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Boltzmann-Hamel equation using quasi-velocities as variable quantities instead of generalized-velocities,is an extending form of the classical Lagrange equation.It is widely used for establishing the motion equations in constrained mechanical systems because of its unique structure.The classical method to solve Boltzmann-Hamel equation includes two steps.The first step is to substitute the relationship between the quasi-velocities and generalized-velocities into the equation to establish the second order equation relating to generalized-coordinates.The second step is to search for the analytical solutions using the method of separating variables or the method of Lie groups.However this method is not very effective in practice.In fact,the majority of studies only focus on the similarity between the quasi-coordinate form and the linear non-holonomic constraint form,without considering the effects of the selection of quasi-coordinates on the Boltzmann-Hamel equation.Because the quasi-coordinates in Boltzmann-Hamel equation can be selected freely,the problem of simplifying the Boltzmann-Hamel equation in holonomic system by choosing the appropriate quasi-coordinates is studied in this paper.Using the method of geometrodynamic analysis,the relationship between quasi-coordinates in the time-invariant configuration space and frame field is indicated based on the frame field theory of manifolds.The Boltzmann-Hamel equation in holonomic system is then derived from the tangle of geometric invariance.It differs from the ordinary methods,such as the action principle or d'Alembert's method.It is demonstrated that Boltzmann-Hamel equation can be simplified into an integrable form in homogenous configuration space with zero generalized-force or zero curvature configuration space with non-zero generalized-force.The process of simplifying the equation is provided in detail and the feasibility of this method is verified through two examples.The result in this paper reveals the close link between the intrinsic curvature of the time-invariant configuration space and the structure of Boltzmann-Hamel equation.The simplest form of Boltzmann-Hamel equation under the generalized-coordinate bases field (Lagrange equation) corresponds to the configuration space of zero curvature,and the simplest form of Boltzmann-Hamel equation under the frame field corresponds to the homogenous configuration space (more often,constant curvature space).For the complex motion equations,it should be transformed first into Boltzmann-Hamel equation,then the intrinsic curvature of the time-invariant configuration space will be calculated.If the conditions mentioned in this paper are satisfied, the Boltzmann-Hamel equation can be simplified into the simplest form by choosing appropriate quasi-coordinates,from which,the analytical solutions can be obtained,furthermore,this frame field derived by the appropriate quasi-coordinates can be used as a tool to study the symmetry and the conserved quantity of this holonomic mechanical system.The results in this paper provide a new way to search for the analytical solution of motion equations.
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Keywords:
- Boltzmann-Hamel equation /
- frame field theory /
- quasi-coordinates
[1] Boltzmann L 1902 Sitz. Math. Natur. Akad. Wiss. B 11 1603
[2] Hamel G 1904 Math. Phys. 50 1
[3] Hamel G 1938 Art. Sitz. Math. Ges. 37 4
[4] Tang C L, Shi R C 1989 J. Beijing. Inst. Tech. 9 35 (in Chinese) [唐传龙, 史荣昌 1989 北京理工大学学报 9 35]
[5] Qiu R 1997 Appl. Math. Mech. 18 1033 (in Chinese) [邱荣 1997 应用数学和力学 18 1033]
[6] Lü Z Q 1994 Jiangxi Sci. 12 195 (in Chinese) [吕哲勤 1994 江西科学 12 195]
[7] Zhou R L, Chen L Q 1993 J. Anshan. Ins. I. S. Tech. 16 46 (in Chinese) [周瑞礼, 陈立群 1993 鞍山钢铁学院学报 16 46]
[8] Zhang J F, Zhang H Z 1990 J. Zhejiang Norm. Univ. (Nat. Sci. Ed.) 13 61 (in Chinese) [张解放, 张洪忠 1990 浙江师范大学学报(自然科学版) 13 61]
[9] Zhang Y, Wu R H, Mei F X 1999 Shanghai J. Mech. 20 196 (in Chinese) [张毅, 吴润衡, 梅凤翔 1999 上海力学 20 196]
[10] Mei F X 1985 The Foundations of Mechanics of Nonholonomic System (Beijing: Beijing Institute of Technology Press) pp87-89 (in Chinese) [梅凤翔 1985 非完整系统力学基础 (北京: 北京工业学院出版社) 第87–89页]
[11] Zhang J F 1990 Huanghuai. J. 6 13 (in Chinese) [张解放 1990 黄淮学刊 6 13]
[12] Fu J L, Liu R W, Mei F X 1998 J. Beijing Inst. Tech. 7 215
[13] Fu J L, Liu R W 2000 Acta Math. Sci. 20 63 (in Chinese) [傅景礼, 刘荣万 2000 数学 20 63]
[14] Fu J L, Chen L Q 2004 The Progress of Research for Mathematics Mechanics Physics and High New Technology (Vol. 2004 (10)) (Chengdu: Southwest Jiaotong University Press) pp124-132 (in Chinese) [傅景礼, 陈立群 2004 数学·力学·物理学·高新技术研究进展 2004 (10) 卷 (成都:西南交通大学出版社) 第124–132页]
[15] Xu X J, Mei F X 2005 Acta Phys. Sin. 54 5521 (in Chinese) [许学军, 梅凤翔 2005 54 5521]
[16] Xue Y, Liu Y Z, Chen L Q 2006 Acta Phys. Sin. 55 3845 (in Chinese) [薛纭, 刘延柱, 陈立群 2006 55 3845]
[17] Zhang Q, Liu Z B, Cai Y 2008 Chin. J. Aeronaut. 21 471 (in Chinese) [战强, 刘增波, 蔡尧 2008 中国航空学报 21 471]
[18] Xie J F, Pang S, Zou J T, Li G F 2012 Acta Phys. Sin. 61 230201 (in Chinese) [谢加芳, 庞硕, 邹杰涛, 李国富 2012 61 230201]
[19] Jarzebowska E M 2015 Selected Papers from CSNDD Agadir, Morocco, May 21-23, 2014 p167
[20] Arnold V I (translated by Qi M Y) 2006 Mathematical Methods of Classical Mechanics (4th Ed.) (Beijing: Higher Education Press) pp59-69 (in Chinese) [阿诺尔德 著 (齐民友 译) 2006 经典力学的数学方法(第四版) (北京:高等教育出版社) 第59–69页]
[21] Chern S S, Chen W H 2001 Lectures on Differential Geometry (2nd Ed.) (Beijing: Peking University Press) pp30-38 (in Chinese) [陈省身, 陈维桓 2001 微分几何讲义(第二版) (北京:北京大学出版社) 第30–38页]
[22] Landau L D, Lifshitz E M (translated by Lu X, Ren L, Yuan B N) 2012 The Classical Theory of Fields (8th Ed.) (Beijing: Higher Education Press) p426 (in Chinese) [朗道, 栗弗席兹 (鲁欣, 任朗, 袁炳南 译) 2012 场论(第八版) (北京:高等教育出版社) 第426页]
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[1] Boltzmann L 1902 Sitz. Math. Natur. Akad. Wiss. B 11 1603
[2] Hamel G 1904 Math. Phys. 50 1
[3] Hamel G 1938 Art. Sitz. Math. Ges. 37 4
[4] Tang C L, Shi R C 1989 J. Beijing. Inst. Tech. 9 35 (in Chinese) [唐传龙, 史荣昌 1989 北京理工大学学报 9 35]
[5] Qiu R 1997 Appl. Math. Mech. 18 1033 (in Chinese) [邱荣 1997 应用数学和力学 18 1033]
[6] Lü Z Q 1994 Jiangxi Sci. 12 195 (in Chinese) [吕哲勤 1994 江西科学 12 195]
[7] Zhou R L, Chen L Q 1993 J. Anshan. Ins. I. S. Tech. 16 46 (in Chinese) [周瑞礼, 陈立群 1993 鞍山钢铁学院学报 16 46]
[8] Zhang J F, Zhang H Z 1990 J. Zhejiang Norm. Univ. (Nat. Sci. Ed.) 13 61 (in Chinese) [张解放, 张洪忠 1990 浙江师范大学学报(自然科学版) 13 61]
[9] Zhang Y, Wu R H, Mei F X 1999 Shanghai J. Mech. 20 196 (in Chinese) [张毅, 吴润衡, 梅凤翔 1999 上海力学 20 196]
[10] Mei F X 1985 The Foundations of Mechanics of Nonholonomic System (Beijing: Beijing Institute of Technology Press) pp87-89 (in Chinese) [梅凤翔 1985 非完整系统力学基础 (北京: 北京工业学院出版社) 第87–89页]
[11] Zhang J F 1990 Huanghuai. J. 6 13 (in Chinese) [张解放 1990 黄淮学刊 6 13]
[12] Fu J L, Liu R W, Mei F X 1998 J. Beijing Inst. Tech. 7 215
[13] Fu J L, Liu R W 2000 Acta Math. Sci. 20 63 (in Chinese) [傅景礼, 刘荣万 2000 数学 20 63]
[14] Fu J L, Chen L Q 2004 The Progress of Research for Mathematics Mechanics Physics and High New Technology (Vol. 2004 (10)) (Chengdu: Southwest Jiaotong University Press) pp124-132 (in Chinese) [傅景礼, 陈立群 2004 数学·力学·物理学·高新技术研究进展 2004 (10) 卷 (成都:西南交通大学出版社) 第124–132页]
[15] Xu X J, Mei F X 2005 Acta Phys. Sin. 54 5521 (in Chinese) [许学军, 梅凤翔 2005 54 5521]
[16] Xue Y, Liu Y Z, Chen L Q 2006 Acta Phys. Sin. 55 3845 (in Chinese) [薛纭, 刘延柱, 陈立群 2006 55 3845]
[17] Zhang Q, Liu Z B, Cai Y 2008 Chin. J. Aeronaut. 21 471 (in Chinese) [战强, 刘增波, 蔡尧 2008 中国航空学报 21 471]
[18] Xie J F, Pang S, Zou J T, Li G F 2012 Acta Phys. Sin. 61 230201 (in Chinese) [谢加芳, 庞硕, 邹杰涛, 李国富 2012 61 230201]
[19] Jarzebowska E M 2015 Selected Papers from CSNDD Agadir, Morocco, May 21-23, 2014 p167
[20] Arnold V I (translated by Qi M Y) 2006 Mathematical Methods of Classical Mechanics (4th Ed.) (Beijing: Higher Education Press) pp59-69 (in Chinese) [阿诺尔德 著 (齐民友 译) 2006 经典力学的数学方法(第四版) (北京:高等教育出版社) 第59–69页]
[21] Chern S S, Chen W H 2001 Lectures on Differential Geometry (2nd Ed.) (Beijing: Peking University Press) pp30-38 (in Chinese) [陈省身, 陈维桓 2001 微分几何讲义(第二版) (北京:北京大学出版社) 第30–38页]
[22] Landau L D, Lifshitz E M (translated by Lu X, Ren L, Yuan B N) 2012 The Classical Theory of Fields (8th Ed.) (Beijing: Higher Education Press) p426 (in Chinese) [朗道, 栗弗席兹 (鲁欣, 任朗, 袁炳南 译) 2012 场论(第八版) (北京:高等教育出版社) 第426页]
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