Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 3, June 2024



Page(s)  273  283  
DOI  https://doi.org/10.1051/wujns/2024293273  
Published online  03 July 2024 
Mathematics
CLC number: O316
Gauss Principle of Least Compulsion for Relative Motion Dynamics and Differential Equations of Motion
College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China
Received:
31
July
2023
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 motion dynamics. By constructing a compulsion function of relative motion, we prove that at any instant, its real motion minimizes the compulsion function under Gaussian variation, compared with the possible motions with the same configuration and velocity but different accelerations. 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 relative motion examples also verify the results' correctness.
Key words: relative motion dynamics / Gauss principle of least compulsion / acceleration energy / compulsion function
Cite this article: ZHANG Yi, XIA Junling. Gauss Principle of Least Compulsion for Relative Motion Dynamics and Differential Equations of Motion[J]. Wuhan Univ J of Nat Sci, 2024, 29(3): 273283.
Biography: ZHANG Yi, male, Ph.D., Professor, research direction: analytical mechanics. Email: zhy@mail.usts.edu.cn
Fundation item: Supported by the National Natural Science Foundation of China (12272248, 11972241)
© Wuhan University 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 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 nonholonomic 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]}, multibody system dynamics^{[614]}, elastic rod dynamics^{[1518]}, 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^{[2028]}.
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^{[3033]} 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^{[3744]}. 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.
1 Gauss Principle of Relative Motion Dynamics
Study a system of particles that comprises a rigid body (carrier) and particles (carried bodies). The carried bodies are moving relative to the carrier. The moving frame of reference is attached to the carrier. We use generalized coordinates to describe the configuration of relative motion . The acceleration of the point in a fixed frame , and the angular velocity of moving frame are the given functions of time . For the ith particle, let be its mass and its position vector relative to . The dynamic equation of relative motion is
where , , , are the active force, the constraint force, the convective inertial force, and the Coriolis inertial force, respectively. is the relative velocity, is the relative acceleration.
By dotting the equation (1) with and summing over , we get
where 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.
2 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 is the relative acceleration in real motion and 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 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, , principles (4) and (7) degenerate to the classical Gauss principle and the least compulsion principle on the absolute motion^{[3]}.
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 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 , then
where is the relative acceleration of the th particle, and is the total mass. Second, in the case of fixedaxis rotation for relative motion, denote the relative angular velocity of the carried body , the relative angular acceleration , and the moment of inertia about the rotation axis as , then
where is the distance between the th particle and axis, and unit vectors and 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 , the relative angular acceleration as , the relative acceleration as , then
where is the distance between the ith particle and C, and and are tangential and normal unit vectors relative to C, respectively. Obviously, from the centroid coordinate formula, we have
Hence, we obtain
where 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
4 Gauss Principle of Relative Motion Dynamics in Generalized Coordinates
Taking the relative derivative of , 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, is the inertia tensor. 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
By the relation between the absolute derivative and the relative derivative of a vector, for any vector , there is
Thus, we have
Secondly, denote as the kinetic energy of relative motion, i.e.,
then we have
By using equations (41) and (42), we can obtain
and
Therefore, we have
From equations (10) and (37), we get
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 , we get
Here, the following relationship is applied, i.e.,
Taking the partial derivative of with respect to , we get
From formula (52) and formula (46), we obtain
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.
5 Dynamical Equations of Relative Motion
For a holonomic system, is independent and arbitrary, so from principle (36), we get
This is the Appell equation of relative motion dynamics, and . From principle (49), we get
This is the Lagrange equation of relative motion dynamics. From principle (54), we get
This is the Nielsen equation of relative motion dynamics. For a nonholonomic system, let ideal twosided 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 is the Lagrange multiplier, . 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
This is the Nielsen equation with multipliers in generalized coordinates for nonholonomic systems in relative motion.
6 Examples
Example 1 A physical pendulum with mass is suspended at point O on a given block AB, as shown in Fig. 1. Let block AB do circumferential translation with radius , whose motion is determined by the angle and known as . The angle describes the position of the pendulum relative to AB, and the distance from its center of mass C to O is , and its radius of gyration to C is . Try to establish the dynamic equation of relative motion using the Gauss principle.
Fig. 1 A physical pendulum in relative motion 
In this example, the carrier is the block AB, and the carried body is the physical pendulum. The carrier's motion is circumferential translation, and the relative motion of the carried body is fixed axis rotation around the axis . The acceleration energy of the relative motion of the pendulum is
The active force is only gravity , and the moment to the axis is
Since the convected motion is translation, there is no Coriolis inertia force, and the convected inertia force is
The moment of convected inertia force about is
Therefore, from formula (22), the compulsion function of relative motion is
To calculate the Gaussian variation and set it to zero, we get
Due to the arbitrariness of , we get
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 and length has one end, A, sliding along the vertical fixed axis and the other end, B, sliding along the horizontal axis . In contrast, rotates around at a uniform angular velocity . The point B is connected to the spring BD, and D is fixed on the axis. Let indicate the angle between AB and the plumb line, when , the spring has its original length. Suppose that the spring stiffness is , friction is ignored, and , find the dynamic equation of relative motion.
Fig. 2 A uniform rod AB in relative motion 
In this scenario, the carrier rotates at a uniform angular speed around . The relative motion of the carried body AB is planar. With the generalized coordinate, the acceleration energy of the relative motion of rod AB is
where . Since
Taking the time derivative of (72) twice, we have
Hence, we have
Substituting equation (74) into equation (71) and noting that , we get
Suppose we take a small segment on AB at a distance from end A; then its mass is . The coordinates in the moving coordinate system are
Then we have
Now let us calculate the relevant terms in the compulsion function formula (11), and we get
In addition, the elastic force 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 and set it to zero, we get
Due to the arbitrariness of , we get
i.e.,
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].
7 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'AlembertLagrange'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.
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All Figures
Fig. 1 A physical pendulum in relative motion  
In the text 
Fig. 2 A uniform rod AB in relative motion  
In the text 
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