1 Introduction

Modern differential geometry theory plays an important role in the development of modern physics. The geometric dynamics of constrained systems can promote the nonholonomic dynamics theory and its applications^[1-16]. In the gravitation theory, the research of generalizing the geometric structure of space-time from Riemann spaces to Riemann-Cartan spaces is very important. By this means, the spin is regarded as the torsion of the Riemann-Cartan spaces. The classical general relativity is generalized to the Einstein-Cartan theory^[17], which was inspired by the torsion theory given by the Cosserat brothers in 1909. The torsion theory is a good idea to solve the problem of singular feature in Euclidean spaces or Riemann spaces. Since the middle of the last century, the singular feature of some physical problems, e.g., the continuum theory of crystal dislocations, Feynman path integrals, distortions of the continua, spin of matter, and non-integrability of constraints, has been related to the non-Euclidean characters of evolution spaces. The singularity of dynamical systems in Euclidean spaces or Riemann spaces was described by the torsion of Riemann-Cartan spaces^[18-20]. It is an effective approach to solve the singularity problems of dynamical systems. In this process, in order to construct a Riemann-Cartan space or a Riemann space, Kleinet and Pelster^[21] and Kleinet and Shabanov^[22] put forward an embedding method, which could embed a Riemann-Cartan space or a Riemann space into a known Euclidean space. In this paper, the above embedding method was further generalized to the first-order linear homogeneous nonholonomic mapping theory. We use the expression of the geometric quantities (including the metric, the connection, the torsion, and the curvature) of the Riemann-Cartan space constructed by the nonholonomic mapping method^[7-10]. The nonholonomic mapping theory not only can be used to study the mapping between spaces with different dimensions, but also can be used to study the mapping between spaces with the same dimension. The above embedding method can be regarded as a special case of the nonholonomic mapping theory. As an application of the nonholonomic mapping theory in nonholonomic dynamics, the differential equations of motion of constrained systems are studied without active forces in its Riemann-Cartan configuration spaces. We prove that the autoparallel and geodesic trajectories in a Riemann-Cartan space can be, respectively, derived from the Lagrange-d'Alembert differential variational principle and the Hamilton's principle. That implies that the autoparallel and geodesic trajectories are, respectively, evolutionary trajectories of the Chetaev equations and the vakonomic equations for a nonholonomic system in its Riemann-Cartan configuration space, respectively.

On this basis, in this paper, the geometric formulation of motion for the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied by the nonholonomic mapping theory. If the coordinate, momentum, and active force on the Euclidean configuration space are, respectively, generalized to the quasi-coordinate, quasi-momentum, and quasi-force in the Riemann-Cartan configuration space, we can obtain the quasi-Newton law of motion, the quasi-momentum theorem, and the second Lagrange equation. In these equations, the nonholonomic constraints of the dynamical systems can be related to the metric and connection of their Riemann-Cartan configuration spaces by the nonholonomic mapping theory. In this way, the constraints of the dynamical systems are implied in the covariant derivative of their differential equations of motion, such that the differential equations of motion of the dynamical systems will be more concise and universal.

A natural idea is that the above theory can be applied to the study of first-order linear homogenous scleronomous nonholonomic systems. By a nonholonomic mapping between two spaces with different dimensions, a first-order linear homogenous scleronomous nonholonomic system on its high-dimensional Euclidean configuration space (or high-dimensional Riemann configuration space) can be reduced to an unconstrained system on its low-dimensional Riemann-Cartan configuration space. Because of the decrease in the dimension of the configuration space, the number of the differential equations of motion for the constrained system will decrease accordingly. More importantly, if the set of quasi-coordinates defined by the nonholonomic mapping is suitable, the differential equations of motion on its Riemann-Cartan configuration space may be simple. Therefore, the nonholonomic mapping theory can solve some nonholonomic problems, which are difficult to be solved by the traditional analytical mechanics method. Two examples (Example 1 and Example 2) will be given in this paper.

However, for holonomic constrained systems, the above theory may be also effective, i.e., by a nonholonomic mapping between two spaces with the same dimension, we can map the Riemann configuration space, which has no torsion and is corresponding to a set of generalized coordinates, of a holonomic constrained system to the Riemann-Cartan configuration space with the same dimension, which has torsion and is corresponding to a set of quasi-coordinates. As a result of the nonholonomic mapping, the differential equations of motion of the constrained system may be simplified, although the geometric structure of the configuration space becomes more complex. A typical example is the rotation of the rigid body with a fixed point. As an illustrated example (Example 3), the rotation of a three-particle-rigid-body system with a fixed point is discussed in this paper. We verify that the angular velocity, angular momentum, and moment are, respectively, the quasi-velocity, quasi-momentum, and quasi-force on the RiemannCartan configuration space of the rigid body, and the Euler equation of the rigid body is exactly the quasi-momentum theorem in the Riemann-Cartan space. Finally, we point out that it is worth further studying how to select a set of suitable quasi-velocities to simplify the study of a constrained system.

The Einstein summation convention is used throughout in this paper, e.g., i, j, k, l = 1, 2, …, n, μ, ν, ρ, σ, λ = 1, 2, …, m, and α, β = 0, 1, 2, …, n − m.

2 Quasi-momentum theorem in Riemann-Cartan spaces

A dynamical system, which is composed of N particles, is discussed. The tangent space of the n-dimensional (n = 3N) flat configuration [X] is denoted by [], whose metric and connection are, respectively, g_ij and Γ_jkⁱ, where

(1)

(2)

m_i (i = 1, 2, …, N) are, respectively, the masses of these particles. The same subscripts do not indicate the summation in Eq. (1). According to Eq. (1), the dual space of the tangent space [] is defined by []. The kinetic energy of the system can be expressed as follows:

(3)

If the dynamical system is unconstrained, the differential equations of motion have the following three equivalent forms:

(4)

(5)

(6)

where f_i is the expression of the force on these particles on the space []. is the dual vector of f_i. For the single free particle, [] and [] are, respectively, the velocity space and the momentum space of the particle. f_i is exactly the force on the particle. Equations (4) and (5) are, respectively, degenerate to the Newton law of motion and the momentum theorem.

For the constrained system composed by N particles, in Eqs. (4), (5), and (6), f_i can be decomposed into the active force F_i and the constrained force R_i, i.e.,

(7)

Suppose that the system is subjected to (n − m) first-order linear homogeneous differential constraints, i.e.,

(8)

Then, the component R_i, which is the resultant force of (n − m) constrained forces, can be expressed as

(9)

in the space [], where C_α is the proportionality factor. If the independent first-order linear differential equations are defined as follows:

(10)

we can obtain the following first-order linear mapping:

(11)

which implies the constraints in Eq. (8). Then, an m-dimensional configuration space [Π] is obtained from the n-dimensional flat configuration space [X]. If we regarded [] as the tangent space of [Π], the metric, the connection, the curvature, and the torsion of space [Π] can be obtained, respectively, as follows^[8]:

(12)

(13)

(14)

(15)

If the constraints in Eq. (8) is non-integrable, Eq. (11) is a nonholonomic mapping. The subscript of the connections in Eq. (13) is asymmetric. Therefore, the configuration space [Π] is a Riemann-Cartan space with the torsion (15). If the constraints in Eq. (8) are integrable, the subscript of connections (13) is symmetric. This means that the configuration space [Π] reduces to a Riemann space [Q] with no torsion.

Substituting the first-order linear mapping (11) into Eqs. (4)–(6), we can, respectively, obtain

(16)

(17)

(18)

where is the covariant derivative defined by the connections in Eq. (13). In Eqs. (16)–(18), if the constraints in Eq. (8) are non-integrable, is the quasi-velocity on the Riemann-Cartan space [Π]. π^μ is the quasi-coordinate according to . If the constraints in Eq. (8) are integrable, and π^μ, respectively, reduce to the generalized velocity and the generalized coordinate q^μ on the Riemann space [Q].

Utilizing Eqs. (9) and (11), we can verify

(19)

Substituting Eq. (19) into Eqs. (16)–(18) yields

(20)

(21)

(22)

which are the motion differential equations on the Riemann-Cartan configuration space of the dynamical system that is only subjected to the first-order linear homogeneous scleronomous ideal differential constraint.

Define

(23)

Then, we, respectively, regard , , and F_μ as the quasi-velocity, quasi-momentum, and quasi-force on the Riemann-Cartan space, which are the generalized concept of the velocity, momentum, and active force, respectively. Then, Eqs. (20)–(22) are the quasi-Newton law, the quasi-momentum theorem, and the second Lagrange equation on the Riemann-Cartan space. It is easy to verify that if the constraints in Eq. (8) are integrable, Eq. (22) will be reduced to the second Lagrange equation of holonomic constrained system, i.e.,

(24)

From the above, we can see that the nonholonomic mapping theory gives a new way to study the first-order linear homogenous scleronomous nonholonomic system. By introducing a proper first-order linear nonholonomic mapping which is equal to the definition of a set of suitable quasi-velocities, the first-order linear homogenous scleronomous nonholonomic constrained system can be reduced to an unconstrained system on its low-dimensional Riemann-Cartan configuration space [Π]. We only need to compute the metric and connection of the RiemannCartan configuration space [Π] by the first-order linear nonholonomic mapping, and put the metric and connection into the quasi-Newton law or the quasi-momentum theorem. Then, the differential equations of motion of the nonholonomic constrained system can be obtained on its Riemann-Cartan configuration space [Π]. If the nonholonomic mapping is suitable, the differential equations of motion on its Riemann-Cartan configuration space will be simpler than the equations on its Riemann configuration space, such that the research of the first-order linear nonholonomic constrained problems may be simplified.

Furthermore, we study the motion of a holonomic constrained system in its generalized coordinate space which is a Riemann space with no torsion. To some special holonomic constrained systems, we can also simplify the problem by introducing a suitable nonholonomic mapping. A typical example is the problem of rotation of the rigid body with a fixed point. Using the first-order linear mapping method to study the motion of a rigid body in Example 3, we can see that the Euler equation is actually the quasi-momentum theorem of the rigid body on its Riemann-Cartan configuration space, which is obviously simpler than the differential equations of motion on its Riemann configuration space.

3 Illustrative examples

Example 1   We study the motion of a particle with unit mass. The equation of constraint is

(25)

The active force is zero.

This is a problem that can be solved by the traditional method of analytical mechanics. The Lagrange equations with a multiplier λ of the particle are

(26)

Combining the constraint equation and Eq. (26), we can obtain the multiplier λ as follows:

(27)

Substituting Eq. (27) into Eq. (26) yields

(28)

The differential equations of motion in Eq. (28) can be solved directly. From the second equation, we can get

(29)

Substituting Eq. (29) into the third equation yields

(30)

Substituting Eqs. (29) and (30) into the first equation yields

(31)

Therefore, the solution of the differential equations in Eq. (28) is

(32)

Since the constants in Eq. (32) must satisfy the constraint equation, we substitute Eq. (32) into the constraint equation. Then, we have

(33)

Therefore, the motion equations of the particle are

(34)

Next, we apply the nonholonomic mapping theory to solve this problem.

We introduce the following nonholonomic mapping:

(35)

Then, from Eqs. (12) and (13), the metric and connection of the Riemann-Cartan space [Π] can be obtained as follows:

(36)

(37)

Obviously, the torsion of space [Π] is not zero. This means that and are two quasivelocities of the particle. Substituting the metric, connection, and active force of the particle into Eq. (20) or Eq. (21), we can get the differential equations of motion on its Riemann-Cartan configuration space [Π] as follows:

(38)

Combining Eqs. (38) and (35), we get

(39)

(40)

It can be seen that the solution (40) is equivalent to the solution (34). The correction of the solution (40) can also be verified by substituting Eq. (40) directly into Eq. (28) and the constraint equation.

Example 2   We study the motion of a particle with unit mass. The constraint equation is

(41)

Three components of the active force are

(42)

At first, we try to solve this problem by the traditional method of analytical mechanics.

The Routh equations with a multiplier λ of the particle are^[23]

(43)

Combining the equation of constraint and Eq. (43), we can obtain the multiplier λ as follows:

(44)

Substituting Eq. (44) into Eq. (43), we can get

(45)

The differential equation of motion in Eq. (45) is on the three-dimensional Euclidean configuration space of the particle, includes three differential equations, and is too complex to be solved directly.

Next, we use the nonholonomic mapping theory to solve this problem.

We introduce the following nonholonomic mapping:

(46)

Then, from Eqs. (12) and (13), the metric and connection of the Riemann-Cartan space [Π] can be obtained as follows:

(47)

(48)

Obviously, the torsion of space [Π] is not zero. This means that and are two quasivelocities of the particle. Substituting the metric, connection, and active force of the particle into Eq. (20) or Eq. (21), we can get the differential equations of motion on its Riemann-Cartan configuration space [Π] as follows:

(49)

Obviously, Eq. (49) is simple, and can be easily solved.

Combining Eqs. (49) and (46), we get

(50)

(51)

Substituting Eq. (51) into Eq. (45) and the constraint equation, we can directly verify the correction of the solution.

Example 3   We study the motion of a rigid body consisting of three particles with the masses m₁, m₂, and m₃. The coordinates of these particles in the body coordinate system, respectively, are (1, 0, 0), (0, 1, 0), and (0, 0, 1). The forces on the three particles, respectively, are , , and . The rigid body rotates around the origin of the body coordinate system.

It is easy to get the velocities of the three particles as follows:

(52)

where

(53)

w_α is the coordinate of the Euler angle space [W], i.e.,

(54)

where ϕ, θ, and ψ are, respectively, the angle of precession, the angle of nutation, and the angle of rotation. The forces on the three particles can be denoted as follows:

(55)

The flat configuration space of the three particles is [X], whose coordinates are xⁱ defined in Eq. (52). The first-order linear mapping from configuration space [X] to the Euler angle space [W] is

(56)

where

(57)

According to the nonholonomic mapping theory, the metric and the connection of the flat Euclidean configuration space [X] are

(58)

and all Γ_ij^k = 0. We can obtain the metric and connection of the Euler angle space [W] as follows:

Obviously, the torsion of the Euler angle space [W] is zero. We can obtain the scalar curvature as follows:

This means that the Euler angle space [W] is a Riemann space with curvature and without torsion. The Euler angles is a set of generalized coordinates of the rigid body. Substituting the metric and connection of the Euler angle space [W] into Eq. (20) or (21), We can obtain the differential equations of motion of the rigid body on its Riemann configuration space [W]. Obviously, the differential equations of motion are very complex.

For simplification, we define the following nonholonomic mapping:

(59)

where

(60)

Substituting the mapping (60) into the mapping (57), we can obtain a nonholonomic mapping from the flat configuration space [X] to a Riemann-Cartan configuration space [Π], i.e.,

(61)

where

(62)

According to Eqs. (11)–(13), we can obtain the metric and the connection of the RiemannCartan configuration space [Π] as follows:

(63)

(64)

Substituting Eqs. (61), (63), and (64) into Eq. (20) or Eq. (21), we can obtain

(65)

If we define that the quasi-velocity is the angular velocity of a rigid body, Eq. (65) is exactly the following Euler equation:

(66)

Therefore, the Euler equation of a rigid body is actually the differential equations of motion on its Riemann-Cartan configuration space [Π], and is obviously simpler than the differential equations of motion on its Riemann configuration space [W]. The rigid-bodies with a fixed point are typical holonomic systems, which can be simplified by a suitable nonholonomic mapping.

4 Discussion

(ⅰ) Equation (19) illustrates that a first-order linear homogenous scleronomous nonholonomic system can be regarded as an unconstrained system on its Riemann-Cartan configuration space [Π], which is constructed by the mapping (11). Equations (20) and (21) illustrate that, when a first-order linear homogenous scleronomous nonholonomic system is not subject to active forces, its trajectory in its Riemann-Cartan configuration space [Π] is autoparallel. This is consistent with the principle of inertia.

(ⅱ) From Eq. (21), we can obtain

(67)

Comparing Eq. (67) with Eq. (66), we can observe that the angular velocity, angular momentum, and moment of the rigid body are, respectively, the quasi-velocity, quasi-momentum, and quasiforce, which are defined on a Riemann-Cartan configuration space [Π] of the rigid body. In Eq. (66), the cross-product between an angular velocity and an angular momentum is defined by the connection of its Riemann-Cartan configuration space.

(ⅲ) Because the rotational inertia matrix of any rigid body can be diagonalized by choosing a suitable coordinate system, any rigid body is equivalent to a three-particle-rigid-body model. Therefore, although we study a special three-particle-rigid-body model in Example 3, the conclusions of Example 3 can be applied to all rotations of rigid bodies with a fixed point.

(ⅳ) The three examples given in this paper illustrate that, an appropriate quasi-velocity space, which is defined by a suitable nonholonomic mapping, may simplify the study of a constrained system. For a constrained system, actually, we can define infinitely different sets of quasi-velocities through different nonholonomic mappings similar to Eq. (11). Therefore, how to select suitable quasi-velocities to simplify the study of constrained systems is a problem which is worth of being studied.

5 Conclusions

Using the first-order linear homogeneous nonholonomic mapping and the concepts of the velocity, we extend the momentum and active force of a particle to the quasi-velocity, quasimomentum, and quasi-force on the Riemann-Cartan configuration space of constrained systems. The differential equations of motion of a constrained system on its Riemann-Cartan configuration space, e.g., the quasi-Newton law, the quasi-momentum theorem, and the second Lagrange equation, can all be obtained. By introducing a suitable nonholonomic mapping, we can compute the metric and connection of the Riemann-Cartan configuration space of a first-order linear homogenous nonholonomic system or a holonomic system, and then directly obtain the differential equations of motion on its Riemann-Cartan configuration space according to the quasi-Newton law or the quasi-momentum theorem. By this way, a first-order linear homogenous nonholonomic system can be reduced to an unconstrained system on its low-dimensional Riemann-Cartan configuration space, from which the nonholonomic problem may be simplified. At the same time, in the study of some holonomic systems, we can also simplify the differential equations of motion by choosing appropriate quasi-velocities.