋

1 2共␰2e ␰2t₋␰1_e␰1t兲xr2₊1 6共e ␰1t_{− e}␰2t兲r3

册

+ r␭共t,x兲 +␮共t,x兲 共16兲

where␭共t,x兲 and␮共t,x兲 should be chosen so that f共t,x,r兲兩

r=x˙共t兲

x=x共t兲in Eq. 共14兲 satisfies the Euler–Lagrange equation. For the sake of simplicity, we can choose the following:

␭共t,x兲 = 0, ␮共t,x兲 = c 3共␰1−␰2兲e

4dt_共␰2e␰2t₋␰1_e␰1t兲x3 共17兲

Finally, the integrand function f共·兲, which yields the force G共·兲 in Eq.共11兲, is given by f共t,x,r兲 = 1 ␰2−␰1e 4dt

冋

1 2共␰2e ␰2t₋␰1_e␰1t兲xr2₊1 6共e ␰1t_{− e}␰2t兲r3

册

− c 3共␰2−␰1兲e 4dt_共␰2e␰2t₋␰1_e␰1t兲x3₌ 1 6共␰2−␰1兲e 4dt_关共␰2e␰2t −␰1e␰1t兲x共3r2_{− 2cx}2兲 + 共e␰1t_{− e}␰2t兲r3兴 共18兲

Likewise, if⌽共t,x,r兲=␺共t,x,r兲 for the linear overdamped sys-tem, one other possible integrand function is given by

f共t,x,r兲 = 1 ␰2−␰1e 4dt

冋

1 2␰1␰2共e ␰2t_{− e}␰1t兲xr2₊1 6共␰2e ␰1t₋␰1_e␰2t兲r3

册

− c 3共␰2−␰1兲e 4dt_␰1␰2共e␰2t_{− e}␰1t兲x3 = 1 6共␰2−␰1兲e 4dt_{关␰1␰2共e}␰2t_{− e}␰1t兲x共3r2_{− 2cx}2兲 + 共␰2_e␰1t −␰1e␰2t兲r3兴 共19兲 If⌽共t,x,r兲=关␾共t,x,r兲兴2_{is chosen, we have} f共t,x,r兲 = 1 共␰2−␰1兲2e 6dt

冋

1 2共␰2e ␰2t₋␰1_e␰1t兲2_x2_r2₊ 1 12共e ␰2t_{− e}␰1t兲2_r4 −1 3共e ␰2t_{− e}␰1t兲共␰2_e␰2t₋␰1_e␰1t兲xr3

册

− c 4共␰2−␰1兲2e 6dt_共␰2e␰2t₋␰1_e␰1t兲2_x4 = 1 12共␰2−␰1兲2e 6dt_关3共␰2e␰2t₋␰1_e␰1t兲2_x2共2r2_{− cx}2兲 + 共e␰2t − e␰1t兲2_r4_{− 4}共e␰2t_{− e}␰1t兲共␰2_e␰2t₋␰1_e␰1t兲xr3兴 共20兲

We can verify that the integrand f共t,x,r兲 given in Eqs. 共18兲–共20兲 is correct by directly substituting it into the Euler– Lagrange equation共3兲. We have, thus, illustrated three 共out of an infinite number兲 different integrand functions f共·兲 that yield the force G共·兲 given in Eq. 共11兲.

3 Variational Principle for Systems With Nonpotential Forces

This section deals with the derivation of some special forms of nonpotential forces that can be obtained through the use of the variational calculus by appropriately choosing the integrand func-tions. Application of the variational calculus will, thus, yield the proper equation of motion for such nonpotential systems. It is noted that in order to find integrand function f共·兲 in Eq. 共1兲 of the variational problem, the general solution a =␾关t,x共t兲,x˙共t兲兴 and b =␺关t,x共t兲,x˙共t兲兴 must be known in order to use a prescribed func-tion ⌽共·兲. However, the use of ⌽共t,x,r兲⬅1 while narrowing down the available functions f共·兲 averts this difficulty 关4兴. It is obvious, then, that for f共t,x,r兲 to be explicitly determined, 兰共⳵G共t,x,r兲/⳵r兲dt in Eq. 共9兲 and 兰兰M共t,x,r˜兲dr˜dr in Eq. 共10兲 should be suitably integrable and that the main difficulties in find-ing the function f共·兲, therefore, analytically originate from these two integrations.

With this in mind, we next consider a general form for ⳵G共t,x,r兲/⳵r given by

011014-2 / Vol. 78, JANUARY 2011 Transactions of the ASME

⳵G共t,x,r兲

⳵r = g1共t兲 + g2共x兲r + g3共r兲G共t,x,r兲 共21兲 so that兰共⳵G共t,x,r兲/⳵r兲dt can be easily obtained and we can avoid the difficulty of integrating 兰兰M共t,x,r˜兲dr˜dr by separating the variables. Equation共21兲 leads us to assume the following form for G共t,x,r兲:

G共t,x,r兲 =␤共t兲r + ␥共x兲r2+␣共t,x兲␦共r兲 +␣¯共t,x兲 共22兲 where␣, ␣¯, ␤, ␥, and␦are the arbitrary共suitably smooth兲 func-tions of their arguments. We, then, explore various cases in which G共t,x,r兲 has the form given in Eq. 共22兲.

As stated before, although the general solution of the nonlinear differential equation共2兲 with G共t,x,r兲 given by Eq. 共22兲 is hard to find, we can avoid the difficulty by setting⌽共t,x,r兲⬅1 in Eq. 共8兲 关4兴. If we choose G共t,x,r兲=␤共t兲r, then, 共⳵/⳵r兲G共t,x,r兲=␤共t兲, which is of the form of Eq.共21兲 in which g1共t兲=␤共t兲, g2共x兲=0, and g3共r兲=0. Thus, for a nonpotential force of the form G共t,x,r兲=␤共t兲r acting on a particle of unit mass, we can find the integrand f共t,x,r兲=exp共−兰␤共t兲dt兲r2_{/2, which when used to make} the integral given in Eq.共1兲 stationary, yields the Euler–Lagrange equation x¨共t兲=G关t,x共t兲,x˙共t兲兴=␤共t兲x˙共t兲. In a similar manner, G共t,x,r兲=␥共x兲r2 _{is another possible nonpotential force since} 共⳵/⳵r兲G共t,x,r兲=2␥共x兲r has the form of Eq. 共21兲 if we let g1共t兲 = 0, g2共x兲=2␥共x兲, and g3共r兲=0. As another example, for G共t,x,r兲=␣共t,x兲␦共r兲, ␦共r兲⫽0, we have 共⳵/⳵r兲G共t,x,r兲=␣共t,x兲 ⫻共d␦共r兲/dr兲=关共1/␦共r兲兲共d␦共r兲/dr兲兴关␣共t,x兲␦共r兲兴, which is of the form of Eq.共21兲 with g3共r兲=共1/␦共r兲兲共d␦共r兲/dr兲. This allows us to solve the inverse problem of finding the integrand f共t,x,r兲, which when used to render stationary the integral given in Eq.共1兲, yields the equation of motion x¨共t兲=␣关t,x共t兲兴␦关x˙共t兲兴. Likewise, exploring other cases, such as G共t,x,r兲=␤共t兲r+␥共x兲r2, G共t,x,r兲=␤共t兲r +␣共t,x兲␦共r兲, G共t,x,r兲=␤共t兲r+␣¯共t,x兲, G共t,x,r兲=␥共x兲r2 +␣共t,x兲␦共r兲, G共t,x,r兲=␥共x兲r2_{+␣¯共t,x兲, G共t,x,r兲=␣共t,x兲␦共r兲} +␣¯共t,x兲, G共t,x,r兲=␤共t兲r+␥共x兲r2_{+␣共t,x兲␦共r兲, G共t,x,r兲=␤共t兲r} +␥共x兲r2_{+␣¯共t,x兲, G共t,x,r兲=␤共t兲r+␣共t,x兲␦共r兲+␣¯共t,x兲, G共t,x,r兲} =␥共x兲r2_{+␣共t,x兲␦共r兲+␣¯共t,x兲, and G共t,x,r兲=␤共t兲r+␥共x兲r}2 +␣共t,x兲␦共r兲+␣¯共t,x兲, a set of nonpotential forces can be found for which a suitable function f共t,x,r兲兩

r=x˙共t兲

x=x共t兲whose integral when made stationary, yields the correct equation of motion. The following six sets of nonpotential forces can be adduced from a suitable function f共·兲 via the variational calculus:

G1关t,x共t兲,x˙共t兲兴 =␣1关t,x共t兲兴 +␤1共t兲x˙共t兲 +␥1关x共t兲兴x˙2_{共t兲 共23a兲} G2关t,x共t兲,x˙共t兲兴 =␣2关t,x共t兲兴␦2关x˙共t兲兴, ␦2关x˙共t兲兴 ⫽ 0 共23b兲 G3关t,x共t兲,x˙共t兲兴 =␤3共t兲x˙共t兲 +␥3关x共t兲兴x˙2共t兲 +␣3关t,x共t兲兴x˙p共t兲, x˙共t兲 ⬎ 0 共23c兲 G4关t,x共t兲,x˙共t兲兴 =␤4共t兲x˙共t兲 +␥4关x共t兲兴x˙2共t兲 +␣4共t兲x˙共t兲ln x˙共t兲, x˙共t兲 ⬎ 0 共23d兲 G5关t,x共t兲,x˙共t兲兴 =␣¯5共t兲 +␣5关t,x共t兲兴eqx˙共t兲, q⫽ 0 共23e兲 G6关t,x共t兲,x˙共t兲兴 =␣¯6关x共t兲兴 +␣6关t,x共t兲兴eqx˙2共t兲, q⫽ 0 共23f兲 Here,␣i,␣¯i,␤i, ␥i, and ␦i共i=1,2,3,4,5,6兲 are the arbitrary

functions of their arguments, as indicated, and p and q are the real numbers. One could relax the condition x˙共t兲⬎0 to x˙共t兲⫽0 in Eq. 共23c兲 if p is restricted to being any real number such that x˙p_{共t兲 is}

a real, well-defined number. Equation共23a兲 and a special form of Eq. 共23b兲, namely, G关t,x共t兲,x˙共t兲兴=␣关t,x共t兲兴x˙2_{共t兲, are already} known and given in Ref.关4兴. Furthermore, the general form given in Eq.共23a兲 includes the special examples 共given in propositions

1–3 and 5兲 in Ref. 关7兴. We can determine f共t,x,r兲 of Eq. 共1兲 using Eq.共10兲. The resulting integrand f共t,x,r兲 of Eq. 共1兲 for the forces described in Eq.共23兲 is, respectively, given by

f1共t,x,r兲 =1 2⌰1共t,x兲r 2₊

冕

_{␣1共t,x兲⌰1共t,x兲dx 共24a兲} f2共t,x,r兲 =

冕冕

1 ␦2共r˜兲dr˜dr +

冕

␣2共t,x兲dx 共24b兲 f3共t,x,r兲 = 1 共1 − p兲共2 − p兲⌰3共t兲⌿3共x兲r2−p +⌰3共t兲

冕

␣3共t,x兲⌿3共x兲dx, p ⫽ 1 and p ⫽ 2 共24c兲 f4共t,x,r兲 = ⌰4共t兲⌿4共x兲共r ln r − r兲 +␤4共t兲⌰4共t兲

冕

⌿4共x兲dx 共24d兲 f5共t,x,r兲 = 1 q2⌰5共t兲e −qx˙_+⌰5共t兲

冕

_{␣5共t,x兲dx 共24e兲} f6共t,x,r兲 =1 2

冑

␲ q⌰6共x兲

冋

r erf共

冑qr

兲 + e−qr2

冑q

␲

册

+

冕

␣6共t,x兲⌰6共x兲dx, q ⬎ 0 共24f兲 f6共t,x,r兲 =1 2

冑

␲ − q⌰6共x兲

冋

r erfi共

冑

− qr兲 − e−qr2

冑

− q␲

册

+

冕

␣6共t,x兲⌰6共x兲dx, q ⬍ 0 共24g兲 where ⌰1共t,x兲 = exp

冋

−

冕

␤1共t兲dt − 2

冕

␥1共x兲dx

册

共25a兲 ⌰3共t兲 = exp

冋

共p − 1兲

冕

␤3共t兲dt

册

, ⌿3共x兲 = exp

冋

共p − 2兲

冕

␥3共x兲dx

册

共25b兲 ⌰4共t兲 = exp

冋

−

冕

␣4共t兲dt

册

, ⌿4共x兲 = exp

冋

−

冕

␥4共x兲dx

册

共25c兲 ⌰5共t兲 = exp

冋

q

冕

␣¯5共t兲dt

册

共25d兲 ⌰6共x兲 = exp

冋

2q

冕

␣¯6共x兲dx

册

共25e兲 and erf共·兲 and erfi共·兲 denote the error function and the imaginary error function, respectively. Equations共24f兲 and 共24g兲 pertain to G6共·兲 given in Eq. 共23f兲. Equation 共24c兲 does not hold when p = 1 or p = 2. When p = 1, Eq.共23c兲 becomes

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G3关t,x共t兲,x˙共t兲兴 =␣3关t,x共t兲兴x˙共t兲 +␥3关x共t兲兴x˙2_{共t兲 共26兲} and

f3共t,x,r兲 = ⌰3共x兲共r ln r − r兲 +

冕

␣3共t,x兲⌰3共x兲dx 共27兲 where⌰3共x兲=exp关−兰␥3共x兲dx兴. In addition, when p=2, Eq. 共23c兲 becomes

G3关t,x共t兲,x˙共t兲兴 =␤3共t兲x˙共t兲 +␣3关t,x共t兲兴x˙2_{共t兲 共28兲} and

f3共t,x,r兲 = − ⌰3共t兲ln r + ⌰3共t兲

冕

␣3共t,x兲dx 共29兲 where⌰3共t兲=exp关兰␤3共t兲dt兴.

By using Eq. 共24b兲, we provide in Table 1 some nonpotential force G关t,x共t兲,x˙共t兲兴 that can be obtained 共␩is constant兲 by using the calculus of variations along with the corresponding integrand f共t,x,r兲. It is noted that if we substitute each integrand f共t,x,r兲兩

r=x˙共t兲

x=x共t兲into the Euler–Lagrange equation共Eq. 共3兲兲, we get the corresponding force G关t,x共t兲,x˙共t兲兴.

Other special cases, such as G关t,x共t兲,x˙共t兲兴=␣x˙3共t兲+␤x˙共t兲, x˙共t兲 ⬎0 共␣,␤⬍0兲, can be obtained from the integrand f共t,x,r兲 =共1/2兲共e2␤t/r兲+␣e2␤tx by using Eq.共24c兲. Another integrand us-ing Eq. 共24b兲 for the same force G共·兲 is given by f共t,x,r兲 = r ln兩r兩/␤−共1/2兲共r ln共−␣r2_{−␤兲/␤兲−共tan}−1_共␣r/

冑

_␤␣兲/

冑

_␤␣兲+x, r⬎0.

As an important example, let us consider a model of the Cou-lomb friction force approximated by

G关t,x共t兲,x˙共t兲兴 =␣ tanh关␤x˙共t兲兴, x˙共t兲 ⬎ 0 共30兲 where ␣⬍0 and ␤⬎0 are the constants. Comparing with Eq. 共23b兲, it is obvious that ␣2关t,x共t兲兴=␣ and␦2关x˙共t兲兴=tanh关␤x˙共t兲兴. Then, Eq.共24b兲 yields the following integrand:

f共t,x,r兲 =

冕冕

coth共␤r˜兲dr˜dr +

冕

␣dx = 1 2␤2关Li2共e

−2␤r_{兲 −␤}2_r2 − 2␤r ln兩1 − e−2␤r兩 + 2␤r ln兩sinh共␤r兲兩兴 + ␣x, r ⬎ 0

共31兲 where Li2is the polylogarithm function, also known as the Jon-quière function, defined as

Lin共z兲

兺

k=1

⬁ zk

kn 共32兲

where n is an integer. To verify that the f共t,x,r兲 in Eq. 共31兲 does yield the Coulomb friction force given by Eq. 共30兲, we directly substitute it into the Euler–Lagrange equation共3兲. First, we have

⳵ f ⳵r= 1 2␤2

冋

− 2␤Li1共e −2␤r_{兲 − 2␤}2_{r − 2␤ ln兩1 − e}−2␤r_{兩 − 4␤}2 re−2␤r 1 − e−2␤r + 2␤ ln兩sinh共␤r兲兩 + 2␤2_{r coth共␤r兲}

册

_共33兲 where we have used the fact共d/dz兲Lin共z兲=共1/z兲Lin−1共z兲. The total

differentiation of Eq.共33兲 with respect to time gives us, after some calculations

d dt

冉

⳵

⳵rf关t,x共t兲,x˙共t兲兴

冊

= x¨共t兲coth关␤x˙共t兲兴 共34兲 Next, it is obvious from Eq.共31兲 that

⳵ f

⳵x=␣ 共35兲

With Eqs. 共34兲 and 共35兲, the Euler–Lagrange equation, then, yields the force G关t,x共t兲,x˙共t兲兴 that is given by Eq. 共30兲 when x˙共t兲⬎0.

4 Conclusions

We have discussed here extensions of variational principles for nonpotential forces, employing the inverse problem of the calcu-lus of variations. While a problem of some considerable interest, few new, general results appear to have been obtained to date beyond those obtained in Refs. 关1,4,5兴. This paper obtains new integrand functions for some general nonpotential forces and de-velops a simple and systematic approach for getting them. We note that our discussion is restricted to the scalar case 共single degree-of-freedom systems兲. The inverse problem for the vector case is considerably more difficult. Thus, the extension to nonpo-tential forces in the multidimensional case remains an open prob-lem.

References

关1兴 Bolza, O., 1931, Vorlesungen über Varationsrechnung, Koehler und Amelang, Leipzig.

关2兴 Douglas, J., 1939, “Solution of the Inverse Problem of the Calculus of Varia-tions,” Proc. Natl. Acad. Sci. U.S.A., 25, pp. 631–637.

关3兴 Douglas, J., 1940, “Theorems in the Inverse Problem in the Calculus of Varia-tions,” Proc. Natl. Acad. Sci. U.S.A., 26, pp. 215–221.

关4兴 Leitmann, G., 1963, “Some Remarks on Hamilton’s Principle,” ASME J. Appl. Mech., 30, pp. 623–625.

关5兴 He, J.-H., 2004, “Variational Principles for Some Nonlinear Partial Differential Equations With Variable Coefficients,” Chaos, Solitons Fractals, 19, pp. 847– 851.

关6兴 Leitmann, G., 1986, The Calculus of Variations and Optimal Control: An Introduction, Springer, New York.

关7兴 Musielak, Z. E., 2008, “Standard and Non-Standard Lagrangians for Dissipa-tive Dynamical Systems With Variable Coefficients,” J. Phys. A: Math. Theor.,

41, p. 055205.

Table 1 The function G†t,x„t…,x˙„t…‡ of Eq. „2… and the

corre-sponding function f„t,x,r… of Eq. „10…

G关t,x共t兲,x˙共t兲兴 f共t,x,r兲 x2_共t兲

冑

_␩2_{+ x˙}2_{共t兲 r ln共r +}

冑

_␩2_{+ r}2_{兲 −}

冑

_␩2_{+ r}2₊1 3x 3 h关t,x共t兲兴e−␩x˙共t兲 1 ␩2e␩r+兰h共t,x兲dx h关t,x共t兲兴 ␩2_{+ x˙}2_共t兲 1 12r 4₊1 2␩ 2_r2_{+兰h共t,x兲dx} h关t,x共t兲兴e−␩2x˙2共t兲

冑

_␲ 2␩

冋

r erfi共␩r兲 − e␩2r2 ␩

冑

␲

册

+兰h共t,x兲dx,

where erfi共z兲ª−i erf共iz兲

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