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

, Q_i = Q_i (t, q_k , q_k'), T =

=Ѕa_ik q_i'q_k'+ a_i q_i'+ a₀ , i = 1,…,n. (35)

Here at left stand generalized forces of inertia expressed through kinetic energy T of the system, Q_i are generalized active forces, and sums are taken from 1 to n. It is clear that Lagrange's equations at left in (35) represent the second Newton's law of motion (32) in generalized coordinates {q_k}, with constant masses and with reactions R_i excluded by (33). If constraints are stationary, i.e., (34) does not depend on t, then a₀ , a_i are zero in (35). Substituting the expression of T in (35) into the Lagrange equations yields

(36)

where stands for terms not containing second derivatives, and the second equation is the unique solution of the first one for q_i", since determinant det (a_ik)ⁿ_i,k=1 0 . This is known as the explicit form of Lagrange's equations [2, pp. 39-40] which define the motion of the system determined by initial values q_i(0), q_i'(0).

The transformation to independent generalized coordinates in configuration space that excludes geometric constraints does not depend on the presence of higher order derivatives in (33). For holonomic systems with variable masses, the generalized coordinates q_i(t) can be introduced to eliminate the constraints (34) even if forces in (33) depend on higher order derivatives. It is interesting and important that the equations thus obtained will also be in the form of Lagrange's equations but with quite different entries, and the order of those equations will be minimal, by construction, with generalized forces depending on left higher order derivatives of generalized coordinates and with kinetic energy corresponding to variable masses.

Indeed, we have according to (5), (8) and (33), after the transformation:

(37)

where v_i(.) = x_i'(.), x^{(r) -} = [x(t, q₁ ,…, q_n )]^{(r) -}, r = 0,…, k, but x_i (t, q₁ ,…, q_n ) are not arbitrary, due to (34). However, all x_i (t, q₁ ,…, q_n ) , x^{(r) -} = [x(t, q₁ ,…, q_n )]^{(r) -}, and x_i (t, q₁ ,…, q_n ) can be expressed through q_j , q_j',…, q_j^(k)- and q_j (j = 1,…, n) yielding

here we retain the same notation for F_i*(q,...) derived from F_i*(x,...) of (37) after the transformation. Changing the order of summation, we obtain

(38)

Since q_j in (38) are arbitrary virtual displacements, we have

(39)

Let us call "generalized forces" the expressions:

, (40)

so that (39) can be rewritten in the form

(41)

The term [m_i v_i]'( x_i / q_j) under summation sign in (41) can be transformed as follows:

[m_i v_i]'( x_i / q_j)=d[m_i v_i( x_i / q_j)] / dt

m_i v_i d( x_i / q_j) / dt . (42)

Since

v_i = dx_i /dt = x_i / t + ( x_i / q_j) q_j',

so v_i / q_j' = x_i / q_j . Also, we have

v_i / q_k = ²x_i / t q_k +

+ ( ²x_i / q_j q_k) q_j' = d( x_i / q_k) / dt . (43)

Relations (42)-(43) imply

[m_i v_i]'( x_i / q_j) = d[m_i v_i( x_i / q_j)] / dt

m_i v_i ( v_i / q_j) = d[ (0.5 m_i v_i²) / q_j'] /

/ dt (0.5 m_i v_i²) / q_j . (44)

Summing up the expression in (44) to obtain the left-hand side of (41), we get

T* = 0.5 m_i(t) v_i²(t, q₁,…,q_n), j = 1,…,n . (45)

Relations at left in (45) have the form of Lagrange's equations (35), with a difference:

1) "generalized forces" Q_j* may depend on higher order derivatives of generalized coordinates, q_j^(k) , thus, explicit form (36) of Lagrange's equations is not preserved;

2) the function T* = 0.5 m_i(t) v_i²(.) of (45) corresponds to variable masses with the same formula as in the case of constant masses, but T* T(.) of (35);

3) the function T* which resembles the expression of kinetic energy may not represent the real kinetic energy of the system. To determine the real kinetic energy of the system, one has to equate left and right derivatives in (40), (45), solve the higher order system with additional initial conditions to obtain the solution in the form of (17), compute the effective velocities v_i*, not those v_i that appear in (37) to (45), and compute the real (effective) kinetic energy of the system.

Preservation of the form of Lagrange's equations for holonomic systems with variable masses and forces depending on higher order derivatives in the right-hand sides presents a useful method allowing one to formally construct the equations of motion excluding geometric constraints, solve the resulting differential system of minimal order with respect to independent generalized coordinates in the configuration space, then return to natural rectangular coordinates, velocities and accelerations, and evaluate by (18) the effective forces in the system that represent the Newtonian forces [1] in this case. The reader can check that with m = const and higher order derivatives absent from (37), equations (45) coincide with Lagrange's equations (35)-(36).

Generalization of the Hamilton equations

If in the Lagrange equations (35) generalized forces Q_i do not depend on generalized velocities, Q_i = Q_i(t, q₁ ,…, q_n), then there exists a potential function P(t, q₁ ,…, q_n) such that Q_i = P/ q_i . Introducing kinetic potential (the Lagrange function) L = T P, the system (35) at left, with constant masses, can be written in the form:

, i = 1,…, n . (46)

If active forces in (37) depend on left higher order derivatives, so do also the generalized forces Q_j*(.) in (40), (41), (45). In this case, consider the “generalized” potential function V(t, q, q'^-,q''^-, …, q^{(k) -}), if it exists, such that Q_i*(.) of (40) can be expressed by the formulae, cf. [2, p. 44]: Q_i* = d( V/ q_i') / dt V/ q_i ,

i = 1, …, n . (47)

With Q_i* from (47), equations (45) can be written in the form (46) if L is substituted by the function L* = T* V with T* from (45). These new equations are not the second order equations, but a higher order system written in the form (46). This form is preserved in mechanical systems with variable masses and higher order derivatives in active forces, if there exists a potential function V(.) with which Q_i* can be expressed in the form (47). An example of such potential is furnished by V = (1+ r' ²/c²) / r which presents the generalized potential in the sense of (47) for Weber's electro-dynamic force of attraction F* cited in Remark 5.1, see [2, p.45].

Equations (46) suggest new coordinates proposed by Hamilton. Denote L / q_i' = p_i (generalized impulses), so that by (46) dp_i /dt = p_i' = L / q_i , and consider new variables p₁ ,…, p_n which together with old variables q₁ ,…, q_n constitute the set of 2n variables of Hamilton. Since ²L / q_i' q_k' = det (a_ik)ⁿ_i,k=1 0, see expression of T in (35), so Jacobian of L / q_i' is nonzero, and equations L / q_i' = p_i can be resolved for q_i' yielding q_i'= _i (t, q_k , p_k), which together with p_i'= L / q_i = _i (t, q_k , p_k) present Hamiltonian system of 2n equations of the first order equivalent to Lagrangian system of n equations (46) of the second order, cf. (32), (36). If the quantity p_i q_i' - L is expressed as function of {t, q_i , p_i} and denoted by H, then equations of motion (46) in the Lagrangian form with constant masses can be represented also in Hamiltonian or canonical form as follows [2, pp. 263-264]:

H = { p_i q_i'- L} = (q_i'p_i p_i'q_i), thus, q_i'= H / p_i , p_i'= H / q_i . (48)

If instead of L in (46), (48), the function L* = T* V(t, q, q'^-,q''^-, …, q^{(k) -}), with T* from (45) is considered, denote L* / q_i'= p_i*(.), so that by (46) with L* instead of L we have dp_i*(.) / dt = p_i*' = L* / q_i . If the Hessian ²L* / q_i' q_k' 0, then Jacobian L* / q_i' is nonzero, and equations L* / q_i' = p_i*(.) can be resolved for q_i' yielding q_i'= _i (t, q_k , p_k*(.)), which together with p_i*'= L* / q_i = _i (t, q_k , p_k*(.)) present generalized Hamiltonian-like system of 2n equations of higher order equivalent to generalized system of n equations (46) with L* substituted for L. If the quantity p_i*(.) q_i' - L*(.) is expressed as function of {t, q_i , p_i*(.)} and denoted by H*(.), then we see that canonical form (48) is preserved, with the understanding that coordinates p_i*(.) contain w_i m_i'(t) and higher order derivatives on which p_i*(.) and Q_i* of (47) depend, so that new equations

H* = { p_i *(.)q_i'- L*(.)} = (q_i'p_i*

p_i*'q_i), thus,

q_i'= H* / p_i*, p_i*' = H* / q_i , (49)

present a higher order system from which the effective forces (18) actually acting in the system can be recovered after integration and passage to Cartesian coordinates.

10. The Method of Integration: Example

Consider a physical pendulum consisting of a rod OC of the length l suspended in a hinge at O with a heavy disc of mass M fixed at its center to the end C of the rod. With such pendulums are equipped free standing clocks that can be seen in furniture or antiquity stores. Friction at the hinge is neutralized by a spring or a battery, and a mass of the rod can be ignored. The moments of inertia of the disc are

I_C = _o^r r²dm = _o^r 2 r³dr = 0.5Mr ²,

I_O = I_C + M l ² = 0.5M(r ² + 2l ²).

The pendulum oscillates in a plane xOy with the axis Ox directed straight down and axis Oy directed to the right. It is required to derive equations of motion.

Case 1. Classical solution. The system has one degree of freedom, and it is convenient to take the angle between Ox and the rod as the generalized coordinate q = . The coordinates of the center of mass are: x_c = l cos , y_c = l sin . The acting force of gravity Mg = (X, 0) is directed straight down, so that generalized force Q =X x_c / = Mg l sin . Kinetic energy is T = 0.5 I_O ' ², so that T /' = I_O ', T / = 0 , yielding the Lagrange equations (35) and (36) for the case as follows:

I_O '' = Q = Mg l sin ,

''+ 2gl sin /(r²+2l²) = 0 ,

(0) =₀ , '(0) = 0. (50)

The equivalent length of the mathematical pendulum with the same period is l*= r²/2l+ l.

Potential function for Q can be taken in the form P = Mgl cos , so that with the Lagrange function (kinetic potential) L = T - P = 0.5 I_O q' ² + Mgl cos q, q , the Lagrange equation in (50) at left can be represented in the form (46). If we denote p = L / q' ( I_O q'), then, due to (46), we have p' = dp/dt = L / q = Mgl sin q, and can define the Hamiltonian H(t, q, p) = pq' - L = I_O q' ² L = p²/2I_O Mgl cos q, yielding canonical equations of the motion, cf. (49):

q' = H / p = p / I_O , p' = H / q =

= Mg l sin q , q , p I_O ', (51)

which are equivalent to (50) since '' p'/I_O = = Mg l sin /I_O = 2g l sin /(r²+2l²).

This classical solution which excludes variable reaction in the hinge can be found in most textbooks on theoretical mechanics.

Case 2. Forces with left higher order derivatives. Consider the same pendulum submerged into aquarium with water. Then the pendulum will be affected by additional force of water resistance F = ' ^-'' ^- (, = const > 0) where ' ^- is Newtonian fluid friction, and '' ^- is the Kirchhoff-Thomson adjoint fluid acceleration resistance [11, 12], see Remark 5.1. Now we have a different generalized force Q* = Mg l sin ' ^- '' ^- with the same kinetic energy of the pendulum. This yields a different equation for the same generalized coordinate q = , see Lemma 5.1,

I_O '' = Q* = Mg l sin ' ^- '' ^-, or

(I_O+) '' + ' ^-+ Mg l sin = 0 . (52)

If =0, then equation (52) can be converted into canonical form with the introduction of generalized potential function

V = Mg l cos + '' ,

such that Q* = V / + d( V/')/dt. It is left to the reader to obtain generalized Hamiltonian equations through the introduction of the L* function with this generalized potential V. Setting also = 0, one would return to the classical canonical equations (51). It is interesting and important that acceleration of a moving body can enter Lagrange's and Hamilton's equations also through generalized forces, not only through kinetic energy which is stipulated by the classical representation (32) of the second Newton's law of motion. The preservation of the form of Lagrange's and Hamilton's equations for generalized systems with left higher order derivatives in the right-hand sides is quite surprising and opens a way for the use of those equations in the large area of soft control with the left higher order derivatives, excluding ideal constraints whose reactions, if needed, can be found afterwards.

Case 3. Forces with left and delayed higher order derivatives. Consider the same pendulum in the air affected by a strong wind from a ventilator in direction of the negative axis Oy (to the left). With a laser, small computer and connecting wires, the ventilator can be controlled to supply a flow of air upon the disc from right to left generating a force Y = (a + b' ^- + h'' ^-+ k'''^-) < 0 depending on the left higher order derivatives of the motion. The generalized force is

Q* = X x_c / + Y y_c / =

= Mg l sin (a + b' ^-+ h'' ^-+

+ k''' ^-) l cos , (53)

where a, b, h, k are some constants. Since ' ^-(t) is measured and '' ^-, ''' ^- are also measured or computed from the measured ' ^-(t), so all three derivatives in (53) are necessarily left and delayed due to a finite speed of information transmittal [16, p.1344]. In this situation, the time delays may play a major role. To fix the ideas, let us consider, for simplicity, that (t) is small, and also b = 0, h > 0, k > 0, a > h'' ^- + k''' ^-. Then in (53) we can set cos = 1, and consider

Q* = al Mg l sin (t) -

- l[ h'' ^-(t₁) + k''' ^-(t ₂)] . (54)

Note that (t) is without delay since it is not a measured and transmitted quantity. With this generalized force and the same kinetic energy, we have the equation, cf. (50), (52):

I_O '' = Q* = al Mg l sin (t) -

- l[ h'' ^-( t ₁) + k''' ^-(t ₂)], t 0 . (55)

In (54), (55), we assume that all three moments of time are within the time interval of the actual motion; out of this interval, the entries are equal zero. Physically, it is clear that always ₁ > 0, ₂ > 0, the question is whether we can ignore both or one of them. It is also clear that oscillations will be distorted and not symmetric with respect to the axis Ox.

Recall [16] that over the length of 100 cm, the information transmittal with the speed of light takes the time 10^-8 sec, whereas the information transmittal with the speed v* 10^-2 cm/s of the ordered motion of electrons over the same length of 100 cm would take * 10 ⁴ sec = 167 min =2.8 hour, which makes quite a difference. For information transmittal over 1 cm, the corresponding delays are 10^-10 sec and 100 sec. For different delays ₁ , ₂ within [10^-10, 1] sec, different dynamics can be obtained for the same system in (55). Equating left and right derivatives, we consider the following cases.

3.1. If ₂ 10 ^-8 sec, small, and ₁ > ₂ , then differential equation (55) is changing its order and right-hand sides over different intervals, and when it is of the third order, initial conditions in (50) are insufficient to define its unique solution. At t = 0, derivatives at right in (55) are not yet in action, so over [0, ₂) we have in (55) the equation as in (50) with term al and same initial conditions, yielding the values (₂) ₀ , ' ^-(₂) 0, '' ^-(₂) (al + Mgl sin ₀ )/I_O . At t = ₂, this value '' ^-(₂) presents initial condition for equation (55) where the second derivative at right is not yet in action. This assures the continuity of the motion over [0, ₁) but with dynamics of the third order over [₂ , ₁) since the third derivative in (55) comes into play and will overtake the motion for small ₂ 10^-8. At the moment t* = ₁, the term h'' ^-(t ₁) at right in (55) comes into play, so we have to replace the value '' ^-(₂) by the new initial condition at t = ₁ according to the equation I_O''(₁) = al Mgl sin(₁) - l[h'' ^-(0)+ k''' ^-(₁₂)], see (55), which is the consistency condition (16), this yielding

''(₁) = (al + Mgl sin(₁)) /I_O - l[ h(al +

+ Mgl sin₀) /I_O + k''' ^-(₁₂)]/I_O ,

where ''' ^-(₁₂) is known from the preceding segment [₂ , ₁) of the motion with h'' ^-(t-₁) in (55) not yet in action. Now, for t ₁ the motion is defined by the third order differential equation, and with the approximation ₂ 0, this equation can be written as ordinary DDE: lk'''(t) = I_O ''(t) al Mg l sin (t) - l h''(t-₁), t ₁ with (₁), '(₁) defined as end-point values in the previous segment of (t) over [0, ₁], and ''(₁) given by the consistency condition.

3.2. If ₁ 10^-8 sec, small, but ₂ is relatively large, then in (55) we have, in fact, the second order differential equation with discontinuity in the right-hand side. Indeed, until after t* > ₂ the third derivative at right of (55) is not in action, thus, setting ₁ 0, we get from (55) the equation (I_O + lh)''(t) = al Mgl (t), different from the equations in (50), due to seemingly heavier disc and additional term al, but with the same initial conditions. This equation exists until t* = ₂ at which moment the third derivative in (55) comes into play, changing the right-hand side for t > ₂ as follows:

I_O '' = al Mg l sin (t) - l[ h''(t ₁) +

+ k'''(t ₂)], t > ₂ . (56)

This is the same equation as (55) with all right derivatives. However, the third derivative at right does not project the motion as it did in Case 3.1, due to a greater delay ₂ > ₁ . It adds an additional force f(t) = lk''' ^-(t₂) depending on the rate of change of the actually realized values of past acceleration ''(t₂) for t > ₂ assuring softer rate of change in acceleration which is good for a vehicle and for the people in the vehicle, if we consider in place of the pendulum a swing with people at entertainment centers.

N.B. In the theory of DDEs, the functions with delays in the right-hand sides must be defined prior to the start of the motion. For example, to define a unique solution in (55) for t 0, cf. (50), the theory requires to define Q*(.) over the prior segment [- , 0] where = max(₁ , ₂). With time delays due to information transmittal [16], delayed terms in forces Q*(.) cannot be "defined" on prior intervals because they physically do not exist in those time intervals. Setting them at zero may bring contradictions. Indeed, if ₁ < ₂ and we set '' ^- = ''' ^- 0 over [0, ₁ ) with (0) = ₀ > 0, then by continuity (Lemma 5.1) we have also '' 0 at left in (55), so that at t = 0 we get in (55): 0 = al Mgl sin₀ <0, an absurdity. For these reasons, we do not mention prior segments of definition for delayed terms which can be dealt with as they come into action.

3.3. The absence of time delays in mathematical descriptions of motion may lead to substantial errors, especially for small particles at high velocities. In deterministic consideration, this can be seen on example of a linear harmonic oscillator by comparison of the magnitude of its period with the order of natural time delays. Suppose that gravitation acts on the electron in the same way as on a metal pendulum and that it is added to other forces according to the parallelogram rule. Then we can imagine that small oscillations are superimposed on the rotational motion of an electron around the nucleus which would distort its uniform rotation. In the oscillatory part of the motion along the bottom arc 2₀ , we can consider the electron as a point-wise mass, so that the second equation in (50) with r = 0, for small ₀ takes the form '' + g /l = 0, irrespective of the mass of the electron, and the solution is = ₀ sin t, where ² = g / l, with the period T = 2(l/g)^0.5. If we take l = a₀ = 0.529 10^-8 cm which is the radius of the first (innermost) Bohr orbit in the hydrogen atom (Bohr radius), then we have T = 1.460 10 ^-5 sec. This is just at the middle of the time uncertainty segment for delays ₁ , ₂ within [10^-10, 1] sec considered above, so that model (50) is inapplicable to the study of harmonic oscillations of the electron in the hydrogen atom. In deterministic studies, time delays should be taken into account, when possible, especially if computations are involved in experiments, or particles move in a field of controlled forces, in which cases time delays due to information transmittal really take place.

Remark 10.1. It is clear that differential equations with the left time derivatives in the right hand sides present the limiting case of functional differential equations (FDEs) with delayed arguments [17-21] if delays tend to zero. However, the limit as 0 cannot be attained if those delayed derivatives are the Newton - Leibniz right time derivatives. In this case, one can consider the connection by reverting to the left time derivatives in the limit, or replacing right derivatives by the left ones when approaching the limit.

Conclusions

This paper presents the causal approach to theoretical mechanics and engineering, different from the current textbook considerations based exclusively on the classical right time derivatives which ignore the natural orientation of increasing time t thus do not physically exist, although are actually used in the mathematical constructs to investigate prospective trajectories as approximations to real motions over some intervals of time.

In contrast to the usual Newton - Leibniz right time derivatives at the left hand side which project the motion into the future, the left and possibly delayed time derivatives are considered in the right hand sides of the equations of motions and processes that take into account the external influences and controls depending on the measured parameters and the immediately preceding exterior actions which are then transmitted into the power train of the motion. The results allow us to introduce the causal corrections into the current representations of the general laws of dynamics, into the Lagrange and Hamilton equations in independent coordinates, and to improve the design of autopilot systems in aviation which is currently based on the use of outboard Pitot tubes that, on occurrence of random wind gusts, give distorted evaluation of the average relative velocity of the airplane, and may get frozen and fail altogether in bad weather which has already happened in the Air France flight 447, Rio de Janeiro - Paris, on May 31st, 2009.

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