Gauss Principle of Least Compulsion for Relative Motion Dynamics and Differential Equations of Motion

: This paper focuses on Gauss principle of least compulsion for relative motion dynamics and derives differential equations of motion from it. Firstly, starting from the dynamic equation of the relative motion of particles, we give the Gauss principle of relative mo‐ tion dynamics. By constructing a compulsion function of relative motion, we prove that at any instant, its real motion minimizes the com‐ pulsion function under Gaussian variation, compared with the possible motions with the same configuration and velocity but different ac‐ celerations. Secondly, the formula of acceleration energy and the formula of compulsion function for relative motion are derived because the carried body is rigid and moving in a plane. Thirdly, the Gauss principle we obtained is expressed as Appell, Lagrange, and Nielsen forms in generalized coordinates. Utilizing Gauss principle, the dynamical equations of relative motion are established. Finally, two rela‐ tive motion examples also verify the results 􀆳 correctness.


Introduction
The Gauss principle is a differential variational principle proposed by Gauss in 1829, a general analytical mechanics principle [1] .Chen pointed out [2] that taking the Gauss principle as the fundamental principle in terms of mechanics concepts seems most appropriate.According to Mei [3] , the Gauss principle can be used as a basis for analytical dynamics.Udwadia and Kalaba [4] took the Gauss principle as a starting point to derive the basic equations of analytical mechanics by using matrix algebra operations and recommended its application to holonomic and non-holonomic mechanics, which reveals the broad applicability of Gauss principle in describing the motion of constrained mechanical systems.Of all differential variational principles, only the Gauss principle is a stationary principle, which shows that the variation of compulsion function in the sense of Gauss is equal to zero [1] .For a system of particles, applying Gauss minimum compulsion principle, its motion equation can be obtained directly by calculating the extremum of the compulsion function [5,6] .Because of this, the Gauss principle of least compulsion is widely used in dynamics modeling and in finding approximate solutions.For example, robot dynamics [5] , multi-body system dynamics [6][7][8][9][10][11][12][13][14] , elastic rod dynamics [15][16][17][18] , and hybrid dynamics [19] , etc.So far, there have been many achievements in the Gauss and least compulsion principles for constrained mechanical systems and their applications [20][21][22][23][24][25][26][27][28] .
Using the analytical mechanics method to study the relative motion dynamics of complex systems can unify the expression forms and show the superiority of analytical mechanics in solving the dynamics problems of complex systems.These complex systems comprise the carrier body and carried bodies moving relative to the former [29] .There are many such systems in practical engineering, such as the relative motion and control of spacecraft, the relative motion of satellites [30][31][32][33] and so on.Whittaker analyzed a holonomic system subject to uniform rotation constraints and derived its Lagrange equations [34] .Lurie studied holonomic mechanics with relative motion [35] .Mei et al extended it to nonholonomic mechanics [36] .Since then, progress has been made in the variational principle, equations of motion, integral theory, and symmetry of relative motion dynamics [37][38][39][40][41][42][43][44] .The Gauss principle of relative motion dynamics is further studied in this paper.Section 1 introduces the establishment of the Gauss principle for relative motion dynamics by analyzing the virtual displacement of acceleration space.In Section 2, the compulsion function of relative motion dynamics is constructed, and it is proved that real motion causes the compulsion function to reach an extreme value under Gaussian variation.Section 3 gives the formulae of acceleration energy and corresponding compulsion function when the carried body is a rigid body whose relative motion is planar motion.In Section 4, we study Gauss principle of relative motion dynamics and give its Appell, Lagrange, and Nielsen forms in generalized coordinates.In Section 5, from Gauss principle we obtained, we deduce dynamical equations with relative motion.In Section 6, two examples are given.Section 7 is the conclusion of the article.

Gauss Principle of Relative Motion Dynamics
Study a system of particles that comprises a rigid body (carrier) and N particles (carried bodies).The carried bodies are moving relative to the carrier.The moving frame of reference Oxyz is attached to the carrier.We use n generalized coordinates q s to describe the configuration of relative motion s = 12n.The acceleration a O of the point O in a fixed frame O 1 x 1 y 1 z 1 , and the angular velocity ω of moving frame are the given functions of time t.For the i-th particle, let m i be its mass and r′ i = r′ i ( q s t ) its position vector relative to Oxyz.The dynamic equation of relative motion is where i are the active force, the constraint force, the convective inertial force, and the Coriolis inertial force, respectively.ṙ͂ ′ i is the relative velocity, r̈͂ ′ i is the relative acceleration.By dotting the equation (1) with δ G r̈͂ ′ i and summing over i, we get where δ G (•) stands for the Gaussian variation [3] .Within acceleration space, the condition of ideal constraints yields Thus, formula (2) becomes Formula (4) is the Gauss principle of relative motion dynamics.

Gauss Principle of Least Compulsion for Relative Motion Dynamics
The compulsion function of relative motion is explained as then we have Therefore, the principle (4) becomes If r̈͂ ′ i is the relative acceleration in real motion and r̈͂ ′ i + δ G r̈͂ ′ i is of possible motion of which the constraints admit, subsequently, the distinction between the compulsion functions is Thus, equation (7) shows that, at any instant, the real motion of a relative motion dynamic system minimizes the compulsion function Z r under Gaussian variation when compared with possible motions with the same configuration and the same velocity but with different accelerations.Equation ( 7) can be called the Gauss principle of least compulsion for relative motion dynamics.When, a O = 0ω = 0, principles (4) and ( 7) degenerate to the classical Gauss principle and the least compulsion principle on the absolute motion [3] .

Calculation of Acceleration Energy and Compulsion Function
Expanding formula (5), we have where the ellipsis "" symbolizes the terms that are independent of relative acceleration.Let S r denote the acceleration energy of relative motion, i.e., The compulsion function gives Next, we study the calculation of acceleration energy of relative motion if the carried body is rigid.First, if the relative motion is translation, denote the center of mass of the carried body as C and its relative acceleration a Cr , then where a ir is the relative acceleration of the i-th particle, and m = ∑ i = 1 N m i is the total mass.Second, in the case of fixedaxis rotation for relative motion, denote the relative angular velocity of the carried body ω r , the relative angular acceleration ε r , and the moment of inertia about the rotation axis Aξ as J ξ , then where ρ i is the distance between the i-th particle and Aξ axis, and unit vectors τ i and n i are along tangential and principal normal directions, respectively.Expanding equation ( 13), we get Third, in the event that the relative motion is planar motion, denote the relative angular velocity of the carried body with planar motion as ω r , the relative angular acceleration as ε r , the relative acceleration as a Cr , then Cr + 2a Cr × ρ i ( ) where ρ i is the distance between the i-th particle and C, and τ i and n i are tangential and normal unit vectors relative to C, respectively.Obviously, from the centroid coordinate formula, we have Hence, we obtain where i represents the moment of inertia.Equation (17) shows that the acceleration energy of the relative motion of the carried body with planar motion equals the sum of the acceleration energy of relative translation with and relative rotation around the center of mass.Let represent the principal vectors of the active forces, the convective inertial forces, and the Coriolis inertial forces, respectively, and represent the principal moment about point C, then Substituting formulas (17) and ( 20) into (11), we get This formula calculates the compulsion function of relative motion for the carried body in planar motion.
If the relative motion is translation, then the compulsion function (21) gives If the relative motion is fixed axis rotation, then the compulsion function (21) provides

Gauss Principle of Relative Motion Dynamics in Generalized Coordinates
Taking the relative derivative of r′ i = r′ i ( q s t ) , we get Hence, we have Calculating the Gaussian variation of equation ( 11), we get Notice that ) ) ) where is the potential energy of a uniform force field [29] , and ) ) is the potential energy of centrifugal forces, And represents the generalized rotational inertia force and is the generalized gyroscopic force.By substituting formulas (28), ( 29) and (30) into formula (27), we get Therefore, principle (7) gives Equation ( 36) is the Appell form of the Gauss principle of relative motion dynamics in generalized coordinates.Now, two alternative forms of the principle are derived: the Lagrange form and the Nielsen form.First of all, we give some formulas for the subsequent derivation.From equations ( 24) and ( 25), we can easily obtain ¶r̈͂ By the relation between the absolute derivative and the relative derivative of a vector, for any vector A, there is Thus, we have ´ ¶r′ i ¶q s (42)   Secondly, denote T r as the kinetic energy of relative motion, i.e., then we have ¶T r ¶q̇s By using equations ( 41) and ( 42), we can obtain From equations ( 10) and (37), we get ¶S r ¶q̈s By comparing formula (47) and formula (48), principle (36) can be expressed as Equation ( 49) is the Lagrange form of the Gauss principle of relative motion dynamics in generalized coordinates.
Calculating the time derivative of T r , we get d dt Here, the following relationship is applied, i.e., Taking the partial derivative of d dt T r with respect to q̇s, we get ¶ ¶q̇s From formula (52) and formula (46), we obtain ¶ ¶q̇s By comparing formula (53) and formula (48), principle (36) can also be expressed as Equation ( 54) is the Nielsen form of the Gauss principle of relative motion dynamics in generalized coordinates.

Dynamical Equations of Relative Motion
For a holonomic system, δ G q̈s is independent and arbitrary, so from principle (36), we get This is the Appell equation of relative motion dynamics, and s = 12n.From principle (49), we get This is the Lagrange equation of relative motion dynamics.From principle (54), we get ¶ ¶q̇s This is the Nielsen equation of relative motion dynamics.For a nonholonomic system, let g ideal two-sided nonholonomic constraints be By differentiating equation (58), we obtain Then we have From the Gauss principle (36) and formula (60) in Appell form, using the Lagrange multiplier method, we get where μ α is the Lagrange multiplier, s = 12n.Formula (61) is the Appell equation with multipliers in generalized coordinates for nonholonomic systems in relative motion.From the Lagrange form of Gauss principle (49) and formula (60), we get This is the Lagrange equation with multipliers in generalized coordinates for nonholonomic systems in relative motion, also known as Routh equation.From the Nielsen form of the Gauss principle (54) and formula (60), we get ¶ ¶q̇s This is the Nielsen equation with multipliers in generalized coordinates for nonholonomic systems in relative motion.

Examples
Example 1 A physical pendulum with mass m is suspended at point O on a given block AB, as shown in Fig. 1.Let block AB do circumferential translation with radius l 0 , whose motion is determined by the angle θ and known as θ = θ (t).The angle describes the position of the pendulum relative to AB φ, and the distance from its center of mass C to O is R, and its radius of gyration to C is ρ.Try to establish the dynamic equation of relative motion using the Gauss principle.
In this example, the carrier is the block AB, and the carried body is the physical pendulum.The carriers motion is circumferential translation, and the relative motion of the carried body is fixed axis rotation around the axis Oξ.The acceleration energy of the relative motion of the pendulum is The active force is only gravity mg, and the moment to the axis Oξ is Since the convected motion is translation, there is no Coriolis inertia force, and the convected inertia force is F Iτ e = ml 0 θ ̈F In e = ml 0 θ ̇2 (66) The moment of convected inertia force about Oξ is Therefore, from formula (22), the compulsion function of relative motion is m ( ) To calculate the Gaussian variation δ G Z r and set it to zero, we get Due to the arbitrariness of δ G φ̈, we get m ( ρ 2 + R 2 ) φ̈+ ml 0 Rθ ̇2 sin ( φ -θ ) + ml 0 Rθ ̈cos ( φ -θ ) + mgR sin φ = 0 (70) This represents the differential equation governing the relative motion of a physical pendulum.Equation ( 70) is consistent with the results obtained using the Lagrange equation in Ref. [29].
Example 2 As shown in Fig. 2, a uniform rod AB with mass m and length l has one end, A, sliding along the vertical fixed axis Oz and the other end, B, sliding along the horizontal axis Ox.In contrast, Ox rotates around Oz at a uniform angular velocity ω.The point B is connected to the spring BD, and D is fixed on the Ox axis.Let θ indicate the angle between AB and the plumb line, when θ = 0, the spring has its original length.Suppose that the spring stiffness is k, friction is ignored, and 0 ≤ θ ≤ π 2 , find the dynamic equation of relative motion.
In this scenario, the carrier rotates at a uniform angular speed around Oz.The relative motion of the carried body AB is planar.With θ the generalized coordinate, the acceleration energy of the relative motion of rod AB is where J C = 1 12 ml 2 .Since Taking the time derivative of (72) twice, we have Hence, we have Substituting equation (74) into equation ( 71) and noting that ε r = θ ̈, we get Suppose we take a small segment dl i on AB at a distance l i from end A; then its mass is dm i = m l dl i .The coordinates in the moving coordinate system Oxyz are Then we have Now let us calculate the relevant terms in the compulsion function formula (11), and we get ω = ωk (79) In addition, the elastic force F BD of the spring and the relative acceleration of its action point B are Thus, we have By substituting equations ( 75) and (87) into equation ( 11), we get To calculate the Gaussian variation δ G Z r and set it to zero, we get This is the dynamic equation of the relative motion of rod AB.It is consistent with the results obtained using the Lagrange equation in Ref. [29].

Conclusion
Complex mechanical systems, including the carrier and the carried bodies, are ubiquitous, so their study is significant.Using the theory of analytical mechanics to study the relative motion dynamics of complex systems not only has the unity of expression form but also shows the superiority of analytical mechanics in solving the problems of complex system dynamics.Unlike other differential variational principles, such as d  Alembert-Lagrange  s or Jourdain  s principle, Gauss principle is an extreme value principle from which the motion of a system can be directly obtained.The work conducted in this article includes the following aspects: ① The Gauss principle of relative motion dynamics and its least compulsion principle were established.Based on the dynamic equation of relative motion and the concept of virtual displacement in acceleration space, the Gauss principle for relative motion dynamics was presented.The compulsion function of relative motion was constructed, and it was proved that real motion makes the compulsion function yield its extreme value under Gaussian variation.
② The formulation of acceleration energy and compulsion function of relative motion was presented.The acceleration energy and compulsion function of relative motion were obtained when the carried rigid body was in planar motion.
③ The Appell, Lagrange, and Nielsen forms in generalized coordinates of the Gauss principle of relative motion were derived.According to the above documents of the Gauss principle, using the Lagrange multiplier method, we established the Appell equation, the Lagrange equation, and the Nielsen equation with multipliers for relative motion dynamics.

Fig. 1 A
Fig. 1 A physical pendulum in relative motion Fig. 2 A uniform rod AB in relative motion