© Wuhan University 2024

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 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^[6-14], elastic rod dynamics^[15-18], 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^[20-28].

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^[30-33] 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^[37-44]. 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.

1 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 $s=\mathrm{1,2},\cdots ,n$. The acceleration ${\mathit{a}}_{\mathrm{O}}$ of the point $O$ in a fixed frame $O_{\mathrm{1}}{x}_{\mathrm{1}}{y}_{\mathrm{1}}{z}_{\mathrm{1}}$, and the angular velocity $\mathit{\omega }$ of moving frame are the given functions of time $t$. For the i-th particle, let $m_i$ be its mass and ${\mathit{r}}_i^{\text{'}}={\mathit{r}}_i^{\text{'}}\left(q_s,t\right)$ its position vector relative to $Oxyz$. The dynamic equation of relative motion is

$-m_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+{\mathit{F}}_i+{\mathit{N}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}=\mathrm{0},i=\mathrm{1,2},\cdots ,N$(1)

where ${\mathit{F}}_{\mathit{i}}$, ${\mathit{N}}_{\mathit{i}}$, ${\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}=-m_i{\mathit{a}}_{\mathrm{O}}-m_i\dot{\mathit{\omega }}\times {\mathit{r}}_i^{\text{'}}-m_i\mathit{\omega }\times \left(\mathit{\omega }\times {\mathit{r}}_i^{\text{'}}\right)$, ${\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}=-\mathrm{2}{m}_i\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}$ are the active force, the constraint force, the convective inertial force, and the Coriolis inertial force, respectively. ${\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}$ is the relative velocity, ${\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}$ is the relative acceleration.

By dotting the equation (1) with ${\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}$ and summing over $i$, we get

${\displaystyle \sum _{i=\mathrm{1}}^N}\left(-m_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+{\mathit{F}}_i+{\mathit{N}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\cdot {\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}=\mathrm{0}$(2)

where ${\delta }_{\mathrm{G}}\left(·\right)$ stands for the Gaussian variation^[3]. Within acceleration space, the condition of ideal constraints yields

${\displaystyle \sum _{i=\mathrm{1}}^N}{\mathit{N}}_i\cdot {\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}=\mathrm{0}$(3)

Thus, formula (2) becomes

${\displaystyle \sum _{i=\mathrm{1}}^N}\left(-m_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+{\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\cdot {\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}=\mathrm{0}$(4)

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

$Z_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}-\frac{{\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}}{m_i}\right)}^{\mathrm{2}}$(5)

then we have

$\begin{array}{l}{\delta }_{\mathrm{G}}{Z}_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}-\frac{{\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}}{m_i}\right)\cdot {\delta }_{\mathrm{G}}\left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}-\frac{{\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}}{m_i}\right)\\ ={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}-\frac{{\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}}{m_i}\right)\cdot {\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\end{array}$(6)

Therefore, the principle (4) becomes

${\delta }_{\mathrm{G}}{Z}_{\mathrm{r}}=\mathrm{0}$(7)

If ${\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}$ is the relative acceleration in real motion and ${\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+{\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}$ is of possible motion of which the constraints admit, subsequently, the distinction between the compulsion functions is

$\begin{array}{l}\mathrm{\Delta }{Z}_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i\left\{{\left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}-\frac{{\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}}{m_i}\right)}^{\mathrm{2}}-{\left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+{\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}-\frac{{\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}}{m_i}\right)}^{\mathrm{2}}\right\}\\ =-{\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\left({\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\right)}^{\mathrm{2}}-{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}-\frac{{\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}}{m_i}\right)\cdot {\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}=-{\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\left({\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\right)}^{\mathrm{2}}<\mathrm{0}\end{array}$(8)

Thus, equation (7) shows that, at any instant, the real motion of a relative motion dynamic system minimizes the compulsion function $Z_{\mathrm{r}}$ 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, ${\mathit{a}}_{\mathrm{O}}=\mathrm{0},\mathit{\omega }=\mathrm{0}$, 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

$Z_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}-{\displaystyle \sum _{i=\mathrm{1}}^N}\left({\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+\cdots$(9)

where the ellipsis "$\cdots$" symbolizes the terms that are independent of relative acceleration.

Let $S_{\mathrm{r}}$ denote the acceleration energy of relative motion, i.e.,

$S_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}$(10)

The compulsion function gives

$Z_{\mathrm{r}}=S_{\mathrm{r}}-{\displaystyle \sum _{i=\mathrm{1}}^N}\left({\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+\cdots$(11)

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 ${\mathit{a}}_{\mathrm{C}\mathrm{r}}$, then

$S_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{a}_{i\mathrm{r}}^{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}m{a}_{\mathrm{C}\mathrm{r}}^{\mathrm{2}}$(12)

where ${\mathit{a}}_{i\mathrm{r}}$ is the relative acceleration of the $i$-th particle, and $m={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i$ is the total mass. Second, in the case of fixed-axis rotation for relative motion, denote the relative angular velocity of the carried body ${\mathit{\omega }}_{\mathrm{r}}$, the relative angular acceleration ${\mathit{\epsilon }}_{\mathrm{r}}$, and the moment of inertia about the rotation axis $A\xi$ as $J_{\xi }$, then

$S_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\mathit{a}}_{i\mathrm{r}}\cdot {\mathit{a}}_{i\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i\left({\rho }_i{\epsilon }_{\mathrm{r}}{\mathit{\tau }}_i+{\rho }_i{\omega }_{\mathrm{r}}^{\mathrm{2}}{\mathit{n}}_i\right)\cdot \left({\rho }_i{\epsilon }_{\mathrm{r}}{\mathit{\tau }}_i+{\rho }_i{\omega }_{\mathrm{r}}^{\mathrm{2}}{\mathit{n}}_i\right)$(13)

where ${\rho }_i$ is the distance between the $i$-th particle and $A\xi$ axis, and unit vectors ${\mathit{\tau }}_i$ and ${\mathit{n}}_i$ are along tangential and principal normal directions, respectively. Expanding equation (13), we get

$S_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\rho }_i^{\mathrm{2}}{\epsilon }_{\mathrm{r}}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\rho }_i^{\mathrm{2}}{\omega }_{\mathrm{r}}^{\mathrm{4}}=\frac{\mathrm{1}}{\mathrm{2}}{J}_{\xi }{\epsilon }_{\mathrm{r}}^{\mathrm{2}}+\cdots$(14)

Third, in the event that the relative motion is planar motion, denote the relative angular velocity of the carried body with planar motion as ${\mathit{\omega }}_{\mathrm{r}}$, the relative angular acceleration as ${\mathit{\epsilon }}_{\mathrm{r}}$, the relative acceleration as ${\mathit{a}}_{\mathrm{C}\mathrm{r}}$, then

$\begin{array}{l}{S}_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i{\mathit{a}}_{i\mathrm{r}}\cdot {\mathit{a}}_{i\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i\left[{\mathit{a}}_{\mathrm{C}\mathrm{r}}+{\rho }_i\left({\epsilon }_{\mathrm{r}}{\mathit{\tau }}_i+{\omega }_{\mathrm{r}}^{\mathrm{2}}{\mathit{n}}_i\right)\right]\cdot \left[{\mathit{a}}_{\mathrm{C}\mathrm{r}}+{\rho }_i\left({\epsilon }_{\mathrm{r}}{\mathit{\tau }}_i+{\omega }_{\mathrm{r}}^{\mathrm{2}}{\mathit{n}}_i\right)\right]\\ ={\displaystyle \sum _{i=\mathrm{1}}^N}\frac{\mathrm{1}}{\mathrm{2}}{m}_i\left[a_{\mathrm{C}\mathrm{r}}^{\mathrm{2}}+\mathrm{2}{\mathit{a}}_{\mathrm{C}\mathrm{r}}\cdot {\rho }_i\left({\epsilon }_{\mathrm{r}}{\mathit{\tau }}_i+{\omega }_{\mathrm{r}}^{\mathrm{2}}{\mathit{n}}_i\right)+{\rho }_i^{\mathrm{2}}\left({\epsilon }_{\mathrm{r}}^{\mathrm{2}}+{\omega }_{\mathrm{r}}^{\mathrm{4}}\right)\right]\end{array}$(15)

where ${\rho }_i$ is the distance between the i-th particle and C, and ${\mathit{\tau }}_i$ and ${\mathit{n}}_i$ are tangential and normal unit vectors relative to C, respectively. Obviously, from the centroid coordinate formula, we have

${\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\rho }_i{\mathit{a}}_{\mathrm{C}\mathrm{r}}\cdot {\mathit{\tau }}_i=\mathrm{0},{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\rho }_i{\mathit{a}}_{\mathrm{C}\mathrm{r}}\cdot {\mathit{n}}_i=\mathrm{0}$(16)

Hence, we obtain

$S_{\mathrm{r}}=\frac{\mathrm{1}}{\mathrm{2}}m{a}_{\mathrm{C}\mathrm{r}}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{J}_{\mathrm{C}}{\epsilon }_{\mathrm{r}}^{\mathrm{2}}$(17)

where $J_{\mathrm{C}}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\rho }_i^{\mathrm{2}}$ 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

$\mathit{F}={\displaystyle \sum _{i=\mathrm{1}}^N}{\mathit{F}}_i,{\mathit{F}}_{\mathrm{e}}^{\mathrm{I}}={\displaystyle \sum _{i=\mathrm{1}}^N}{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}},{\mathit{F}}_{\mathrm{c}}^{\mathrm{I}}={\displaystyle \sum _{i=\mathrm{1}}^N}{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}$(18)

represent the principal vectors of the active forces, the convective inertial forces, and the Coriolis inertial forces, respectively, and

$M_{\mathrm{C}}={\displaystyle \sum _{i=\mathrm{1}}^N}{M}_{\mathrm{C}}\left({\mathit{F}}_i\right),M_{\mathrm{C}\mathrm{e}}^{\mathrm{I}}={\displaystyle \sum _{i=\mathrm{1}}^N}{M}_{\mathrm{C}}\left({\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}\right),M_{\mathrm{C}\mathrm{c}}^{\mathrm{I}}={\displaystyle \sum _{i=\mathrm{1}}^N}{M}_{\mathrm{C}}\left({\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)$(19)

represent the principal moment about point C, then

${\displaystyle \sum _{i=\mathrm{1}}^N}\left({\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{i=\mathrm{1}}^N}\left({\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\cdot \left[{\mathit{a}}_{\mathrm{C}\mathrm{r}}+{\rho }_i\left({\epsilon }_{\mathrm{r}}{\mathit{\tau }}_i+{\omega }_{\mathrm{r}}^{\mathrm{2}}{\mathit{n}}_i\right)\right]$

$\begin{array}{l}={\mathit{a}}_{\mathrm{C}\mathrm{r}}\cdot {\displaystyle \sum _{i=\mathrm{1}}^N}\left({\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)+{\epsilon }_{\mathrm{r}}{\displaystyle \sum _{i=\mathrm{1}}^N}{\rho }_i{\mathit{\tau }}_i\cdot \left({\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)+{\omega }_{\mathrm{r}}^{\mathrm{2}}{\displaystyle \sum _{i=\mathrm{1}}^N}{\rho }_i{\mathit{n}}_i\cdot \left({\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\\ ={\mathit{a}}_{\mathrm{C}\mathrm{r}}\cdot \left(\mathit{F}+{\mathit{F}}_{\mathrm{e}}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}}^{\mathrm{I}}\right)+{\epsilon }_{\mathrm{r}}\left(M_{\mathrm{C}}+M_{\mathrm{C}\mathrm{e}}^{\mathrm{I}}+M_{\mathrm{C}\mathrm{c}}^{\mathrm{I}}\right)+\cdots \end{array}$(20)

Substituting formulas (17) and (20) into (11), we get

$Z_{\mathrm{r}}=\frac{\mathrm{1}}{\mathrm{2}}m{a}_{\mathrm{C}\mathrm{r}}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}{J}_{\mathrm{C}}{\epsilon }_{\mathrm{r}}^{\mathrm{2}}-{\mathit{a}}_{\mathrm{C}\mathrm{r}}\cdot \left(\mathit{F}+{\mathit{F}}_{\mathrm{e}}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}}^{\mathrm{I}}\right)-{\epsilon }_{\mathrm{r}}\left(M_{\mathrm{C}}+M_{\mathrm{C}\mathrm{e}}^{\mathrm{I}}+M_{\mathrm{C}\mathrm{c}}^{\mathrm{I}}\right)+\cdots$(21)

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

$Z_{\mathrm{r}}=\frac{\mathrm{1}}{\mathrm{2}}m{a}_{\mathrm{C}\mathrm{r}}^{\mathrm{2}}-{\mathit{a}}_{\mathrm{C}\mathrm{r}}\cdot \left(\mathit{F}+{\mathit{F}}_{\mathrm{e}}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}}^{\mathrm{I}}\right)+\cdots$(22)

If the relative motion is fixed axis rotation, then the compulsion function (21) provides

$Z_{\mathrm{r}}=\frac{\mathrm{1}}{\mathrm{2}}{J}_{\xi }{\epsilon }_{\mathrm{r}}^{\mathrm{2}}-{\epsilon }_{\mathrm{r}}\left(M_{\xi }+M_{\xi \mathrm{e}}^{\mathrm{I}}+M_{\xi \mathrm{c}}^{\mathrm{I}}\right)+\cdots$(23)

4 Gauss Principle of Relative Motion Dynamics in Generalized Coordinates

Taking the relative derivative of ${\mathit{r}}_i^{\text{'}}={\mathit{r}}_i^{\text{'}}\left(q_s,t\right)$, we get

${\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{s=\mathrm{1}}^n}\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}{\dot{q}}_s+\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial t}$(24)

${\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{s=\mathrm{1}}^n}\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}{\ddot{q}}_s+{\displaystyle \sum _{s=\mathrm{1}}^n}{\displaystyle \sum _{k=\mathrm{1}}^n}\frac{{\partial }^{\mathrm{2}}{\mathit{r}}_i^{\text{'}}}{\partial q_s\partial q_k}{\dot{q}}_s{\dot{q}}_k+\mathrm{2}{\displaystyle \sum _{s=\mathrm{1}}^n}\frac{{\partial }^{\mathrm{2}}{\mathit{r}}_i^{\text{'}}}{\partial t\partial q_s}{\dot{q}}_s+\frac{{\partial }^{\mathrm{2}}{\mathit{r}}_i^{\text{'}}}{\partial t^{\mathrm{2}}}$(25)

Hence, we have

${\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{s=\mathrm{1}}^n}\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}{\delta }_{\mathrm{G}}{\ddot{q}}_s$(26)

Calculating the Gaussian variation of equation (11), we get

${\delta }_{\mathrm{G}}{Z}_{\mathrm{r}}={\delta }_{\mathrm{G}}{S}_{\mathrm{r}}-{\displaystyle \sum _{i=\mathrm{1}}^N}\left({\mathit{F}}_i+{\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\cdot {\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}$(27)

Notice that

${\delta }_{\mathrm{G}}{S}_{\mathrm{r}}={\displaystyle \sum _{s=\mathrm{1}}^n}\frac{\partial S_r}{\partial {\ddot{q}}_s}{\delta }_{\mathrm{G}}{\ddot{q}}_s$(28)

${\displaystyle \sum _{i=\mathrm{1}}^N}{\mathit{F}}_i\cdot {\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{s=\mathrm{1}}^n}{\displaystyle \sum _{i=\mathrm{1}}^N}{\mathit{F}}_i\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}{\delta }_{\mathrm{G}}{\ddot{q}}_s={\displaystyle \sum _{s=\mathrm{1}}^n}{Q}_s{\delta }_{\mathrm{G}}{\ddot{q}}_s$(29)

$\begin{array}{l}{\displaystyle \sum _{i=\mathrm{1}}^N}\left({\mathit{F}}_{\mathrm{e}i}^{\mathrm{I}}+{\mathit{F}}_{\mathrm{c}i}^{\mathrm{I}}\right)\cdot {\delta }_{\mathrm{G}}{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{i=\mathrm{1}}^N}\left[-m_i{\mathit{a}}_{\mathrm{o}}-m_i\dot{\mathit{\omega }}\times {\mathit{r}}_i^{\text{'}}-m_i\mathit{\omega }\times \left(\mathit{\omega }\times {\mathit{r}}_i^{\text{'}}\right)-\mathrm{2}{m}_i\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\right]\cdot {\displaystyle \sum _{s=\mathrm{1}}^n}\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}{\delta }_{\mathrm{G}}{\ddot{q}}_s\\ ={\displaystyle \sum _{s=\mathrm{1}}^n}{\displaystyle \sum _{i=\mathrm{1}}^N}\{-m_i{\mathit{a}}_{\mathrm{o}}\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}-m_i\left(\dot{\mathit{\omega }}\times {\mathit{r}}_i^{\text{'}}\right)\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}-m_i\left[\mathit{\omega }\times \left(\mathit{\omega }\times {\mathit{r}}_i^{\text{'}}\right)\right]\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}-\mathrm{2}{m}_i\left(\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\right)\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}\}{\delta }_{\mathrm{G}}{\ddot{q}}_s\\ ={\displaystyle \sum _{s=\mathrm{1}}^n}\left[-\frac{\partial }{\partial q_s}\left({\mathrm{\Pi }}^{\mathrm{o}}+{\mathrm{\Pi }}^{\mathit{\omega }}\right)+Q_s^{\dot{\mathit{\omega }}}+{\mathrm{\Gamma }}_s\right]{\delta }_{\mathrm{G}}{\ddot{q}}_s\end{array}$(30)

where

${\mathrm{\Pi }}^{\mathrm{o}}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\mathit{a}}_{\mathrm{o}}\cdot {\mathit{r}}_i^{\text{'}}=m{\mathit{a}}_{\mathrm{o}}\cdot {\mathit{r}}_{\mathrm{c}}^{\text{'}}$(31)

is the potential energy of a uniform force field^[29], and

${\mathrm{\Pi }}^{\mathbf{\omega }}=\frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\left[\mathit{\omega }\times \left(\mathit{\omega }\times {\mathit{r}}_i^{\text{'}}\right)\right]\cdot {\mathit{r}}_i^{\text{'}}=-\frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\left(\mathit{\omega }\times {\mathit{r}}_i^{\text{'}}\right)\cdot \left(\mathit{\omega }\times {\mathit{r}}_i^{\text{'}}\right)=-\frac{\mathrm{1}}{\mathrm{2}}\mathit{\omega }\cdot {\mathit{\theta }}^{\mathrm{o}}\cdot \mathit{\omega }$(32)

is the potential energy of centrifugal forces, ${\mathit{\theta }}^{\mathrm{o}}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\left[{\left(r_i^{\text{'}}\right)}^{\mathrm{2}}\mathit{E}-{\mathit{r}}_i^{\text{'}}{\mathit{r}}_i^{\text{'}}\right]$ is the inertia tensor. And

$Q_s^{\dot{\mathit{\omega }}}=-{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\left(\dot{\mathit{\omega }}\times {\mathit{r}}_i^{\text{'}}\right)\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}$(33)

represents the generalized rotational inertia force and

${\mathrm{\Gamma }}_s=-{\displaystyle \sum _{i=\mathrm{1}}^N}\mathrm{2}{m}_i\left(\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\right)\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}$(34)

is the generalized gyroscopic force. By substituting formulas (28), (29) and (30) into formula (27), we get

${\delta }_{\mathrm{G}}{Z}_{\mathrm{r}}={\displaystyle \sum _{s=\mathrm{1}}^n}\left[\frac{\partial S_{\mathrm{r}}}{\partial {\ddot{q}}_s}-Q_s+\frac{\partial }{\partial q_s}\left({\mathrm{\Pi }}^{\mathrm{o}}+{\mathrm{\Pi }}^{\mathit{\omega }}\right)-Q_s^{\dot{\mathit{\omega }}}-{\mathrm{\Gamma }}_s\right]{\delta }_{\mathrm{G}}{\ddot{q}}_s$(35)

Therefore, principle (7) gives

${\displaystyle \sum _{s=\mathrm{1}}^n}\left[\frac{\partial S_{\mathrm{r}}}{\partial {\ddot{q}}_s}-Q_s+\frac{\partial }{\partial q_s}\left({\mathrm{\Pi }}^{\mathrm{o}}+{\mathrm{\Pi }}^{\mathit{\omega }}\right)-Q_s^{\dot{\mathit{\omega }}}-{\mathrm{\Gamma }}_s\right]{\delta }_{\mathrm{G}}{\ddot{q}}_s=\mathrm{0}$(36)

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

$\frac{\partial {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}}{\partial {\ddot{q}}_s}=\frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial {\dot{q}}_s}=\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}$(37)

$\frac{\partial {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}}{\partial {\dot{q}}_s}=\mathrm{2}{\displaystyle \sum _{k=\mathrm{1}}^n}\frac{{\partial }^{\mathrm{2}}{\mathit{r}}_i^{\text{'}}}{\partial q_s\partial q_k}{\dot{q}}_k+\mathrm{2}\frac{{\partial }^{\mathrm{2}}{\mathit{r}}_i^{\text{'}}}{\partial t\partial q_s}=\frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial q_s}$(38)

$\frac{\tilde{\mathrm{d}}}{\mathrm{d}t}\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}=\frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial q_s}$(39)

By the relation between the absolute derivative and the relative derivative of a vector, for any vector $\mathit{A}$, there is

$\frac{\mathrm{d}}{\mathrm{d}t}\mathit{A}=\frac{\tilde{\mathrm{d}}}{\mathrm{d}t}\mathit{A}+\mathit{\omega }\times \mathit{A}$(40)

Thus, we have

$\frac{\mathrm{d}}{\mathrm{d}t}{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}=\frac{\tilde{\mathrm{d}}}{\mathrm{d}t}{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}+\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}={\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}+\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}$(41)

$\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}=\frac{\tilde{\mathrm{d}}}{\mathrm{d}t}\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}+\mathit{\omega }\times \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}=\frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial q_s}+\mathit{\omega }\times \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}$(42)

Secondly, denote $T_{\mathrm{r}}$ as the kinetic energy of relative motion, i.e.,

$T_{\mathrm{r}}=\frac{\mathrm{1}}{\mathrm{2}}{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}$(43)

then we have

$\frac{\partial T_{\mathrm{r}}}{\partial {\dot{q}}_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial {\dot{q}}_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}$(44)

By using equations (41) and (42), we can obtain

$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T_{\mathrm{r}}}{\partial {\dot{q}}_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\frac{\mathrm{d}}{\mathrm{d}t}{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}+{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\right)\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}+{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \left(\frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial q_s}+\mathit{\omega }\times \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}\right)\\ ={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}+{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial q_s}\end{array}$(45)

and

$\frac{\partial T_{\mathrm{r}}}{\partial q_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial q_s}$(46)

Therefore, we have

$\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T_{\mathrm{r}}}{\partial {\dot{q}}_s}-\frac{\partial T_{\mathrm{r}}}{\partial q_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}$(47)

From equations (10) and (37), we get

$\frac{\partial S_{\mathrm{r}}}{\partial {\ddot{q}}_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}}{\partial {\ddot{q}}_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}$(48)

By comparing formula (47) and formula (48), principle (36) can be expressed as

${\displaystyle \sum _{s=\mathrm{1}}^n}\left[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T_{\mathrm{r}}}{\partial {\dot{q}}_s}-\frac{\partial T_{\mathrm{r}}}{\partial q_s}-Q_s+\frac{\partial }{\partial q_s}\left({\mathrm{\Pi }}^{\mathrm{o}}+{\mathrm{\Pi }}^{\mathit{\omega }}\right)-Q_s^{\dot{\mathit{\omega }}}-{\mathrm{\Gamma }}_s\right]{\delta }_{\mathrm{G}}{\ddot{q}}_s=\mathrm{0}$(49)

Equation (49) is the Lagrange form of the Gauss principle of relative motion dynamics in generalized coordinates.

Calculating the time derivative of $T_{\mathrm{r}}$, we get

$\frac{\mathrm{d}}{\mathrm{d}t}{T}_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\mathrm{d}}{\mathrm{d}t}{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \left({\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}+\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\right)={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}$(50)

Here, the following relationship is applied, i.e.,

${\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \left(\mathit{\omega }\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\right)=\mathit{\omega }\cdot \left({\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\times {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\right)=\mathrm{0}$(51)

Taking the partial derivative of $\frac{\mathrm{d}}{\mathrm{d}t}{T}_{\mathrm{r}}$ with respect to ${\dot{q}}_s$, we get

$\frac{\partial }{\partial {\dot{q}}_s}\frac{\mathrm{d}}{\mathrm{d}t}{T}_{\mathrm{r}}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial {\dot{q}}_s}+{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i\frac{\partial {\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}}{\partial {\dot{q}}_s}\cdot {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}+\mathrm{2}{\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\widetilde{\dot{\mathit{r}}}}_i^{\text{'}}}{\partial q_s}$(52)

From formula (52) and formula (46), we obtain

$\frac{\partial }{\partial {\dot{q}}_s}\frac{\mathrm{d}}{\mathrm{d}t}{T}_{\mathrm{r}}-\mathrm{2}\frac{\partial T_{\mathrm{r}}}{\partial q_s}={\displaystyle \sum _{i=\mathrm{1}}^N}{m}_i{\widetilde{\ddot{\mathit{r}}}}_i^{\text{'}}\cdot \frac{\partial {\mathit{r}}_i^{\text{'}}}{\partial q_s}$(53)

By comparing formula (53) and formula (48), principle (36) can also be expressed as

${\displaystyle \sum _{s=\mathrm{1}}^n}\left[\frac{\partial }{\partial {\dot{q}}_s}\frac{\mathrm{d}}{\mathrm{d}t}{T}_{\mathrm{r}}-\mathrm{2}\frac{\partial T_{\mathrm{r}}}{\partial q_s}-Q_s+\frac{\partial }{\partial q_s}\left({\mathrm{\Pi }}^{\mathrm{o}}+{\mathrm{\Pi }}^{\mathit{\omega }}\right)-Q_s^{\dot{\mathit{\omega }}}-{\mathrm{\Gamma }}_s\right]{\delta }_{\mathrm{G}}{\ddot{q}}_s=\mathrm{0}$(54)

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, ${\delta }_{\mathrm{G}}{\ddot{q}}_s$ is independent and arbitrary, so from principle (36), we get

$\frac{\partial S_{\mathrm{r}}}{\partial {\ddot{q}}_s}=Q_s-\frac{\partial }{\partial q_s}\left({\mathrm{\Pi }}^{\mathrm{o}}+{\mathrm{\Pi }}^{\mathit{\omega }}\right)+Q_s^{\dot{\mathit{\omega }}}+{\mathrm{\Gamma }}_s$(55)

This is the Appell equation of relative motion dynamics, and $s=\mathrm{1,2},\cdots ,n$. From principle (49), we get

$\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T_{\mathrm{r}}}{\partial {\dot{q}}_s}-\frac{\partial T_{\mathrm{r}}}{\partial q_s}=Q_s-\frac{\partial }{\partial q_s}\left({\mathrm{\Pi }}^{\mathrm{o}}+{\mathrm{\Pi }}^{\mathit{\omega }}\right)+Q_s^{\dot{\mathit{\omega }}}+{\mathrm{\Gamma }}_s$(56)