Generalized equations of motion for mechanical sys-tems with variable masses and forces depending on higher order derivatives

Generalized equations of motion for mechanical systems…

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2014 Математика. Механика. Информатика Вып. 2 (25)

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Generalized equations of motion for mechanical systems with variable masses and forces depending on higher order derivatives

E.A. Galperin

The Buquoy generalization of Newton's second law of motion for systems of bodies with variable masses driven by reactive forces produced by ejected burnt fuel (Mestschersky) is considered, with its extension for motions subject to external forces depending on accelerations and higher order derivatives of velocities. Such forces are exhibited in Weber's electro-dynamic law of attraction; they are produced by the Kirchhoff-Thomson adjoint fluid acceleration resistance acting on a body moving in a fluid and are also involved in manual control of aircrafts and spacecrafts that depends on acceleration of the craft itself. The causality of systems driven by such forces is assured by consideration of the left higher order derivatives in the right-hand sides of the equations of motion. The consistency condition and a new solution method are presented, and the existence and uniqueness of solutions for equations of motion driven by such forces is proved. The notion of effective forces is discussed, and the parallelogram law is verified for the effective forces in mechanical systems with left higher order derivatives in controls. On this basis, the new autopilot design is proposed for added security in civil aviation, independent of the currently used Pitot tubes which may fail or render the local measurements of wind gusts instead of the correct estimates for the average relative velocity of the aircraft with respect to the wind in flight or to the airstrip at landing.

Key words: Motion of bodies with variable masses; Forces with the left higher order derivatives of velocity; Generalized equations of minimum order; Autopilot design in aviation.

1. Introduction

buquoy generalization newton fuel

In analytical mechanics, the attention is directed to the study of motion of bodies with constant masses under the forces depending on time, space coordinates, and velocities, according to the classical representation of the second law of Newton [1]. For such motion, various forms of the generalized equations (Lagrange, Hamilton, etc.) were developed, see, e.g., [2-5] and references therein. With the advent of jet propulsion, it is important to consider the motion of bodies with variable masses under forces which may depend on accelerations and higher order derivatives of velocity. However, such systems have different dynamics, follow different laws of motion, and require different forms of the minimum order equations with specific solution methods. These equations and their solution are considered, with application to the autopilot design for added security which may be compromised if based solely on the Pitot tubes currently used in aviation.

First, we reproduce the symbolic representations of the second law of Newton as given in textbooks on mechanics, followed by its generalization by G. Buquoy [6] and later by I.V. Me-stschersky [7]. Another important point is related to the orientation of time and the misconception concerning time-derivatives in the right-hand sides of differential equations which are routinely used in mathematical description of processes. The use of right time-derivatives severely restricts the possibility of control of processes according to the formally written representations of classical laws and some currently accepted forms of generalized equations of motion. New representations are considered, and causality of differential systems is studied in relation to the orientation of time. Then, geometry and time phenomena in classical mechanics are revisited, and the new forms of generalized equations are derived with the left (possibly delayed) higher order time derivatives in the right-hand sides of the equations of motion. The method for their integration is demonstrated by an example of a physical pendulum.

The paper is organized as follows. In Section 2, the classical forms of Newton's second law of motion are presented. Section 3 describes the generalization of this law for bodies with variable masses due to Buquoy [6] and later Mestschersky [7] and Levi-Civita [8]. Problems related to time and causality are considered in Section 4. Section 5 presents a generalization for systems driven by forces with left higher order derivatives in the right-hand sides. In Section 6, the existence of solution is proved under certain consistency condition related to the left highest order derivative in the right-hand side. In Section 7, the notion of effective forces is presented, and the parallelogram law is verified for effective forces, with application to the autopilot design. Section 8 presents the space shuttle example of motion with variable mass to expose the need for acceleration assisted control. Section 9 presents generalized equations in independent coordinates for motion of bodies with variable masses and left higher order derivatives in the right-hand sides. Section 10 describes a method of integration, and in Section 11, some points of interest are summarized, followed by references immediately relative to the problems considered.

2. Representations of the Second

Newton's Law of Motion

The second law of Newton states: "Law II. The change of motion is proportional to the motive force impressed and is made in the direction of the right line in which that force is impressed" [1], see also [5, p. 259]. In high school textbooks, this law is written in the form: ma = F where m means a constant mass, a - the acceleration, and F is "the motive force impressed" or simply "a force", a self-explanatory notion known from life experience. In university textbooks, the Law II is specified in more exact terms:

m x'' = F(t, x(t), v(t)), v(t)=x'(t),

x''(t)=v'(t)=a(t), x(0)=x₀ , v(0)=v₀ , t 0, (1)

which define a particular motion starting at x₀ , v₀ with velocity v(t) defined as time derivative

v(t) = x'(t) = dx/dt = lim [x(t + t) - x(t)] / t (2)

as t 0, t > 0 . Widely used representations (1)-(2) impose heavy restrictions in mechanics and control theory which restrictions are not necessary and can be removed.

3. Generalization for Variable Masses by G. Buquoy [6]

When m = const, the first formula in (1) can be written as follows:

m x'' = m v'(t) = m dv/dt = d(mv)/dt =

= F(t, x(t), v(t)), t 0 . (3)

The last equality in (3) can be written in a more general form:

d(mv) = m dv + v dm = F(t, x(t), v(t)) dt,

t 0, dt > 0, (4)

where differentials can be viewed as small increments, this leading to the well known interpretation: "the change of momentum, d(mv), equals the impulse of force (or simply impulse), Fdt". If m = const, then dm = 0, and (4) coincides with (1) as dt 0. If the mass m = m(t) const, then (4) accounts for the changing mass of a moving body when dm, moving with the same velocity v(t), separates from the body. However, if elementary mass dm is ejected from the body (e.g., as burnt fuel) with a different velocity w(t) v(t), it will impress an additional force upon the body which force must be proportional to the additional "change of motion" (see the second Newton's law cited above), i.e., to the additional change of momentum which is itself proportional to the relative velocity v - w with which dm is ejected from the body. Thus, the quantity v dm shown in (4) should be replaced by the quantity (v w) dm, yielding the equation

m dv + (v w) dm = F(t, x(t), v(t)) dt,

t [0, T ) , (5)

where velocities v and w are absolute velocities of the body and the ejected mass dm respectively, in a coordinate frame at rest in which the motion of a body is considered. The reader can see the change in the force impressed on the body by the mass dm being ejected, if (5) is rewritten in the form which corresponds to the form in (1), (3)

m dv = F(t, x(t), v(t)) dt + (w - v) dm,

t [0, T ) . (6)

Here the change in momentum of a body is at left, and all impulses are at right of the equation. Now, if m is constant (dm = 0) or is being separated from the body without ejection (w = v, dm < 0), then the force is not changing, only the mass m(t) of the body is decreasing and acceleration increasing since the same force is acting on decreasing mass of the body. In this case, the last term at right is zero, and (6) coincides with (1). However, in a spacecraft with jet engine, the burnt fuel mass is ejected, dm < 0, with velocity w different from the velocity v of the spacecraft. To explain the action of ejected mass dm in (6), we assume, for simplicity, that w, v are collinear vectors. If the burnt fuel mass is ejected in the same direction in which the spacecraft moves, so that w > v, then additional reactive force exerts the braking effect upon the spacecraft since (w - v)dm < 0. If it is ejected in the opposite direction, so that w v < 0, then additional reactive force accelerates the motion since (w - v)dm > 0. However, the entries in (1)-(6) can be considered as 3D vectors (except time t, dt and mass m, dm which are scalars), so that turning the funnel ejecting the burnt fuel mass allows one to control also direction of the motion. The term (w - v)dm added to the nominal impulse F(.)dt in (6) represents, in fact, the control impulse u(t)dt in the resulting total impulse F*(.)dt = [F(.)+ u(t)]dt, yielding the equation of controlled motion:

m(t) dv/dt = F*(.) = F(t, x(t), v(t)) + u(t),

u(t) = [w(t) - v(t)] dm/dt, t [0, T ). (7)

Burnt fuel generates not only the reactive force of ejected masses but also a direct active force of heated gas pressure which is considered a part of F(.) in (5)-(7), but can be studied as separate action, see Space Shuttle example, Section 8. It is worth noting that equation (7), quite different from (1)-(4), can be included in the original Newton's statement of the Law II above since it is not specified what "the motive force impressed" actually is. This emphasizes the importance of particular symbolic representations.

Equations (5), (6) represent a fundamental generalization of the classical equations of motion (1)-(4), very important for applications (as we know today). However, when published in 1815, see [6], this generalization was not properly recognized, not entered in textbooks, and thus, quickly forgotten. So, it was rediscovered by I. Mestschersky in 1897, see [7] where many special cases are also studied. Then in 1928, the equation d(mv)/dt = F, cf. the right equality in (3), was independently derived by T. Levi-Civita [8], representing the case w = 0, that corresponds to the motion of a body with variable mass m(t) when dm(t) is being separated from the body without any impulse of force upon the body, thus excluding the control of motion by means of ejected burnt fuel mass.

4. Time Orientation and Causality

In the literature, velocity v(t) on which the motive force F(.) in (1) may depend is defined as right derivative through the limit in (2). However, at the moment t of actual motion, the value v(t + t) does not exist for any t > 0. This means that the limit in (2) also does not exist, so that equation (1) refers, in fact, to some prospective values of v(t) in future, being thus non-causal. The reader may object: well, then what is shown on the speedometer of a car? Yes, the velocity is shown which is actually measured as left time derivative v(t) = lim [v(t) v(tt)]/t, t 0, t > 0, not right derivative as written in (2). This reflects the positive orientation of time: suppose that x(t) in (1) is a distance of the moving mass m from the origin if the motion has started at time t = 0 with initial conditions indicated in (1). If we consider a moment t* > 0 with the past history of motion registered in a measuring device or in a computer over the segment [0, t*], then over the interval (0, t*) there exist both right and left derivatives; at the moment t = 0, there exists only right derivative; at t = t* there exists only left derivative, and over the future interval (t*, T), T , there is no motion yet, thus, no derivatives exist, and the same on the interval (-, 0) when there was no motion at all. This concerns all natural processes (physical, biological, etc.) developing in time: right time derivatives may exist only in the registered past history of a process.

Of course, right derivatives at the current moment, as well as future situations and/or decisions (called rational expectations), can be postulated (imagined as desired) and taken into account, which is routinely done in economy and finance; but in engineering and technology it may be improper and needless to do so. In natural sciences, there is another way to include current accelerations and other higher order time derivatives into process equations, thereby retaining their causality.

In control of motion, the effect of time orientation is compounded by time uncertainty. Indeed, velocity v(t) as left derivative continuously measured by speedometer in a car appears on driver's panel with a delay > 0 due to a finite speed of information transmittal. Hence, at the moment t = t*, a driver sees the velocity v(t*-), not the actual velocity v(t*). However, in the equation (1) of the motion, the force F(.) is impressed (not measured by a device, but felt as are, e.g., gravitational or resistance forces), thus, at a moment t*, we have the force F(t*, x(t*), v(t*)) acting without delay if there is no information transmittal for the values x(t), v(t), in which case time-uncertainty is not implicated in the motion governed by the laws of mechanics such as Law II above. In contrast, if the control u(.) in (7) depends on certain parameters which are measured on the trajectory and transmitted into the power train of the motion, then u(t-) actually depends on > 0, at each moment t > 0, through those measured parameters. Thorough consideration of time-uncertainty is beyond the scope of the paper, so we assume here that = 0, except of certain cases of special interest, see below.

5. Generalization for Controls with Left Higher Order Derivatives

Consider the specification of Newton's Law II presented by equation (1) which can be found in all books on mechanics and related subjects. Distinctive feature of this equation is that "the motive force impressed" F(.) is defined for the moment t and depends only on t and/or x(t) and/or v(t). In some textbooks, it is explained that force F(.) does not depend on acceleration a(t) = x''(t), since if it did, we would have the equation m x'' = F(t, x, v, x'') which, if solved for x'', would render x'' = F*(t, x, v, m), hence, the right-hand side F(.) would not be “the motive force impressed” in the sense of Newton's Law II, but rather it would be F*(t, x, v, m) which does not depend on x''(t) again. What would happen if F(.) = F(t, x, v, x'', x''') is not even mentioned since such a consideration is taken as an obvious blunder.

However, equations of motion with variable mass contain controls: w in (5)-(6), or u(t) in (7), and it is not clear why w and u(t) must not depend on acceleration x''(t) and its rate of change x'''(t). In fact, they can, and the so called acceleration assisted control is widely used in practice for soft regulation, despite its contradiction with (1)-(4). Indeed, consider the following railway construction principle. If to change direction of motion, a perfect circular arc is joined to a right line segment of a railway, then at the connection point the train will receive a hard impact of centripetal force, and the train may derail if its speed is high enough. If a person is standing on the platform of a coach with a door open, he will be thrown out of the train by centrifugal force. To avoid such eventualities, the railway connection must be designed as a cubic or higher degree curve in order to soften the turn and eliminate hard impacts by means of a correct profile of the railway. Obviously, the same concerns the profile of a highway. Whatever the actual profile of a road, experienced drivers always soften a turn by crossing the lanes while continuously turning the steering wheel (this cannot be done by a train because of the rails on which it runs). With manual control, the pilot of an aircraft or spacecraft does the same by making a turn along some higher degree curve following his personal feeling of the centrifugal force that appears during the turn. As a matter of fact, in all manually controlled vehicles, a turn is being done by a control u(.) which is called, in theory, "open loop control u(t)", being, in reality, a feedback control u(t, x(t), x''(t), x'''(t)) depending on actual acceleration x''(t) and its rate of change x'''(t) felt by the pilot, and, maybe, on higher order derivatives if they are felt by a human being (an open question for medicine). In manually controlled aircraft, the pilot always employs a feedback control of the form u(t-, x(t-), x''(t-), x'''(t-)) which depends on time t (with delay > 0 due to a finite speed of information transmittal in human senses) and distance x(t), if it is seen during landing, but does not depend on v since constant velocity is not felt by a human being nor by instruments on board, according to the postulate of physical equivalence of all inertial systems [3]. Dependence on velocity v(t) means, in fact, dependence on acceleration dv/dt which accompanies a varying velocity v(t). A manual control u(.) always depends on acceleration x''(t-) and its rate of change x'''(t-), no matter that they are theoretically excluded by a choice of representation in the equations of motion (1), (7). Therefore, it is important to extend the real life situation in manual control onto automatic control systems by removing the existing restriction with a new choice of representation for Newton's Law II, which would allow higher order time derivatives in the right-hand side of (1), (3)-(7).

Consider equation (7) where w and/or dm/dt, thus u(t), may depend on acceleration and higher order derivatives. Dividing (7) by m(t)>0 and using the left time derivatives at the right-hand side for t > 0, we can write the causal representation of the general equation of motion in the form [9, 10]:

x''= dv/dt = [F(t,x(t),v(t))+ u(t)]/m(t)=

=F*(t, x, x'^-, x''^-, …, x^(k)-),

x(0)=x₀, x'^-(0)=v₀, (8)

where superscript ( ^- ) indicates the left time derivative of corresponding variable which is written in normal script for better visibility. The only right time derivative is x'' = dv/dt, at left in (8) due to forward propagation of motion. It is clear that F*(.) at right in (8) is well defined for all t > 0. The highest order k 2 in (8) depends on the control u(t) employed. For simplicity, the time-uncertainty is omitted from further considerations as well as m(t) which is not shown explicitly as a variable of F*(.) in equation (8). For a natural phenomenon with resistance in F*(.) depending on acceleration of a solid falling into a viscous liquid, see [9, p. 181] and [10, p. 34]. For an application to acceleration assisted hovercraft control, see [9, pp. 179-180] and [10, pp. 39-41].

Remark 5.1. Forces containing left higher order derivatives can appear in equations of motion not only through controls. Such forces depending on accelerations have been considered by Sir Horace Lamb in equations of motion of a solid in ideal liquid, see [11, p.168, § 124, Equations (1)] with reference to Kirchhoff and Sir W. Thomson (1871), where forces of the fluid pressure linearly depended on the acceleration of the solid itself, see [11, p.168, Equations (2); p.169, Equations (3)]. Such forces usually can be taken into account by the introduction of adjoint masses, see example given in [11, p.190, § 137, Equations (2)] with reference to Thomson and Tait [12, Art. 321]. The author is grateful to V.V.Rumyantsev for these references. Another example is furnished by Weber's electrodynamic law of attraction, with the force per unit mass F*=[1 (r' ² 2rr'')/c²]/r² where r is the distance of the particle from the center of force ( W. Weber, Annalen der Phys. LXXIII, 1848, p. 193), see also [2, p. 45], and r', r'' should be understood as left time derivatives.

The causal equation (8) can be solved by standard methods of ordinary differential equations, for which we need the following

Lemma 5.1 [9, 10]. If a function x(t) is defined on an open interval (a, b) and has continuous left derivative on (a, b), then x(t) is continuously differentiable on (a, b).

Proof. By hypothesis, for every t (a, b) there is a limit

,

, (9)

which, as a function of t, is continuous on (a, b), that is

, . (10)

Let t - t = t₀ , then (9) can be rewritten as follows, yielding the right derivative at t₀ :

,

. (11)

Since by construction,

[x(t) - x(t - t)] /t [x(t₀ + t) - x(t₀)] /t,

t (a, b), t₀ = t - t (a, b), (12)

so, from (9), (11), (12), we have x'^-(t) = x'⁺(t₀) x'(t₀), which, due to (10), implies

x'^-(t₀) = x'⁺(t₀) x'(t₀), (13)

as t +0, t t₀ for every t₀ (a, b).

Remark 5.2. Left and right derivatives considered above are special cases of Dini derivatives and the Lemma, in a more general setting, corresponds to the Denjoy-Young-Saks Theorem [13] where only finiteness of a one-sided derivative is required for every t (a, b), implying differentiability of x(t) almost everywhere in (a, b).

Remark 5.3. As follows from (12) with t = (t₀ + t) t₀ + 0, as t +0, left derivatives in (8) can be regarded as delayed right derivatives: x^(k)-(t) x^(k)+(t₀) = lim x^(k)+(t - t), as t +0. This, however, leads to theoretical complications and may result in the loss of stability which might not be the case for the original equation (8), see Section 8. For these reasons, we do not use such representations.

6. Consistency Condition and Existence of Solutions

The continuity of motion x(t), v(t) = x'(t) does not imply that the right-hand side of (8) is continuous. However, in this research we are concerned with the existence and mechanical properties of motions affected by higher order derivatives in the right-hand side. With this issue in mind and in order to get clear of other issues and complications caused by possible discontinuities [14], we assume henceforth that the function F*(…) in (8) and all its entries including all higher order derivatives are continuous on [0, T), T . In this case, equation (8) is mathematically identical, by the Lemma, to the similar equation with all right derivatives, and we assume, for the same reasons, that this equation with all right derivatives has no singular solutions, is solvable for the highest derivative, and in its normal form

x^(k)(t) = (t, x, x', …, x^(k-1)), t [0, T),

k 2 (14)

the function (.) of (14) satisfies the standard conditions that guarantee the existence, uniqueness and extendibility of solutions over the entire interval [0, T ). Under these regularity conditions, there is a unique solution of (14) which depends on the initial data

x(0) = x₀ , x'(0) = v₀ ,

x''(0) = p₂ , … , x^(k-1)(0) = p_k-1 , (15)

where x₀ , v₀ are given and the values p₂ ,…, p_k-1 can be considered as control parameters. Since derivatives in F*(.) of (8) are, in fact, left derivatives, one has to assign initial values for p₂ and p_k = x^(k)(0) in such a way that (8), (14) hold for t = 0 :

p₂ = F*(0, x₀ , v₀ , p₂ ,…, p_k-1 , p_k ),

p_k = x^(k)(0), k 2, (16)

which we call the consistency condition. If k = 2 and x''^-(t) actually enters F*(.), then there are no free control parameters, due to (16), and the same if F*(.) does not contain higher order derivatives which renders the usual 2^nd order equation with two initial conditions in (8). If k > 2, then there are exactly k - 2 free control parameters in (15) plus two initial conditions x₀ , v₀ for the total of k initial conditions as required by the theory of ODEs.

For example, if k = 3, then from (16) we compute p₂ = h(x₀ , v₀ , p₃), and in (15) we obtain p_k-1 = x''(0) = p₂ = h(x₀ , v₀ , p₃), as required, whereby p₂ is the initial condition for (14) depending on a free parameter p₃ which defines also initial data x'' ^-(0) = p₂ = h(x₀ , v₀ , p₃) and x''' ^-(0) = p₃ in (8). If F*(…) of (8) is linear in higher order derivatives, k 2, the calculations are simple, see Section 7.2 below and other examples in [9, 10].

7. Effective Forces, the Parallelogram Law, and Autopilot Design

Equation (14) with initial data (15) and consistency condition (16) has a unique solution in the form

x(t) = (t, t₀ , x₀ , v₀ , p₂ ,…, p_k-1 ),

t [t₀ ,T ), t₀ 0, T , (17)

x(t₀) = (t₀ , .) = x₀ , dx(t₀ )/dt = d(t₀ ,.)/dt = v₀ .

The second derivative of this solution defines the function

f(t, t₀ , x₀ , v₀ , p₂ ,…, p_k-1 ) = d ² / dt ² = x''(t),

t [t₀ , T). (18)

With this function, we can write the equation of motion (8) in the usual form of the second Newton's law as x'' = f(t,…). For this reason, we call f(t,…) the effective force.

Consider (8) as a vector equation. At the initial moment t = t₀ , the vector F*(t₀ , .) of (8) defines the vector F₀ = F* (t₀ , x₀ , v₀ , p₂ ,…, p_k ) due to (15)-(16). If the solution (17) is known, then the vector

F*(t, .) = F*(t, , ',…, ^(k)) = x''(t) =

= f(t, t₀ , x₀,, v₀ , p₂ ,…, p_k-1 ), t [t₀ , T) (19)

is also specified and equal to the effective force f(t,…) for each t [t₀ , T).

Fields of effective forces

Imagine that equation (8) is integrated for all possible initial data in (15)-(16). Then we have all possible solutions (17) which create a field of effective forces f(t,…), see (18), (19), identical to the field F*(t, x, x'^-, x''^-, …, x^(k)-) in (8) with respect to its action on a moving body m(t) in (6)-(8). The field f(t,…) does not depend on higher order derivatives implying that over this field of effective forces the second Newton's law has the same form as described by Newton [1] and symbolically specified in (5), (6). This means that effective force (18), (19) embodies "the motive force impressed" mentioned by Newton in his Law II. The original feedback relation (8) represents a force in the sense of Newton only on curves of (17), that is, for such higher order derivatives of x(t) that correspond to parametric equations (17). Outside those curves, i.e., with unrelated x, x'^-, x''^-, …, x^(k)- considered as free or partially free parameters, equation (8) does not represent any mechanical motion at all.

This observation means that the inclusion of left higher order derivatives in the right-hand side of (8), i.e., application of controls with higher order derivatives (which are measured or computed derivatives, thus, automatically left derivatives), does not violate any of Newton's laws, if we consider the trajectories defined by (15)-(18). With higher order derivatives, relation (8), due to Lemma 5.1 and assumed solvability of (8) with respect to its higher order derivative, introduces a field of effective forces f(t,…) over which a body moves along the curves (17) as if acted upon by the genuine Newton forces. Therefore, the application of the parallelogram law (Corollary I in [1], also called Law IV of Newton) to the right-hand side of (8) with respect to the vector F*(.) is incorrect, as indicated in [4]; this is understandable since that right-hand side F*(.) is, in general for k >1, not a force in the sense of Newton, but a feedback liaison of higher order defining certain motion in space for which the vector F*(t, x, x'^-, …, x^(k)-) of (8) does not define an acceleration, but the vector f(t,…) = d ² / dt ² = x''(t) defines it.

Fields of effective forces exist also if equation (8) contains terms with natural time delays due to finite speed of information transmittal. Effective forces are recovered after the integration of equation (8) and act along its solutions obtained with consideration of time delays if they are known. If delays are bounded but not exactly known, then corresponding bands can be evaluated within which the real trajectories are located with effective forces acting along those trajectories. A method of integration in this general case is demonstrated in Section 10, Example, Case 3.

Verification of the parallelogram law for effective forces

Consider a motion in a plane x₁ 0 x₂ defined by differential equations (8) over a small interval t [0, ) with initial conditions x_i (0) = 0, x_i' ^-(0) = 0, i = 1, 2. Over this interval, the mass m(t) in (8) can be considered constant and the components F₁ , F₂ of F* in (8) with x_i' ^-(0) = 0 can be approximated as linear functions, yielding the system

mx₁'' = a₁ + u₁(t) = a₁ b₁ x₁''^- - c₁ x₂''^- = F₁ ,

t [0, ) (20)

mx₂'' = a₂ + u₂(t) = a₂ b₂ x₁''^- - c₂ x₂''^- = F₂ ,

t [0, ) (21)

where a_i , b_i , c_i are constants. This approximation is valid for any F*(.), m(t) continuous over a small interval t [0, ). Equating left and right derivatives in (20)-(21), see Lemma 5.1, we can write the system (20)-(21) in the form

(m + b₁) x₁'' + c₁ x₂'' = a₁ , t [0, ) (22)

b₂ x₁'' + (m + c₂) x₂'' = a₂ , t [0, ) . (23)

Setting t = 0 defines the values u_i(0) in (20)-(21) and the consistency parameters p_2i = x_i''(0), i = 1, 2, of (16) which can be determined from (22)-(23) assuming that its principal determinant is nonzero. Determinants are:

D = (m+b₁)(m+c₂) b₂ c₁ 0,

D₁ = a₁ (m+c₂) a₂ c₁ , D₂ = (m+b₁)a₂ b₂ a₁ ,

so we have x₁'' = D₁ /D, x₂'' = D₂ /D, and with zero initial data, the solutions are:

x₁(t) = t ²D₁ / 2D, x₂(t) = t ²D₂ / 2D,

t [0, ) , (24)

yielding a strait line trajectory in the plane x₁ 0 x₂ with the angle

tan = x₂''(t) / x₁''(t) = D₂ / D₁ = const,

if D₁ 0, or (25)

tan = x₁''(t) / x₂''(t) = D₁ / D₂ = const,

if D₂ 0 .

According to the second Newton's law, this line should be the line of "the motive force impressed". If we considered the right-hand sides F₁, F₂ of (20)-(21) as components of the motive force F*(t) = (F₁, F₂) before integration, then F*(t) would be undefined for t 0, since accelerations in the left-hand sides of (20)-(21) are yet unknown. If we considered right-hand sides of the transformed system (22)-(23) as components of the force, then its direction would be tan = a₂ /a₁ tan , or tan = a₁ /a₂ tan , so it is not "the motive force impressed" in the sense of the second Newton's law. However, if we consider F₁, F₂ in (20)-(21) as components of the effective force f(t, .), after the integration of equations (20)-(21), then we have at t = 0, due to (24) used in (20)-(21):

F₁ = a₁ b₁ x₁''- c₁ x₂'' = a₁ b₁ D₁ /D -

- c₁ D₂ /D = mx₁'' = mD₁ /D ,

F₂ = a₂ b₂ x₁''- c₂ x₂'' = a₂ b₂ D₁ /D -

- c₂ D₂ /D = mx₂'' = mD₂ /D ,

yielding "the direction of the right line in which that force is impressed" (Law II):

tan = F₂ /F₁ = (a₂ b₂ D₁ /D - c₂ D₂ /D) /

/ (a₁ b₁ D₁ /D - c₁ D₂ /D) = D₂ /D₁ = tan ,

identical to the line in (25) of the "change of motion" (Law II) according to (24), in full compliance with the second Newton's law of motion. This demonstrates that effective forces obey the parallelogram law. Clearly, the same is valid under any initial conditions since they are eliminated by derivation of variables in (24). It also shows that consistency condition (16) is essential since otherwise u(0) would be undefined and the motion in (20)-(21) could not start.

Application to the landing of hovercrafts and airplanes

Consider vertical landing of a hovercraft in still air. For this case, the horizontal coordinate x₂(t) 0, so the coordinate system x₁0x₂ reduces to the vertical axis 0x₁ directed downward, corresponding to equation (20) with x₂(t) 0, and yielding

m(t)x'' = a₁ + u₁(t) = g - k x' + u₁(t) ,

x(0) = h, x'(0) = v₀ , t 0 , (20-1)

where g is the acceleration of gravity, and k x' is the resistance of air proportional to the velocity of motion. In the velocity coordinate v = x'(t), the equation (20-1) reduces to the first order equation

m(t) v' = g - k v + u(t), k > 0,

v(0) = x'(0) = v₀ , t 0 , (20-2)

where the sub-index in u₁(t) of (20-1) is dropped.

The vertical acceleration x'' ^-(t) = v' ^-(t) can be readily measured during flight and landing, so that for soft landing and for suspended hovercraft we can apply the control

u(t) = g + x'' ^- g + v' ^- , t 0 ,(20-3)

yielding, instead of (20-2), the equation

m(t) v' = k v + v' ^- , v(0) = v₀ .(20-4)

For smooth landing, we have v' = v' ^- by Lemma 5.1, so that (20-4) reduces to the simple equation, with its solution for m = const , = const , as follows

[m(t) ] v' / v = k , v(0) = v₀ , or

v(t) = v₀ exp [ k / ( m)], t 0 , (20-5)

which should be applied for landing of hovercrafts, given resistance coefficient k, measured acceleration v' ^- = x'' ^-(t) and computed velocity v(t) which allows to determine the correct reactive braking force in (20-5) for the soft landing at v* = 0 for some t* > 0.

Consider the landing of an airplane. In this case, we have to use both equations (20)-(21) with the landing conditions: v₁ (t*) = x₁'(t*) = 0, x₁''(t*) = 0 for the vertical axis 0x₁ in (20-1) to assure soft landing for undefined values of the horizontal parameters v₂(t) = x₂'(t*) and x₂''(t*) at the moment t* of touchdown at x₁(t*) [ l₁ , l₂ ) along the axis 0x₂ . Equations (20)-(21) for horizontal landing without vertical braking jet are

m(t)x₁'' = a₁ + u₁(t) = g k₁ x₁' + u₁ (t) ,

t [0, t* ), (20-5)

m(t)x₂'' = a₂ + u₂(t) = k₂(t)x₂' ^- + u₂ (t) ,

t [0, t* ), x₂(t*) [ l₁ , l₂ ) , (20-6)

where k₁ x₁' is the vertical resistance of the air for the airplane with u₁(t) < 0 added to it by the plane ailerons at landing, and k₂ x₂' ^- is the horizontal resistance of the air with u₂(t) < 0 added to it by the plane controls. Here the values x₂' ^-(t) are measured by the outboard Pitot tubes, if so equipped and precise enough, and x₂'' ^-(t) is measured by a special device onboard and fed into the autopilot system of the plane, cf. [15, Section 8]. If there are essential time delays [16] in transmission of signals to ailerons, it should be taken into account by the autopilot system of the airplane.

8. Space Shuttle Example

The ascending vertical motion of a rocket with the axis 0x directed straight up was considered by Mestschersky in 1897 and described by the equation [7, p. 114, Eq. (1)]:

mx''(t) = mg + (p* - p_x) - m'(t) w*

- R(x'(t)), (26)

Here m(t) is the variable mass of the rocket, x(t) is its vertical coordinate (height) and g = 9.8 m/s² is gravitational acceleration; is the area of the funnel opening that ejects burnt gases, p* is the pressure of ejected gases, p_x is the air pressure at the height x(t); m'(t) < 0 is the rate of change of mass of the rocket due to combustion, and w* = v - w is "geometric difference between velocities of separating mass and the body directed straight down" [7, pp.113-114] where w is absolute velocity of "separating mass"(gases) and v(t) = x'(t) is the absolute velocity of the rocket; R(x'(t)) is resistance of the air.

A solution of equation (26) is given in [7, pp. 114-115], assuming resistance of the air R(x') = R(v) = a + bv, uniform combustion m = m₀(1 t) m* > 0, > 0 over some time 0 t T = (1- m*/m₀)/ where m₀ , m* are initial and final mass of a rocket, and the constancy of parameters: w* = const and p = (p* p_x) = const .At higher velocities of a rocket, the air resistance is quadratic, R(x') = R(v) = a v². With a finite volume of fuel in a roket, equation (26) and also (27)-(30) below are valid over finite periods of time when combustion takes place, and over periods of free flight one has to set m'(t) 0, p = 0 in (27) returning to Newton's equation (1) or its generalization (8) with mass m = const, different and differently distributed over different periods of free flight. Many other examples of motion with variable masses are presented in [7].

To consider the acceleration assisted control, see Section 5 above, we adopt, for simplicity, the assumptions of Mestschersky, except for the resistance of air which we take in the form R(x') = R(v) = a v². With the notation x'(t) = v(t), this renders a differential equation of the first order (Riccati equation) for v(t):

m(t) v'(t) = m(t)g + p - m'(t) w* a v²,

0 t T . (27)

Clearly, the same equation governs the launch of a space shuttle, but with a difference.

A rocket is launched like a bullet, but a shuttle ascends slowly which can be seen on T.V. showing a shuttle launch. This is due not only to a greater weight of a shuttle, but mainly to the presence of humans in it. Indeed, the health of a human being requires certain gravitational conditions with total acceleration v'(t) + g in the range [kg , ng], where the numbers 0 k 1, 1 n n* < 9 depend on personal health and flight duration, and are determined by medical considerations since v'(t)+ g > n*g or v'(t)+ g < kg for a long time may cause sickness and incapacity of a person in the shuttle.

This means that m'(t), if used to control dangerous gravitational load on people in the shuttle, must depend on acceleration v'(t). Suppose that m'(t) = b+ qv', where b, q are positive constants. With small q, we have m'(t) < 0, thus dm < 0, due to combustion, so that qv' > 0 acts as regulator and moderates the thrust in order to prevent too high accelerations. Substituting the feedback m'(t) = b + qv' into (27) and assembling the terms that do not depend on v(t), we obtain the equation with control parameter q :

m(t) v'(t) = h(t) q w* v'(t) a v²(t),

h(t) = m(t)g+ p+ bw* . (28)

However, the feedback qw*v'(t) in (28) depends on measured acceleration which carries a small time delay > 0, yielding the equation

m(t) v'(t) = h(t) q w* v'(t-) a v²(t) . (29)

This is not an ordinary delay differential equation, DDE, since delay affects the highest order derivative, and there is no theory yet for such equations. An attempt to formally expand v'(t-) into Taylor series taking the first two terms in it to obtain a normal ODE, yields an equation with a small parameter at the highest derivative, and the method fails. Indeed, we have v'(t-) = v'(t) v''(t) + v'''(t) ²…, rapidly converging for small if all derivatives are uniformly bounded. Putting the first two terms into (29) for v'(t-), we get the equation

q w* v''(t) = m(t) v'(t) h(t) +

+ q w* v'(t) + a v²(t), (30)

which may be extremely unstable. Indeed, assuming that the length of information transmittal is 1 mm and its speed equals the speed of light, we have = 0.1 cm / 310¹⁰ cm/sec 0.310^-11 sec, so that the rate of change in acceleration v''(t) = dv'/dt 10¹¹[…], the bracket standing for the right-hand side of (30) divided by qw*, which, if nonzero, would cause the acceleration to explode. To illustrate this effect, consider an example obtained from (29) by setting m(t) 1, h(t) 2, qw* = 1, a = 0, yielding an equation similar to (29) but much simpler:

v'(t) = 2 v'(t-), v(0) = v₀ (31)

If = 0, then v'(t) = 1 and the solution is v(t) = v₀ + t. If 0 small, then, using the first two terms of the Taylor series above, we obtain the equation v' = 2 v'+ v'', that is, v'' 2 v'+ 2 = 0. For this equation of the second order, we have to add one more initial condition, and to comply with (31) for = 0, t = 0, we should set v'(0) = 1. Characteristic equation is r ² - 2r + 2 = 0, with roots r _1,2 = [1 (1 2) ^0.5] / . For small 10 ^-11, we have (1 2) ^0.5 =1 + ² …, yielding r₁ = (2-) / 2/, r₂ =( ²)/ =1 1, and the general solution is v(t) = a exp(2t/) + be^t. Using initial conditions v(0) = a + b = v₀ , v'(0) = 2a/ + b = 1, we get a = (1-v₀) /(2-) (1-v₀)/2, b =(2v₀ -) /(2-) v₀ so that v(t) 0.5 (1-v₀)exp(2t/)+ v₀ e^t , and very fast for 10 ^-11 if v₀ 1. Hence, we have to get rid of delay in (29). One way is to set = 0, which renders one and the same equation in (28) to (30) with acceleration assisted control whose action provides smoothing effect on a flight with seemingly increased mass of the shuttle corresponding to actually decreased acceleration, with lesser gravitational load on the people in the shuttle. There is another approach to account for time delays, see Section 10, the example of a physical pendulum, Case 3.

9. Generalized Equations in Independent Coordinates

Let us consider differential systems of Newtonian mechanics in relation to variable masses and higher order derivatives in the right-hand sides. First, we reproduce some well known concepts of analytical mechanics [2-4] with constant masses and without higher order derivatives, thus, considering only the geometry of Newtonian motion as presented in the classical theory.

Newtonian equations of motion for a constrained mechanical system of N point-wise masses are written in the form:

m_i x_i'' =F_i(t, x, v)+R_i(t, x, v), i =1,...,N;

x'' = v' =d ²x/dt², v = dx/dt, t 0, x R^3N, (32)

where m_i are constant masses, F_i are active forces and R _i reactions of constraints acting on the masses m_i .The variables x_i ,v_i , x_i'' are state, velocity and acceleration vectors in Cartesian (rectangular) coordinate system (phase space). Since the mass m_i can be subject to forces acting from other masses, the 3N-vector x composed of N 3D-vectors x_i (that denote coordinates of masses m_i) is included in the forces F_i and R_i together with velocity v. This is a short form to avoid a double index writing m_i x_i'' =F_i(t, x_k , v_k) + R_i(t, x_k , v_k), cf. (36) below, where x_k means {x₁ ,…, x_N} with index k not included in subsequent summations. If equations (32) are divided by masses and reduced to the normal form by writing dv_i /dt instead of x_i'' with vector equations dx_i /dt = v_i added, we obtain 6N dimensional vector equation (32) of the first order whereby F_i can be regarded as controls (or containing controls). Constraints are assumed ideal which means that the total work of constraint reactions is zero, R_ix_i = 0, where x_i are any possible, i.e., allowed by the constraints (virtual, t fixed) displacements. Using this equation to exclude the unknown reactions of constraints yields the general equation of motion (principle of D'Alembert):

(33)

for the N - mass system (32). At rest x_i'' ? 0, and in this case equation (33) renders the criterion (necessary and sufficient condition) for the equilibrium of active forces F_i (principle of virtual displacements, J. Bernoulli, 1717).

At the time of Newton (1687) [1], and for more than two centuries thereafter, only constant masses m_i were considered, with “the motive force impressed” F_i(.) being known functions of time, coordinates and velocities. It means that trajectories of motion were considered as fixed geometric curves x(t) known for all t[0, T). Time t was perceived as absolute, and reactive forces were irrelevant with the consideration of constant masses. Since the results of Buquoy (1815) [6] were not properly recognized, thus, unknown for more than a century, and reactive forces were ignored even after the publications of Mestschersky [7] and Levi-Civita [8], it is not surprising that equations (32), (33) are still written in their maiden form of eighteenth century, without reactive forces nor control forces depending on left higher order derivatives.

With the advent of jet propulsion, the forces in (32)-(33) should be replaced by F*(.) of (7), with variable masses m_i(t), which would, of course, alter the classical formulae (32)-(33). With the use of acceleration assisted control, there is no problem, if left and right time derivatives are considered identical and equations of motion are resolved for actual accelerations as in (32), requiring nonzero Jacobian with respect to accelerations. For controls with left higher order derivatives, equations (32)-(33) are propelled by effective forces generated by expressions F_i(.) in (32) with ideal reactions R_i(.) excluded by (33). It means that F_i(.) can still be formally written in (33) with higher order derivatives of v(t) included therein for subsequent integration which would render the trajectories of motion (17), and identify the actual effective forces (18), (19) in the system which are to be recognized as "the motive force impressed" mentioned by Newton in his Law II [1].

Generalization of the Lagrange equations

The general equation of motion (33) excludes reaction forces of ideal constraints, but not the constraints themselves which are still restricting coordinates and velocities of the motion, no matter if forces of reaction are excluded. Let us consider the exclusion of constraints altogether, i.e., elimination of kineto-statical relations [2] of the problem.

Analytically, constraints are expressed by several independent equations:

l_k (t, x_i , v_i ) = 0 , k = 1,…, s . (34)

If equations (34) do not contain velocities v_i or can be integrated not to contain them, the constraints are called geometric, and the system (32)-(34) is called holonomic. In this case, from equations l_k (t, x_i )=0 one can express certain s coordinates as functions of 3N s other coordinates and time t and consider those 3N - s coordinates as independent variables that define the state of the system at time t. However, it is not binding to take Cartesian coordinates as independent variables. It may be convenient to express all 3N Cartesian coordinates as functions of n = 3N - s independent parameters q₁ ,…, q_n and time t which define the so called configuration space. Substituting thus obtained x_i = x_i (t, q₁ ,…, q_n ) into (32) and using (33), a new system of second order equations with respect to independent parameters q₁,…,q_n and time t is derived. Those parameters (called generalized coordinates) do not have transparent meaning of Cartesian coordinates, but they present a differential system without constraints, of minimal order with respect to independent variables q_i(t) which define all Cartesian (rectangular) coordinates and velocities, thus, the state of original system (32) subject to constraints (34). Using (33), and noting that elementary work of active forces A =_j=1^NF_jx_j =_i=1ⁿQ_iq_i, the minimal order system of the Lagrange equations of the second kind is obtained in the form: