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由 一星 2014-05-27, 02:54
弱同位旋与弱超荷[编辑]
主条目：弱超荷
标准模型中的左手费米子[11]
弱同位旋（T3）是所有粒子的一种性质（量子数），决定粒子在弱相互作用下该如何反应[12]。对于弱相互作用来说，弱同位旋的作用跟电磁中的电荷一样，也跟强相互作用中的色荷一样。其他基本粒子的弱同位旋为±1⁄2。例如，上型夸克（上、魅及顶）的T3 = +1⁄2，它们总是会变换成下型夸克（下、奇及底），而它们的T3 = −1⁄2，反之亦然。另一方面，夸克在弱衰变后，T3永远跟衰变前不一样。就像电荷一样，弱同位旋的两种不同值，大小一样，正负相反。
通过弱相互作用衰变的π+
介子
弱同位旋是守恒的：反应产物的弱同位旋总和，等于反应物的弱同位旋总和。例如，一左手π+
介子，弱同位旋为+1，一般衰变成一ν
μ（+1⁄2）及一μ+
（+1⁄2，因为是右手反粒子）[10]。
在电弱理论的开发后，有一个新的性质，叫弱超荷。它的数值由粒子的电荷及弱同位旋决定：
，
其中YW为某一种粒子的弱超荷，Q为电荷（基本电荷单位）及T3为弱同位旋。弱超荷是U(1)部份生成元的规范群[13]。
对称破缺[编辑]
长久以来，人们以为自然定律在镜像反射后会维持不变，镜像反射等同把所有空间轴反转。也就是说在镜中看实验，跟把实验设备转成镜像方向后看实验，两者的实验结果会是一样的。这条所谓的定律叫宇称守恒，经典引力、电磁及强相互作用都遵守这条定律；它被假定为一条万物通用的定律[14]。然而，在1950年代中期，杨振宁与李政道提出弱相互作用可能会破坏这一条定律[15]。吴健雄与同事于1957年发现了弱相互作用的宇称不守恒[16]，为杨振宁与李政道带来了1957年的诺贝尔物理学奖[17]。
尽管以前用费米理论就能描述弱相互作用，但是在发现宇称不守恒及重整化理论后，弱相互作用需要一种新的描述手法。在1957年罗伯特·马沙克（Robert Marshak）与乔治·苏达尚（George Sudarshan）[18]，及稍后理查德·费曼与默里·盖尔曼[19]，提出了弱相互作用的V−A（矢量V减轴矢量A或左手性）拉格朗日量。在这套理论中，弱相互作用只作用于左手粒子（或右手反粒子）。由于左手粒子的镜像反射是右手粒子，所以这解释了宇称的最大破坏。有趣的是，由于V−A开发时还未有发现Z玻色子，所以理论并没有包括进入中性流相互作用的右手场。
然而，该理论允许复合对称CP守恒。CP由两部份组成，宇称P（左右互换）及电荷共轭C（把粒子换成反粒子）。1964年的一个发现完全出乎物理学家的意料，詹姆斯·克罗宁与瓦尔·菲奇以K介子衰变，为弱相用作用下CP对称破缺提供了明确的证据，二人因此获得1980年的诺贝尔物理学奖[20]。小林诚与益川敏英于1972年指出，弱相互作用的CP破坏，需要两代以上的粒子[21]，因此这项发现实际上预测了第三代粒子的存在，而这个预测在2008年为他们带来了半个诺贝尔物理学奖[22]。跟宇称不守恒不一样，CP破坏的发生概率并不高，但是它仍是解答宇宙间物质反物质失衡的一大关键；它因此成了安德烈·萨哈罗夫的重子产生过程三条件之一[23]。
相互作用类型[编辑]
弱相互作用共有两种。第一种叫“载荷流相互作用”，因为负责传递它的粒子带电荷（W+
或W−
），β衰变就是由它所引起的。第二种叫“中性流相互作用”，因为负责传递它的粒子，Z玻色子，是中性的（不带电荷）。
载荷流相互作用[编辑]
在其中一种载荷流相互作用中，一带电荷的轻子（例如电子或μ子，电荷为−1）可以吸收一W+
玻色子（电荷为+1），然后转化成对应的中微子（电荷为0），而中微子（电子、μ及τ）的类型（代）跟相互作用前的轻子一致，例如：
同样地，一下型夸克（电荷为−1⁄3）可以通过发射一W−
玻色子，或吸收一W+
玻色子，来转化成一上型夸克（电荷为+2⁄3）。更准确地，下型夸克变成了上型夸克的量子叠加态：也就是说，它有着转化成三种上型夸克中任何一种的可能性，可能性的大小由CKM矩阵所描述。相反地，一上型夸克可以发射一W+
玻色子，或吸收一W−
玻色子，然后转化成一下型夸克：
由于W玻色子很不稳定，所以它寿命很短，很快就发生衰变。例如：
W玻色子可以衰变成其他产物，可能性不一[24]。
在中子所谓的β衰变中（见上图），中子内的一下夸克，发射出一虚W−
玻色子，并因此转化成一上夸克，中子亦因此转化成质子。由于过程中的能量（即下夸克与上夸克间的质量差），W−
只能转化成一电子及一反电中微子[25]。在夸克的层次，过程可由下式所述：
中性流相互作用[编辑]
在中性流相互作用中，一夸克或一轻子（例如一电子或μ子）发射或吸收一中性Z玻色子。例如：
跟W玻色子一样，Z玻色子也会迅速衰变[24]，例如：
电弱理论[编辑]
主条目：弱电相互作用
在粒子物理学的标准模型描述中，弱相互作用与电磁相互作用是同一种相互作用的不同方面，叫弱电相互作用，这套理论在1968年发表，开发者为谢尔登·格拉肖[26]、阿卜杜勒·萨拉姆[27]与史蒂文·温伯格[28]。他们的研究在1979年获得了诺贝尔物理学奖的肯定[29]。希格斯机制解释了三种大质量玻色子（弱相互作用的三种载体）的存在，还有电磁相互作用的无质量光子[30]。
根据电弱理论，在能量非常高的时候，宇宙共有四种无质量的规范玻色子场，它们跟光子类似，还有一个复矢量希格斯场双重态。然而在能量低的时候，规范对称会出现自发破缺，变成电磁相互作用的U(1)对称（其中一个希格斯场有了真空期望值）。虽然这种对称破缺会产生三种无质量玻色子，但是它们会与三股光子类场融合，这样希格斯机制会为它们带来质量。这三股场就成为了弱相互作用的W+
、W−
及Z玻色子，而第四股规范场则继续保持无质量，也就是电磁相互作用的光子[30]。
虽然这套理论作出好几个预测，包括在Z及W玻色子发现前预测到它们的质量，但是希格斯玻色子本身仍未被发现。欧洲核子研究组织辖下的大型强子对撞机，它其中一项主要任务，就是要生产出希格斯玻色子[31]。 2013年3月14日，欧洲核子研究组织发布新闻稿，正式宣布探测到新的粒子即希格斯玻色子。
主条目：弱超荷
| 电子 |  |  | μ子 |  | | τ子 |  | |
| 电中微子 |  |  | μ中微子 |  | | τ中微子 |  | |
| 上夸克 |  | | 粲夸克 |  | | 顶夸克 |  | |
| 下夸克 |  | | 奇夸克 |  | | 底夸克 |  | |
| 所有左手反粒子的弱同位旋均为零。右手反粒子的弱同位旋与左手粒子相反。 | ||||||||
弱同位旋（T3）是所有粒子的一种性质（量子数），决定粒子在弱相互作用下该如何反应[12]。对于弱相互作用来说，弱同位旋的作用跟电磁中的电荷一样，也跟强相互作用中的色荷一样。其他基本粒子的弱同位旋为±1⁄2。例如，上型夸克（上、魅及顶）的T3 = +1⁄2，它们总是会变换成下型夸克（下、奇及底），而它们的T3 = −1⁄2，反之亦然。另一方面，夸克在弱衰变后，T3永远跟衰变前不一样。就像电荷一样，弱同位旋的两种不同值，大小一样，正负相反。
通过弱相互作用衰变的π+
介子
弱同位旋是守恒的：反应产物的弱同位旋总和，等于反应物的弱同位旋总和。例如，一左手π+
介子，弱同位旋为+1，一般衰变成一ν
μ（+1⁄2）及一μ+
（+1⁄2，因为是右手反粒子）[10]。
在电弱理论的开发后，有一个新的性质，叫弱超荷。它的数值由粒子的电荷及弱同位旋决定：
，其中YW为某一种粒子的弱超荷，Q为电荷（基本电荷单位）及T3为弱同位旋。弱超荷是U(1)部份生成元的规范群[13]。
对称破缺[编辑]
长久以来，人们以为自然定律在镜像反射后会维持不变，镜像反射等同把所有空间轴反转。也就是说在镜中看实验，跟把实验设备转成镜像方向后看实验，两者的实验结果会是一样的。这条所谓的定律叫宇称守恒，经典引力、电磁及强相互作用都遵守这条定律；它被假定为一条万物通用的定律[14]。然而，在1950年代中期，杨振宁与李政道提出弱相互作用可能会破坏这一条定律[15]。吴健雄与同事于1957年发现了弱相互作用的宇称不守恒[16]，为杨振宁与李政道带来了1957年的诺贝尔物理学奖[17]。
尽管以前用费米理论就能描述弱相互作用，但是在发现宇称不守恒及重整化理论后，弱相互作用需要一种新的描述手法。在1957年罗伯特·马沙克（Robert Marshak）与乔治·苏达尚（George Sudarshan）[18]，及稍后理查德·费曼与默里·盖尔曼[19]，提出了弱相互作用的V−A（矢量V减轴矢量A或左手性）拉格朗日量。在这套理论中，弱相互作用只作用于左手粒子（或右手反粒子）。由于左手粒子的镜像反射是右手粒子，所以这解释了宇称的最大破坏。有趣的是，由于V−A开发时还未有发现Z玻色子，所以理论并没有包括进入中性流相互作用的右手场。
然而，该理论允许复合对称CP守恒。CP由两部份组成，宇称P（左右互换）及电荷共轭C（把粒子换成反粒子）。1964年的一个发现完全出乎物理学家的意料，詹姆斯·克罗宁与瓦尔·菲奇以K介子衰变，为弱相用作用下CP对称破缺提供了明确的证据，二人因此获得1980年的诺贝尔物理学奖[20]。小林诚与益川敏英于1972年指出，弱相互作用的CP破坏，需要两代以上的粒子[21]，因此这项发现实际上预测了第三代粒子的存在，而这个预测在2008年为他们带来了半个诺贝尔物理学奖[22]。跟宇称不守恒不一样，CP破坏的发生概率并不高，但是它仍是解答宇宙间物质反物质失衡的一大关键；它因此成了安德烈·萨哈罗夫的重子产生过程三条件之一[23]。
相互作用类型[编辑]
弱相互作用共有两种。第一种叫“载荷流相互作用”，因为负责传递它的粒子带电荷（W+
或W−
），β衰变就是由它所引起的。第二种叫“中性流相互作用”，因为负责传递它的粒子，Z玻色子，是中性的（不带电荷）。
载荷流相互作用[编辑]
在其中一种载荷流相互作用中，一带电荷的轻子（例如电子或μ子，电荷为−1）可以吸收一W+
玻色子（电荷为+1），然后转化成对应的中微子（电荷为0），而中微子（电子、μ及τ）的类型（代）跟相互作用前的轻子一致，例如：
同样地，一下型夸克（电荷为−1⁄3）可以通过发射一W−
玻色子，或吸收一W+
玻色子，来转化成一上型夸克（电荷为+2⁄3）。更准确地，下型夸克变成了上型夸克的量子叠加态：也就是说，它有着转化成三种上型夸克中任何一种的可能性，可能性的大小由CKM矩阵所描述。相反地，一上型夸克可以发射一W+
玻色子，或吸收一W−
玻色子，然后转化成一下型夸克：
由于W玻色子很不稳定，所以它寿命很短，很快就发生衰变。例如：
W玻色子可以衰变成其他产物，可能性不一[24]。
在中子所谓的β衰变中（见上图），中子内的一下夸克，发射出一虚W−
玻色子，并因此转化成一上夸克，中子亦因此转化成质子。由于过程中的能量（即下夸克与上夸克间的质量差），W−
只能转化成一电子及一反电中微子[25]。在夸克的层次，过程可由下式所述：
中性流相互作用[编辑]
在中性流相互作用中，一夸克或一轻子（例如一电子或μ子）发射或吸收一中性Z玻色子。例如：
跟W玻色子一样，Z玻色子也会迅速衰变[24]，例如：
电弱理论[编辑]
主条目：弱电相互作用
在粒子物理学的标准模型描述中，弱相互作用与电磁相互作用是同一种相互作用的不同方面，叫弱电相互作用，这套理论在1968年发表，开发者为谢尔登·格拉肖[26]、阿卜杜勒·萨拉姆[27]与史蒂文·温伯格[28]。他们的研究在1979年获得了诺贝尔物理学奖的肯定[29]。希格斯机制解释了三种大质量玻色子（弱相互作用的三种载体）的存在，还有电磁相互作用的无质量光子[30]。
根据电弱理论，在能量非常高的时候，宇宙共有四种无质量的规范玻色子场，它们跟光子类似，还有一个复矢量希格斯场双重态。然而在能量低的时候，规范对称会出现自发破缺，变成电磁相互作用的U(1)对称（其中一个希格斯场有了真空期望值）。虽然这种对称破缺会产生三种无质量玻色子，但是它们会与三股光子类场融合，这样希格斯机制会为它们带来质量。这三股场就成为了弱相互作用的W+
、W−
及Z玻色子，而第四股规范场则继续保持无质量，也就是电磁相互作用的光子[30]。
虽然这套理论作出好几个预测，包括在Z及W玻色子发现前预测到它们的质量，但是希格斯玻色子本身仍未被发现。欧洲核子研究组织辖下的大型强子对撞机，它其中一项主要任务，就是要生产出希格斯玻色子[31]。 2013年3月14日，欧洲核子研究组织发布新闻稿，正式宣布探测到新的粒子即希格斯玻色子。
一星- 帖子数 : 3787
注册日期 : 13-08-07
回复： 万有理论
由 一星 2014-05-27, 02:56
参考资料[编辑]
大众书籍[编辑]
科学书籍[编辑]
注释[编辑]
大众书籍[编辑]
- Oerter, R. The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics. Plume. 2006.ISBN 9780132366786.
- Schumm, B.A. Deep Down Things: The Breathtaking Beauty of Particle Physics. Johns Hopkins University Press. 2004. ISBN 0-8018-7971-X.
科学书籍[编辑]
- Bromley, D.A. Gauge Theory of Weak Interactions. Springer. 2000.ISBN 3-540-67672-4.
- Coughlan, G.D.; Dodd, J.E.; Gripaios, B.M. The Ideas of Particle Physics: An Introduction for Scientists 3rd. Cambridge University Press. 2006. ISBN 978-0521677752.
- Cottingham, W.N.; Greenwood, D.A. An introduction to nuclear physics 2. Cambridge University Press. 2001. 30.ISBN 9780521657334.
- Griffiths, D.J. Introduction to Elementary Particles. John Wiley & Sons. 1987. ISBN 0-471-60386-4.
- Kane, G.L. Modern Elementary Particle Physics. Perseus Books. 1987.ISBN 0-201-11749-5.
- Perkins, D.H. Introduction to High Energy Physics. Cambridge University Press. 2000. ISBN 0-521-62196-8.
注释[编辑]
- ^ The Nobel Prize in Physics 1979: Press Release. NobelPrize.org. Nobel Media.[22 March 2011].
- ^ W.-M. Yao et al. (Particle Data Group). Review of Particle Physics: Quarks. Journal of Physics G. 2006, 33: 1.arXiv:astro-ph/0601168.Bibcode:2006JPhG...33....1Y.doi:10.1088/0954-3899/33/1/001.
- ^ Peter Watkins. Story of the W and Z. Cambridge: Cambridge University Press. 1986. 70.ISBN 9780521318754.
- ^ 4.0 4.1 Coupling Constants for the Fundamental Forces. HyperPhysics. Georgia State University. [2 March 2011].
- ^ 5.0 5.1 5.2 J. Christman. The Weak Interaction. Physnet. Michigan State University. 2001.
- ^ Electroweak. The Particle Adventure. Particle Data Group.[3 March 2011].
- ^ Walter Greiner; Berndt Müller.Gauge Theory of Weak Interactions. Springer. 2009. 2. ISBN 9783540878421.
- ^ Cottingham & Greenwood 2001，第29页
- ^ 9.0 9.1 Cottingham & Greenwood 2001，第28页
- ^ 10.0 10.1 Cottingham & Greenwood 2001，第30页
- ^ John C. Baez and John Huerta, The Algebra of Grand Unified Theories, Department of Mathematics, University of California. 2009 [7 March 2011]
- ^ Griffiths 1987，第344页
- ^ T-P Chang; L-F Li. Gauge Theory of Elementary Particle Physics. Oxford: Clarendon Press. 1984: 346.ISBN 0198519613.
- ^ Charles W. Carey. Lee, Tsung-Dao//American scientists. Facts on File Inc. 2006. 225.
- ^ C.N. Yang; T.D. Lee. Question of Parity Conservation in Weak Interactions. Physcial Review. 1956, 104: 254.doi:10.1103/PhysRev.104.254.
- ^ C.S. Wu; E. Ambler; R.W. Haywood; D.D. Hoppes; R.P. Hudson. Experimental Test of Parity Conservation in Beta Decay. Physcial Review. 1957,105: 1413.doi:10.1103/PhysRev.105.1413.
- ^ The Nobel Prize in Physics 1957. NobelPrize.org. Nobel Media. [26 February 2011].
- ^ E.C.G. Sudarshan; R.E. Marshak. Proceedings of the Padua-Venice Conference on Mesons and Recently Discovered Particles, September, 1957. Padua-Venice: Società Italiana di Fisica. 1958.
- ^ R.P. Feynman; M. Gell-Mann. Theory of the Fermi Interaction. Physical Review. 1958, 109: 193.doi:10.1103/PhysRev.109.193.
- ^ The Nobel Prize in Physics 1980. NobelPrize.org. Nobel Media. [26 February 2011].
- ^ M. Kobayashi, T. Maskawa. CP-Violation in the Renormalizable Theory of Weak Interaction. Progress of Theoretical Physics. 1973, 49(2): 652–657.Bibcode:1973PThPh..49..652K.doi:10.1143/PTP.49.652.
- ^ The Nobel Prize in Physics 1980. NobelPrize.org. Nobel Media. [17 March 2011].
- ^ Paul Langacker. Cp Violation and Cosmology//In Cecilia Jarlskog. CP violation. London, River Edge]: World Scientific Publishing Co. 1989, 2001. 552.
- ^ 24.0 24.1 K. Nakamura et al. (Particle Data Group). Gauge and Higgs Bosons. Journal of Physics G. 2010, 37.
- ^ K. Nakamura et al. (Particle Data Group). n. Journal of Physics G. 2010, 37: 7.
- ^ S.L. Glashow. Partial-symmetries of weak interactions. Nuclear Physics. 1961, 22: 579.doi:10.1016/0029-5582(61)90469-2.
- ^ A. Salam. Weak and electromagnetic interactions//N. Svartholm. Elementary Particle Theory, Proceedings of the 8th Nobel Symposium. Stockholm: Almquist and Wiksell. 1968.
- ^ S. Weinberg. A model of leptons. Physics Review Letters. 1967, 19: 1264.doi:10.1103/PhysRevLett.19.1264.
- ^ The Nobel Prize in Physics 1979. NobelPrize.org. Nobel Media. [26 February 2011].
- ^ 30.0 30.1 C. Amsler et al.(Particle Data Group). Review of Particle Physics – Higgs Bosons: Theory and Searches. Physics Letters B. 2008, 667: 1.Bibcode:2008PhLB..667....1P.doi:10.1016/j.physletb.2008.07.018.
- ^ Missing Higgs. European Organization for Nuclear Research. 2008 [1 March 2011].
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注册日期 : 13-08-07
回复： 万有理论
由 一星 2014-06-05, 03:27
In the setting of noncommutative geometry, physical states are primary and space-time is secondary.
Most physicists expect that the creation of the final theory of quantum gravity will dramatically change our knowledge about space and time.
*Compact Riemann sur faces correspond to string states; conformally equivalent Riemann sur faces represent the same string state.
In this modern terminology, the space of string states corresponds to Riemann’s
classical moduli space. If g denotes the genus of the string state, then the following
are met:
(i) If g ≥ 2, then the space of string states can be parametrized by 3g −3 complex
parameters.
(ii) If g = 0, then there exists a unique string state which corresponds to the
Riemann sphere.
(ii) If g = 1, then the string states correspond to tori. The space of string states is
in one-to-one correspondence to the subsetM1 of the complex upper half-plane
pictured . Therefore, the string states can be parameterized by one
complex parameter.
For genus g ≥ 1, the string state space is not a manifold, but has singularities.
The theory of Riemann sur faces and its generalizations has played a key role in the
development of modern algebraic geometry.
黎曼曲面[编辑]
函数
的黎曼曲面
[ltr]
数学上，特别是在复分析中，一个黎曼曲面是一个一维复流形。黎曼曲面可以被视为是一个复平面的变形版本：在每一点局部看来，他们就像一片复平面，但整体的拓扑可能极为不同。例如，他们可以看起来像球或是环，或者两个页面粘在一起。
黎曼曲面的精髓在于在曲面之间可以定义全纯函数。黎曼曲面现在被认为是研究这些函数的整体行为的自然选择，特别是像平方根和自然对数这样的多值函数。
每个黎曼曲面都是二维实解析流形（也就是曲面），但它有更多的结构（特别是一个复结构），因为全纯函数的无歧义的定义需要用到这些结构。一个实二维流形可以变成为一个黎曼曲面（通常有几种不同的方式）当且仅当它是可定向的。所以球和环有复结构，但是莫比乌斯带，克莱因瓶和射影平面没有。
黎曼曲面的几何性质是最妙的，它们也给与其它曲线，流形或簇上的推广提供了直观的理解和动力。黎曼-罗赫定理就是这种影响的最佳例子。[/ltr]
[ltr]
形式化定义[编辑]
令X为一个豪斯多夫空间。一个从开子集U⊂C到X的子集的同胚称为坐标卡。两个有重叠区域的坐标卡 f 和 g 称为相容的，如果映射f o g-1 和g o f-1 是在定义域上全纯的。若A一组相容的图，并且每个X中的x都在某个f的定义域中，则称A为一个图册'。当我们赋予X一个图册A,我们称（X,A）为一个黎曼曲面。如果知道有图册，我们简称X为黎曼曲面。
不同的图册可以在X上给出本质上相同的黎曼曲面结构；为避免这种模糊性，我们有时候要求X为极大的，也就是它不是任何一个更大的图集的子集。根据佐恩引理每个图集A包含于一个唯一的最大图集中。
例子[编辑][/ltr]
[ltr]
属性和更多的定义[编辑]
两个黎曼曲面M和N之间的 函数f : M → N称为全纯，如果对于M的图集中的每个图g和N的图集中的每个图h，映射h o f o g-1 在所有有定义的地方是全纯的(作为从C到C的函数) 。两个全纯函数的复合是全纯的。两个黎曼曲面M和N称为保角等价（或共形等价），如果存在一个双射的从M到N的全纯函数并且其逆也是全纯的（最后一个条件是自动满足的所以可以略去）。两个保角等价的黎曼曲面对于所有的实际应用来讲是完全相同的。
每个单连通的黎曼曲面和C或黎曼球C ∪ {∞}或开圆盘{z ∈ C : |z| < 1}保角等价。这个命题称为单值化定理。
每个连通黎曼曲面可以转成有常数曲率-1，0或1 的完备实黎曼流形。这个黎曼结构除了度量的缩放外是唯一。有曲率-1的黎曼曲面称为双曲的;开圆盘是个经典的例子。有曲率0的黎曼曲面称为抛物的；C是典型的抛物黎曼曲面。最后，有曲率+1的黎曼曲面称为椭圆的；黎曼球C ∪ {∞}是这样的一个例子.
对于每个闭抛物黎曼曲面，基本群同构于2阶格群，因而曲面可以构造为C/Γ,其中C是复平面而Γ 是格群。陪集的代表的集合叫做基本域。
类似的，对每个双曲黎曼曲面，基本群同构于富克斯群，因而曲面可以由富克斯模型H/Γ 构造，其中H是上半平面而Γ是富克斯群。H/Γ陪集的代表是自由正则集，可以作为度量基本多边形。
当一个双曲曲面是紧的，则曲面的总面积是
, 其中 g 是曲面的亏格；面积可由把高斯-博内定理应用到基本多边形的面积上来算出。
前面我们提到黎曼曲面，象所有复流形，象实流形一样可定向。因为复图f和g有变换函数h = f(g-1(z))，我们 可以认为h是从R2开集到R2的映射，在点z的雅可比矩阵也就是由乘以复数h'(z)的运算给出的实线性变换。但是，乘以复数α的行列式等于|α|^2, 所以h的雅可比阵有正的行列式值。所以，复图集是可定向图集。
历史[编辑]
黎曼最早开始研究黎曼曲面。黎曼曲面以他命名。
相关主题[编辑][/ltr]
[ltr]
参考[编辑][/ltr]
A Glance at Analytic S-Matrix Theory
In the 1950s and 1960s, physicists thoroughly studied scattering processes for elementary particles by using analytic continuation for the S-matrix and the Green’s functions. If these functions have a singularity at the complex energy
E = E0 + iΔE,
then there exists an elementary particle with rest mass m0 = E0/c2 and mean
lifetime Δt = /ΔE.
Most physicists expect that the creation of the final theory of quantum gravity will dramatically change our knowledge about space and time.
*Compact Riemann sur faces correspond to string states; conformally equivalent Riemann sur faces represent the same string state.
In this modern terminology, the space of string states corresponds to Riemann’s
classical moduli space. If g denotes the genus of the string state, then the following
are met:
(i) If g ≥ 2, then the space of string states can be parametrized by 3g −3 complex
parameters.
(ii) If g = 0, then there exists a unique string state which corresponds to the
Riemann sphere.
(ii) If g = 1, then the string states correspond to tori. The space of string states is
in one-to-one correspondence to the subsetM1 of the complex upper half-plane
pictured . Therefore, the string states can be parameterized by one
complex parameter.
For genus g ≥ 1, the string state space is not a manifold, but has singularities.
The theory of Riemann sur faces and its generalizations has played a key role in the
development of modern algebraic geometry.
黎曼曲面[编辑]
函数
的黎曼曲面[ltr]
数学上，特别是在复分析中，一个黎曼曲面是一个一维复流形。黎曼曲面可以被视为是一个复平面的变形版本：在每一点局部看来，他们就像一片复平面，但整体的拓扑可能极为不同。例如，他们可以看起来像球或是环，或者两个页面粘在一起。
黎曼曲面的精髓在于在曲面之间可以定义全纯函数。黎曼曲面现在被认为是研究这些函数的整体行为的自然选择，特别是像平方根和自然对数这样的多值函数。
每个黎曼曲面都是二维实解析流形（也就是曲面），但它有更多的结构（特别是一个复结构），因为全纯函数的无歧义的定义需要用到这些结构。一个实二维流形可以变成为一个黎曼曲面（通常有几种不同的方式）当且仅当它是可定向的。所以球和环有复结构，但是莫比乌斯带，克莱因瓶和射影平面没有。
黎曼曲面的几何性质是最妙的，它们也给与其它曲线，流形或簇上的推广提供了直观的理解和动力。黎曼-罗赫定理就是这种影响的最佳例子。[/ltr]
[ltr]
形式化定义[编辑]
令X为一个豪斯多夫空间。一个从开子集U⊂C到X的子集的同胚称为坐标卡。两个有重叠区域的坐标卡 f 和 g 称为相容的，如果映射f o g-1 和g o f-1 是在定义域上全纯的。若A一组相容的图，并且每个X中的x都在某个f的定义域中，则称A为一个图册'。当我们赋予X一个图册A,我们称（X,A）为一个黎曼曲面。如果知道有图册，我们简称X为黎曼曲面。
不同的图册可以在X上给出本质上相同的黎曼曲面结构；为避免这种模糊性，我们有时候要求X为极大的，也就是它不是任何一个更大的图集的子集。根据佐恩引理每个图集A包含于一个唯一的最大图集中。
例子[编辑][/ltr]
- 复平面C可能是最平凡的黎曼曲面了。映射f(z) = z (恒等映射)定义了C的一个图,而{f} 是C的一个图集. 映射g(z) = z* (共轭)映射也定义了C的一个图而{g}也是C的一个图集. 图f和g不相容，所以他们各自给了C一个黎曼曲面结构。事实上，给定黎曼曲面X及其图集A, 共轭图集B = {f* : f ∈ A} 总是不和A相容, 因此赋予X一个不同的黎曼曲面结构。
- 类似的，每个复平面的开子集可以自然的视为黎曼曲面。更一般的，每个黎曼曲面的开子集是一个黎曼曲面。
- 令S = C ∪ {∞} 并令f(z) = z 其中z 属于S \ {∞} 并且令g(z) = 1 / z 其中z属于S \ {0} 以及 定义1/∞为0. 则f 和g为图，它们相容，而{ f, g }是S图集, 使S成为黎曼曲面。这个特殊的曲面称为黎曼球因为它可以解释为把复平面裹在一个球上。不像复平面，它是一个紧空间。
- 紧黎曼曲面可以视为和定义在复数上的非奇异代数曲线等效。非紧黎曼曲面的重要例子由解析连续给出(见下面)
[ltr]
属性和更多的定义[编辑]
两个黎曼曲面M和N之间的 函数f : M → N称为全纯，如果对于M的图集中的每个图g和N的图集中的每个图h，映射h o f o g-1 在所有有定义的地方是全纯的(作为从C到C的函数) 。两个全纯函数的复合是全纯的。两个黎曼曲面M和N称为保角等价（或共形等价），如果存在一个双射的从M到N的全纯函数并且其逆也是全纯的（最后一个条件是自动满足的所以可以略去）。两个保角等价的黎曼曲面对于所有的实际应用来讲是完全相同的。
每个单连通的黎曼曲面和C或黎曼球C ∪ {∞}或开圆盘{z ∈ C : |z| < 1}保角等价。这个命题称为单值化定理。
每个连通黎曼曲面可以转成有常数曲率-1，0或1 的完备实黎曼流形。这个黎曼结构除了度量的缩放外是唯一。有曲率-1的黎曼曲面称为双曲的;开圆盘是个经典的例子。有曲率0的黎曼曲面称为抛物的；C是典型的抛物黎曼曲面。最后，有曲率+1的黎曼曲面称为椭圆的；黎曼球C ∪ {∞}是这样的一个例子.
对于每个闭抛物黎曼曲面，基本群同构于2阶格群，因而曲面可以构造为C/Γ,其中C是复平面而Γ 是格群。陪集的代表的集合叫做基本域。
类似的，对每个双曲黎曼曲面，基本群同构于富克斯群，因而曲面可以由富克斯模型H/Γ 构造，其中H是上半平面而Γ是富克斯群。H/Γ陪集的代表是自由正则集，可以作为度量基本多边形。
当一个双曲曲面是紧的，则曲面的总面积是
, 其中 g 是曲面的亏格；面积可由把高斯-博内定理应用到基本多边形的面积上来算出。前面我们提到黎曼曲面，象所有复流形，象实流形一样可定向。因为复图f和g有变换函数h = f(g-1(z))，我们 可以认为h是从R2开集到R2的映射，在点z的雅可比矩阵也就是由乘以复数h'(z)的运算给出的实线性变换。但是，乘以复数α的行列式等于|α|^2, 所以h的雅可比阵有正的行列式值。所以，复图集是可定向图集。
历史[编辑]
黎曼最早开始研究黎曼曲面。黎曼曲面以他命名。
相关主题[编辑][/ltr]
[ltr]
参考[编辑][/ltr]
- Hershel M. Farkas and Irwin Kra, Riemann Su***ces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4
- Jurgen Jost, Compact Riemann Su***ces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X
- Riemann Su***ce on Planet Math
A Glance at Analytic S-Matrix Theory
In the 1950s and 1960s, physicists thoroughly studied scattering processes for elementary particles by using analytic continuation for the S-matrix and the Green’s functions. If these functions have a singularity at the complex energy
E = E0 + iΔE,
then there exists an elementary particle with rest mass m0 = E0/c2 and mean
lifetime Δt = /ΔE.
一星- 帖子数 : 3787
注册日期 : 13-08-07
回复： 万有理论
由 一星 2014-07-02, 11:17
Richard Talman
Geometric Mechanics
Introduction
The first edition of this text was envisaged as a kind of Mathematical Methods
of Classical Mechanics for Pedestrians, with geometry playing a more important
role than in the traditional pedagogy of classical mechanics. Part of the rationale
was to prepare the student for subsequent geometry-intensive physics
subjects, especially general relativity. Subsequently I have found that, as a
text for physics courses, this emphasis was somewhat misplaced. (Almost by
definition) students of physics want to learn “physics” more than they want
to learn “applied mathematics.” Consistent with this, there has been a tendency
for classical mechanics to be squeezed out of physics curricula in favor
of general relativity or, more recently, string theory. This second edition has
been revised accordingly. Instead of just laying the groundwork for subjects
such as electromagnetic theory, string theory, and general relativity, it subsumes
these subjects into classical mechanics. After these changes, the text
has become more nearly a Classical Theory of Fields for Pedestrians.
Geometric approaches have contributed importantly to the evolution of
modern physical theories. The best example is general relativity; the most
modern example is string theory. In fact general relativity and string theory
are the theories for which the adjective “geometric” is most unambiguously
appropriate. There is now a chapter on each of these subjects in this text,
along with material on (classical) field theory basic to these subjects. Also,
because electromagnetic theory fits the same template, and is familiar to most
students, that subject is here also formulated as a “branch” of classical mechanics.
In grandiose terms, the plan of the text is to arrogate to classical mechanics
all of classical physics, where “classical” means nonquantum-mechanical and
“all” means old-fashioned classical mechanics plus the three physical theories
mentioned previously. Other classical theories, such as elasticity and hydrodynamics,
can be regarded as having already been subsumed into classical
mechanics, but they lie outside the scope of this text.
In more technical terms, the theme of the text is that all of classical physics
starts from a Lagrangian, continues with Hamilton’s principle (also known
as the principle of least action) and finishes with solving the resultant equations
and comparison with experiment. This program provides a unification
of classical physics. General principles, especially symmetry and special relativity,
limit the choices surprisingly when any new term is to be added to the
Lagrangian. Once a new term has been added the entire theory and its predictions
are predetermined. These results can then be checked experimentally.
The track record for success in this program has been astonishingly good. As
far as classical physics is concerned the greatest triumphs are due to Maxwell
and Einstein. The philosophic basis for this approach, apparently espoused
by Einstein, is not that we live in the best of all possible worlds, but that we
live in the only possible world. Even people who find this philosophy silly
find that you don’t have to subscribe to this philosophy for the approach to
work well.
There is an ambitious program in quantum field theory called “grand unification”
of the four fundamental forces of physics. The present text can be regarded
as preparation for this program in that it describes classical physics in
ways consistent with this eventual approach. As far as I know, any imagined
grand unification scheme will, when reduced to the classical level, resemble
the material presented here. (Of course most of the essence of the physics is
quantum mechanical and cannot survive the reduction to classical physics.)
Converting the emphasis from applied mathematics to pure physics required
fewer changes to the text than might be supposed. Much of the earlier
book emphasized specialized mathematics and computational descriptions
that could be removed to make room for the “physics” chapters already
mentioned. By no means does this mean that the text has been gutted of practical
worked examples of classical mechanics. For example, most of the long
chapters on perturbation theory and on the application of adiabatic invariants
(both of which are better thought of as physics than as mathematics) have
been retained. All of the (admittedly unenthusiastic) discussion of canonical
transformation methods has also been retained.
Regrettably, some material on the boundary between classical and quantum
mechanics has had to be dropped. As well as helping to keep the book
length within bounds, this deletion was consistent with religiously restricting
the subject matter to nothing but classical physics. There was a time when classical
Hamiltonian mechanics seemed like the best introduction to quantum
mechanics but, like the need to study Latin in school, that no longer seems to
be the case. Also, apart from its connections to the Hamilton–Jacobi theory
(which every educated physicist has to understand) quantum mechanics is
not very geometric in character. It was relatively painless therefore, to remove
unitary geometry, Bragg scattering (illustrating the use of covariant tensors),
and other material on the margin between classical and quantum mechanics.
In this book’s first manifestation the subject of mechanics was usefully, if
somewhat artificially, segmented into Lagrangian, Hamiltonian, and Newtonian
formulations. Much was made of Poincaré’s extension to the Lagrangian
approach. Because this approach advances the math more than the physics, it
now has had to be de-emphasized (though most of the material remains). On
the other hand, asmentioned already, the coverage of Lagrangian field theory,
and especially its conservation laws, needed to be expanded. Reduced weight
also had to be assigned to Hamiltonian methods (not counting Hamilton’s
principle.) Those methods provide the most direct connections to quantum
mechanics but, with quantum considerations now being ignored, they are less
essential to the program. Opposite comments apply to Newtonian methods,
which stress fictitious forces (centrifugal and Coriolis), ideas that led naturally
to general relativity. Gauge invariant methods, which play such an important
role in string theory, are also naturally introduced in the context of direct Newtonian
methods. The comments in this paragraph, taken together, repudiate
much of the preface to the first edition which has, therefore, been discarded.
Everything contained in this book is explained with more rigor, or more
depth, or more detail, or (especially) more sophistication, in at least one of the
books listed at the end of this introduction. Were it not for the fact that most
of those books are fat, intimidating, abstract, formal, mathematical and (for
many) unintelligible, the reader’s time would be better spent reading them
(in the right order) than studying this book. But if this text renders books like
these both accessible and admirable, it will have achieved its main purpose. It
has been said that bridge is a simple game; dealt thirteen cards, one has only
to play them in the correct order. In the same sense mechanics is easy to learn;
one simply has to study readily available books in a sensible order. I have tried
to chart such a path, extractingmaterial fromvarious sources in an order that I
have found appropriate. At each stage I indicate (at the end of the chapter) the
reference my approach most closely resembles. In some cases what I provide
is a kind of Reader’s Digest of a more general treatment and this may amount
to my having systematically specialized and made concrete, descriptions that
the original author may earlier have systematically labored to generalize and
make abstract. The texts to which these statements are most applicable are
listed at the end of each chapter, and keyed to the particular section to which
they relate. It is not suggested that these texts should be systematically referred
to as they tend to be advanced and contain much unrelated material.
But if particular material in this text is obscure, or seems to stop short of some
desirable goal, these texts should provide authoritative help.
Not very much is original in the text other than the selection and arrangement
of the topics and the style of presentation. Equations (though not text)
have been “borrowed,” in some cases verbatim, from various sources. This
is especially true of Chapters 9, on special relativity, 11, on electromagnetic
theory, and 13, on general relativity. These chapters follow Landau and Lifschitz
quite closely. Similarly, Chapter 12 follows Zwiebach closely. There
are also substantial sections following Cartan, or Arnold, or others. As well
as occasional reminders in the text, of these sources, the bibliography at the
end of each chapter lists the essential sources. Under “General References”
are books, like the two just mentioned, that contain one or more chapters discussing
much the same material in at least as much, and usually far more,
detail, than is included here. These references could be used instead of the
material in the chapter they are attached to, and should be used to go deeper
into the subject. Under “References for Further Study” are sources that can
be used as well as the material of the chapter. In principle, none of these references
should actually be necessary, as the present text is supposed to be
self-sufficient. In practice, obscurities and the likelihood of errors or misunderstandings,
make it appropriate, or even necessary, to refer to other sources
to obtain anything resembling a deep understanding of a topic.
The mathematical level strived for is only high enough to support a persuasive
(to a nonmathematician) trip through the physics. Still, “it can be shown
that” almost never appears, though the standards of what constitutes “proof”
may be low, and the range of generality narrow. I believe that much mathematics
is made difficult for the less-mathematically-inclined reader by the absence
of concrete instances of the abstract objects under discussion. This text
tries to provide essentially correct instances of otherwise hard to grasp mathematical
abstractions. I hope and believe that this will provide a broad base
of general understanding from which deeper, more specialized, more mathematical
texts can be approached with a respectable general comprehension.
This statement is most applicable to the excellent books by Arnold, who tries
hard, but not necessarily successfully, to provide physical lines of reasoning.
Much of this book was written with the goal of *** one or another of his
discussions comprehensible.
In the early days of our weekly Laboratory of Nuclear Studies Journal Club,
our founding leader, RobertWilson, imposed a rule – though honored asmuch
in the breach as in the observance, it was not intended to be a joke – that the
Dirac γ-matrices never appear. The (largely unsuccessful) purpose of this rule
was to force the lectures to be intelligible to us theory-challenged experimentalists.
In this text there is a similar rule. It is that hieroglyphics such as
φ : {x ∈ R2 : |x| = 1} → R
not appear. The justification for this rule is that a “physicist” is likely to skip
such a statement altogether or, once having understood it, regard it as obvious.
Like the jest that the French “don’t care what they say as long as they
pronounce it properly” one can joke that mathematicians don’t care what
their mappings do, as long as the spaces they connect are clear. Physicists
Geometric Mechanics
Introduction
The first edition of this text was envisaged as a kind of Mathematical Methods
of Classical Mechanics for Pedestrians, with geometry playing a more important
role than in the traditional pedagogy of classical mechanics. Part of the rationale
was to prepare the student for subsequent geometry-intensive physics
subjects, especially general relativity. Subsequently I have found that, as a
text for physics courses, this emphasis was somewhat misplaced. (Almost by
definition) students of physics want to learn “physics” more than they want
to learn “applied mathematics.” Consistent with this, there has been a tendency
for classical mechanics to be squeezed out of physics curricula in favor
of general relativity or, more recently, string theory. This second edition has
been revised accordingly. Instead of just laying the groundwork for subjects
such as electromagnetic theory, string theory, and general relativity, it subsumes
these subjects into classical mechanics. After these changes, the text
has become more nearly a Classical Theory of Fields for Pedestrians.
Geometric approaches have contributed importantly to the evolution of
modern physical theories. The best example is general relativity; the most
modern example is string theory. In fact general relativity and string theory
are the theories for which the adjective “geometric” is most unambiguously
appropriate. There is now a chapter on each of these subjects in this text,
along with material on (classical) field theory basic to these subjects. Also,
because electromagnetic theory fits the same template, and is familiar to most
students, that subject is here also formulated as a “branch” of classical mechanics.
In grandiose terms, the plan of the text is to arrogate to classical mechanics
all of classical physics, where “classical” means nonquantum-mechanical and
“all” means old-fashioned classical mechanics plus the three physical theories
mentioned previously. Other classical theories, such as elasticity and hydrodynamics,
can be regarded as having already been subsumed into classical
mechanics, but they lie outside the scope of this text.
In more technical terms, the theme of the text is that all of classical physics
starts from a Lagrangian, continues with Hamilton’s principle (also known
as the principle of least action) and finishes with solving the resultant equations
and comparison with experiment. This program provides a unification
of classical physics. General principles, especially symmetry and special relativity,
limit the choices surprisingly when any new term is to be added to the
Lagrangian. Once a new term has been added the entire theory and its predictions
are predetermined. These results can then be checked experimentally.
The track record for success in this program has been astonishingly good. As
far as classical physics is concerned the greatest triumphs are due to Maxwell
and Einstein. The philosophic basis for this approach, apparently espoused
by Einstein, is not that we live in the best of all possible worlds, but that we
live in the only possible world. Even people who find this philosophy silly
find that you don’t have to subscribe to this philosophy for the approach to
work well.
There is an ambitious program in quantum field theory called “grand unification”
of the four fundamental forces of physics. The present text can be regarded
as preparation for this program in that it describes classical physics in
ways consistent with this eventual approach. As far as I know, any imagined
grand unification scheme will, when reduced to the classical level, resemble
the material presented here. (Of course most of the essence of the physics is
quantum mechanical and cannot survive the reduction to classical physics.)
Converting the emphasis from applied mathematics to pure physics required
fewer changes to the text than might be supposed. Much of the earlier
book emphasized specialized mathematics and computational descriptions
that could be removed to make room for the “physics” chapters already
mentioned. By no means does this mean that the text has been gutted of practical
worked examples of classical mechanics. For example, most of the long
chapters on perturbation theory and on the application of adiabatic invariants
(both of which are better thought of as physics than as mathematics) have
been retained. All of the (admittedly unenthusiastic) discussion of canonical
transformation methods has also been retained.
Regrettably, some material on the boundary between classical and quantum
mechanics has had to be dropped. As well as helping to keep the book
length within bounds, this deletion was consistent with religiously restricting
the subject matter to nothing but classical physics. There was a time when classical
Hamiltonian mechanics seemed like the best introduction to quantum
mechanics but, like the need to study Latin in school, that no longer seems to
be the case. Also, apart from its connections to the Hamilton–Jacobi theory
(which every educated physicist has to understand) quantum mechanics is
not very geometric in character. It was relatively painless therefore, to remove
unitary geometry, Bragg scattering (illustrating the use of covariant tensors),
and other material on the margin between classical and quantum mechanics.
In this book’s first manifestation the subject of mechanics was usefully, if
somewhat artificially, segmented into Lagrangian, Hamiltonian, and Newtonian
formulations. Much was made of Poincaré’s extension to the Lagrangian
approach. Because this approach advances the math more than the physics, it
now has had to be de-emphasized (though most of the material remains). On
the other hand, asmentioned already, the coverage of Lagrangian field theory,
and especially its conservation laws, needed to be expanded. Reduced weight
also had to be assigned to Hamiltonian methods (not counting Hamilton’s
principle.) Those methods provide the most direct connections to quantum
mechanics but, with quantum considerations now being ignored, they are less
essential to the program. Opposite comments apply to Newtonian methods,
which stress fictitious forces (centrifugal and Coriolis), ideas that led naturally
to general relativity. Gauge invariant methods, which play such an important
role in string theory, are also naturally introduced in the context of direct Newtonian
methods. The comments in this paragraph, taken together, repudiate
much of the preface to the first edition which has, therefore, been discarded.
Everything contained in this book is explained with more rigor, or more
depth, or more detail, or (especially) more sophistication, in at least one of the
books listed at the end of this introduction. Were it not for the fact that most
of those books are fat, intimidating, abstract, formal, mathematical and (for
many) unintelligible, the reader’s time would be better spent reading them
(in the right order) than studying this book. But if this text renders books like
these both accessible and admirable, it will have achieved its main purpose. It
has been said that bridge is a simple game; dealt thirteen cards, one has only
to play them in the correct order. In the same sense mechanics is easy to learn;
one simply has to study readily available books in a sensible order. I have tried
to chart such a path, extractingmaterial fromvarious sources in an order that I
have found appropriate. At each stage I indicate (at the end of the chapter) the
reference my approach most closely resembles. In some cases what I provide
is a kind of Reader’s Digest of a more general treatment and this may amount
to my having systematically specialized and made concrete, descriptions that
the original author may earlier have systematically labored to generalize and
make abstract. The texts to which these statements are most applicable are
listed at the end of each chapter, and keyed to the particular section to which
they relate. It is not suggested that these texts should be systematically referred
to as they tend to be advanced and contain much unrelated material.
But if particular material in this text is obscure, or seems to stop short of some
desirable goal, these texts should provide authoritative help.
Not very much is original in the text other than the selection and arrangement
of the topics and the style of presentation. Equations (though not text)
have been “borrowed,” in some cases verbatim, from various sources. This
is especially true of Chapters 9, on special relativity, 11, on electromagnetic
theory, and 13, on general relativity. These chapters follow Landau and Lifschitz
quite closely. Similarly, Chapter 12 follows Zwiebach closely. There
are also substantial sections following Cartan, or Arnold, or others. As well
as occasional reminders in the text, of these sources, the bibliography at the
end of each chapter lists the essential sources. Under “General References”
are books, like the two just mentioned, that contain one or more chapters discussing
much the same material in at least as much, and usually far more,
detail, than is included here. These references could be used instead of the
material in the chapter they are attached to, and should be used to go deeper
into the subject. Under “References for Further Study” are sources that can
be used as well as the material of the chapter. In principle, none of these references
should actually be necessary, as the present text is supposed to be
self-sufficient. In practice, obscurities and the likelihood of errors or misunderstandings,
make it appropriate, or even necessary, to refer to other sources
to obtain anything resembling a deep understanding of a topic.
The mathematical level strived for is only high enough to support a persuasive
(to a nonmathematician) trip through the physics. Still, “it can be shown
that” almost never appears, though the standards of what constitutes “proof”
may be low, and the range of generality narrow. I believe that much mathematics
is made difficult for the less-mathematically-inclined reader by the absence
of concrete instances of the abstract objects under discussion. This text
tries to provide essentially correct instances of otherwise hard to grasp mathematical
abstractions. I hope and believe that this will provide a broad base
of general understanding from which deeper, more specialized, more mathematical
texts can be approached with a respectable general comprehension.
This statement is most applicable to the excellent books by Arnold, who tries
hard, but not necessarily successfully, to provide physical lines of reasoning.
Much of this book was written with the goal of *** one or another of his
discussions comprehensible.
In the early days of our weekly Laboratory of Nuclear Studies Journal Club,
our founding leader, RobertWilson, imposed a rule – though honored asmuch
in the breach as in the observance, it was not intended to be a joke – that the
Dirac γ-matrices never appear. The (largely unsuccessful) purpose of this rule
was to force the lectures to be intelligible to us theory-challenged experimentalists.
In this text there is a similar rule. It is that hieroglyphics such as
φ : {x ∈ R2 : |x| = 1} → R
not appear. The justification for this rule is that a “physicist” is likely to skip
such a statement altogether or, once having understood it, regard it as obvious.
Like the jest that the French “don’t care what they say as long as they
pronounce it properly” one can joke that mathematicians don’t care what
their mappings do, as long as the spaces they connect are clear. Physicists
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on the other hand, care primarily what their functions represent physically
and are not fussy about what spaces they relate. Another “rule” has just been
followed; the word function will be used in preference to the (synonymous)
word mapping. Other terrifying mathematical words such as flow, symplectomorphism,
and manifold will also be avoided except that, to avoid long-winded
phrases such as “configuration space described by generalized coordinates,”
the word manifold will occasionally be used. Of course one cannot alter the
essence of a subject by denying the existence ofmathematics that is manifestly
at its core. In spite of the loss of precision, I hope that sugar-coating the material
in this way will make it more easily swallowed by nonmathematicians.
Notation: “Notation isn’t everything, it’s the only thing.” Grammatically
speaking, this statement, like the American football slogan it paraphrases,
makes no sense. But its clearly intended meaning is only a mild exaggeration.
After the need to evaluate some quantity has been expressed, a few straightforwardmathematical
operations are typically all that is required to obtain the
quantity. But specifying quantities is far from simple. The conceptual depth
of the subject is substantial and ordinary language is scarcely capable of defining
the symbols, much less expressing the relations among them. This makes
the introduction of sophisticated symbols essential. Discussion of notation
and the motivation behind its introduction is scattered throughout this text
– probably to the point of irritation for some readers. Here we limit discussion
to the few most important, most likely to be confusing, and most deviant
from other sources: the qualified equality q = , the vector, the preferred reference
system, the active/passive interpretation of transformations, and the terminology
of differential forms.
A fairly common occurrence in this subject is that two quantities A and B
are equal or equivalent from one point of view but not from another. This
circumstance will be indicated by “qualified equality” A q = B. This notation
is intentionally vague (the “q” stands for qualified, or questionable, or query?
as appropriate) and may have different meanings in different contexts; it only
warns the reader to be wary of the risk of jumping to unjustified conclusions.
Normally the qualification will be clarified in the subsequent text.
Next vectors. Consider the following three symbols or collections of symbols:
−→, x, and (x, y, z)T. The first, −→, will be called an arrow (because it
is one) and this word will be far more prevalent in this text than any other of
which I am aware. This particular arrow happens to be pointing in a horizontal
direction (for convenience of typesetting) but in general an arrow can point
in any direction, including out of the page. The second, bold face, quantity,
x, is an intrinsic or true vector; this means that it is a symbol that “stands for”
an arrow. The word “intrinsic” means “it doesn’t depend on choice of coordinate
system.” The third quantity, (x, y, z)T, is a column matrix (because the T
stands for transpose) containing the “components” of x relative to some preestablished coordinate system. From the point of view of elementary physics
these three are equivalent quantities, differing only in the ways they are to
be manipulated; “addition” of arrows is by ruler and compass, addition of
intrinsic vectors is by vector algebra, and addition of coordinate vectors is
component wise. Because of this multiplicity of meanings, the word “vector”
is ambiguous in some contexts. For this reason, we will often use the word
arrow in situations where independence of choice of coordinates is being emphasized
(even in dimensionality higher than 3.) According to its definition
above, the phrase intrinsic vector could usually replace arrow, but some would
complain of the redundancy, and the word arrow more succinctly conveys the
intended geometric sense. Comments similar to these could bemade concerning
higher order tensors but they would be largely repetitive.
Avirtue of arrows is that they can be plotted in figures. This goes a longway
toward *** their meaning unambiguous but the conditions defining the
figure must still be made clear. In classical mechanics “inertial frames” have
a fundamental significance and we will almost always suppose that there is a
“preferred” reference system, its rectangular axes fixed in an inertial system.
Unless otherwise stated, figures in this text are to be regarded as “snapshots”
taken in that frame. In particular, a plotted arrow connects two points fixed in
the inertial frame at the instant illustrated. As mentioned previously, such an
arrow is symbolized by a true vector such as x.
It is, of course, essential that these vectors satisfy the algebraic properties
defining a vector space. In such spaces “transformations” are important; a
“linear” transformation can be represented by a matrix symbolized, for example,
by M, with elements Mi
j. The result of applying this transformation to
vector x can be represented symbolically as the “matrix product” y q = Mx
of “intrinsic” quantities, or spelled out explicitly in components yi = Mi
jxj.
Frequently both formswill be given. This leads to a notational difficulty in distinguishing
between the “active” and “passive” interpretations of the transformation.
The new components yi can belong to a new arrow in the old frame
(active interpretation) or to the old arrow in a new frame (passive interpretation).
On the other hand, the intrinsic form y q = Mx seems to support only an
active interpretation according to which M “operates” on vector x to yield a
different vector y. To avoid this problem, when we wish to express a passive
interpretation we will ordinarily use the form x q = Mx and will insist that x
and x stand for the same arrow. The significance of the overhead bar then is that
x is simply an abbreviation for an array of barred-frame coordinates xi. When
the active interpretation is intended the notation will usually be expanded to
clarify the situation. For example, consider a moving point located initially at
r(0) and at r(t) at later time t. These vectors can be related by r(t) = O(t) r(0)
where O(t) is a time-dependent operator. This is an active transformation.
established coordinate system. From the point of view of elementary physics
these three are equivalent quantities, differing only in the ways they are to
be manipulated; “addition” of arrows is by ruler and compass, addition of
intrinsic vectors is by vector algebra, and addition of coordinate vectors is
component wise. Because of this multiplicity of meanings, the word “vector”
is ambiguous in some contexts. For this reason, we will often use the word
arrow in situations where independence of choice of coordinates is being emphasized
(even in dimensionality higher than 3.) According to its definition
above, the phrase intrinsic vector could usually replace arrow, but some would
complain of the redundancy, and the word arrow more succinctly conveys the
intended geometric sense. Comments similar to these could bemade concerning
higher order tensors but they would be largely repetitive.
Avirtue of arrows is that they can be plotted in figures. This goes a longway
toward *** their meaning unambiguous but the conditions defining the
figure must still be made clear. In classical mechanics “inertial frames” have
a fundamental significance and we will almost always suppose that there is a
“preferred” reference system, its rectangular axes fixed in an inertial system.
Unless otherwise stated, figures in this text are to be regarded as “snapshots”
taken in that frame. In particular, a plotted arrow connects two points fixed in
the inertial frame at the instant illustrated. As mentioned previously, such an
arrow is symbolized by a true vector such as x.
It is, of course, essential that these vectors satisfy the algebraic properties
defining a vector space. In such spaces “transformations” are important; a
“linear” transformation can be represented by a matrix symbolized, for example,
by M, with elements Mi
j. The result of applying this transformation to
vector x can be represented symbolically as the “matrix product” y q = Mx
of “intrinsic” quantities, or spelled out explicitly in components yi = Mi
jxj.
Frequently both formswill be given. This leads to a notational difficulty in distinguishing
between the “active” and “passive” interpretations of the transformation.
The new components yi can belong to a new arrow in the old frame
(active interpretation) or to the old arrow in a new frame (passive interpretation).
On the other hand, the intrinsic form y q = Mx seems to support only an
active interpretation according to which M “operates” on vector x to yield a
different vector y. To avoid this problem, when we wish to express a passive
interpretation we will ordinarily use the form x q = Mx and will insist that x
and x stand for the same arrow. The significance of the overhead bar then is that
x is simply an abbreviation for an array of barred-frame coordinates xi. When
the active interpretation is intended the notation will usually be expanded to
clarify the situation. For example, consider a moving point located initially at
r(0) and at r(t) at later time t. These vectors can be related by r(t) = O(t) r(0)
where O(t) is a time-dependent operator. This is an active transformation.
The beauty and power of vector analysis as it is applied to physics is that
a bold face symbol such as V indicates that the quantity is intrinsic and also
abbreviates its multiple components Vi into one symbol. Though these are
both valuable purposes, they are not the same. The abbreviation works in
vector analysis only because vectors are the only multiple component objects
occurring. That this will no longer be the case in this book will cause considerable
notational difficulty because the reader, based on experience with
vector analysis, is likely to jump to unjustified conclusions concerning bold
face quantities.1 We will not be able to avoid this problem however since we
wish to retain familiar notation. Sometimes we will be using bold face symbols
to indicate intrinsically, sometimes as abbreviation, and sometimes both.
Sometimes the (redundant) notation v will be used to emphasize the intrinsic
aspect. Though it may not be obvious at this point, notational insufficiency
was the source of the above-mentioned need to differentiate verbally between
active and passive transformations. In stressing this distinction the text differs
from a text such as Goldstein that, perhaps wisely, de-emphasizes the issue.
According to Arnold “it is impossible to understand mechanics without the
use of differential forms.” Accepting the validity of this statement only grudgingly
(and trying to corroborate it) but knowing from experience that typical
physics students are innocent of any such knowledge, a considerable portion
of the text is devoted to this subject. Briefly, the symbol dx will stand for an
old-fashioned differential displacement of the sort familiar to every student of
physics. But a new quantity dx to be known as a differential form, will also
be used. This symbol is distinguished from dx both by being bold face and
having an overhead tilde. Displacements dx1, dx2, . . . in spaces of higher dimension
will have matching forms dx1,dx2, . . . . This notation is mentioned
at this point only because it is unconventional. In most treatments one or the
other form of differential is used, but not both at the same time. I have found it
impossible to cause classical formulations tomorph into modern formulations
without this distinction (and others to be faced when the time comes.)
It is hard to avoid using termswhose meanings are vague. (See the previous
paragraph, for example.) I have attempted to acknowledge such vagueness, at
least in extreme cases, by placing such terms in quotationmarks when they are
first used. Since quotationmarks are also usedwhen the term is actually being
defined, a certain amount of hunting through the surrounding sentences may
be necessary to find if a definition is actually present. (If it is not clear whether
or not there is a definition then the term is without any doubt vague.) Italics
are used to emphasize key phrases, or pairs of phrases in opposition, that are
central to the discussion. Parenthesized sentences or sentence fragments are
supposedly clear only if they are included right there but they should not be
allowed to interrupt the logical flow of the surrounding sentences. Footnotes,
though sometimes similar in intent, are likely to be real digressions, or technical
qualifications or clarifications.
The text contains at least enough material for a full year course and far more
than can be covered in any single term course. At Cornell the material has
been the basis for several distinct courses: (a) Junior/senior level classical mechanics
(as that subject is traditionally, and narrowly, defined.) (b) First year
graduate classical mechanics with geometric emphasis. (c) Perturbative and
adiabatic methods of solution and, most recently, (d) “Geometric Concepts in
Physics.” Course (d) was responsible for the math/physics reformulation of
this edition. The text is best matched, therefore, to filling a curricular slot that
allows variation term-by-term or year-by-year.
Organization of the book: Chapter 1, containing review/examples, provides
appropriate preparation for any of the above courses; it contains a brief
overview of elementary methods that the reader may wish (or need) to review.
Since the formalism (primarily Lagrangian) is assumed to be familiar,
this review consists primarily of examples,manyworked out partially or completely.
Chapter 2 and the first half of Chapter 3 contain the geometric concepts
likely to be both “new” and needed. The rest of Chapter 3 as well as
Chapter 4 contain geometry that can be skipped until needed. Chapters 5, 6,
7, and 8 contain, respectively, the Lagrangian, Newtonian, Hamiltonian and
Hamilton–Jacobi, backbone of course labeled (a) above. The first half of Chapter
10, on conservation laws, is also appropriate for such a course, and methods
of solution should be drawn from Chapters 14, 15, and 16.
The need for relativistic mechanics is what characterizes Chapters 9, 11, 12,
and 13. These chapters can provide the “physics” content for a course such
as (d) above. The rest of the book does not depend on the material in these
chapters. A course should therefore include either none of this material or all
of it, though perhaps emphasizing either, but not both, of general relativity
and string theory.
Methods of solution are studied in Chapters 14, 15, and 16. These chapters
would form an appreciable fraction of a course such as (c) above.
Chapter 17 is concerned mainly with the formal structure of mechanics in
Hamiltonian form. As such it is most likely to be of interest to students planning
to take a subsequent courses in dynamical systems, chaos, plasma or accelerator
physics. Somehow the most important result of classical mechanics
– Liouville’s theorem – has found its way to the last section of the book.
The total number of problems has been almost doubled compared to the
first edition. However, in the chapters covering areas of physics not traditionally
regarded as classical mechanics, the problems are intended to require no
special knowledge of those subjects beyond what is covered in this text.
and are not fussy about what spaces they relate. Another “rule” has just been
followed; the word function will be used in preference to the (synonymous)
word mapping. Other terrifying mathematical words such as flow, symplectomorphism,
and manifold will also be avoided except that, to avoid long-winded
phrases such as “configuration space described by generalized coordinates,”
the word manifold will occasionally be used. Of course one cannot alter the
essence of a subject by denying the existence ofmathematics that is manifestly
at its core. In spite of the loss of precision, I hope that sugar-coating the material
in this way will make it more easily swallowed by nonmathematicians.
Notation: “Notation isn’t everything, it’s the only thing.” Grammatically
speaking, this statement, like the American football slogan it paraphrases,
makes no sense. But its clearly intended meaning is only a mild exaggeration.
After the need to evaluate some quantity has been expressed, a few straightforwardmathematical
operations are typically all that is required to obtain the
quantity. But specifying quantities is far from simple. The conceptual depth
of the subject is substantial and ordinary language is scarcely capable of defining
the symbols, much less expressing the relations among them. This makes
the introduction of sophisticated symbols essential. Discussion of notation
and the motivation behind its introduction is scattered throughout this text
– probably to the point of irritation for some readers. Here we limit discussion
to the few most important, most likely to be confusing, and most deviant
from other sources: the qualified equality q = , the vector, the preferred reference
system, the active/passive interpretation of transformations, and the terminology
of differential forms.
A fairly common occurrence in this subject is that two quantities A and B
are equal or equivalent from one point of view but not from another. This
circumstance will be indicated by “qualified equality” A q = B. This notation
is intentionally vague (the “q” stands for qualified, or questionable, or query?
as appropriate) and may have different meanings in different contexts; it only
warns the reader to be wary of the risk of jumping to unjustified conclusions.
Normally the qualification will be clarified in the subsequent text.
Next vectors. Consider the following three symbols or collections of symbols:
−→, x, and (x, y, z)T. The first, −→, will be called an arrow (because it
is one) and this word will be far more prevalent in this text than any other of
which I am aware. This particular arrow happens to be pointing in a horizontal
direction (for convenience of typesetting) but in general an arrow can point
in any direction, including out of the page. The second, bold face, quantity,
x, is an intrinsic or true vector; this means that it is a symbol that “stands for”
an arrow. The word “intrinsic” means “it doesn’t depend on choice of coordinate
system.” The third quantity, (x, y, z)T, is a column matrix (because the T
stands for transpose) containing the “components” of x relative to some preestablished coordinate system. From the point of view of elementary physics
these three are equivalent quantities, differing only in the ways they are to
be manipulated; “addition” of arrows is by ruler and compass, addition of
intrinsic vectors is by vector algebra, and addition of coordinate vectors is
component wise. Because of this multiplicity of meanings, the word “vector”
is ambiguous in some contexts. For this reason, we will often use the word
arrow in situations where independence of choice of coordinates is being emphasized
(even in dimensionality higher than 3.) According to its definition
above, the phrase intrinsic vector could usually replace arrow, but some would
complain of the redundancy, and the word arrow more succinctly conveys the
intended geometric sense. Comments similar to these could bemade concerning
higher order tensors but they would be largely repetitive.
Avirtue of arrows is that they can be plotted in figures. This goes a longway
toward *** their meaning unambiguous but the conditions defining the
figure must still be made clear. In classical mechanics “inertial frames” have
a fundamental significance and we will almost always suppose that there is a
“preferred” reference system, its rectangular axes fixed in an inertial system.
Unless otherwise stated, figures in this text are to be regarded as “snapshots”
taken in that frame. In particular, a plotted arrow connects two points fixed in
the inertial frame at the instant illustrated. As mentioned previously, such an
arrow is symbolized by a true vector such as x.
It is, of course, essential that these vectors satisfy the algebraic properties
defining a vector space. In such spaces “transformations” are important; a
“linear” transformation can be represented by a matrix symbolized, for example,
by M, with elements Mi
j. The result of applying this transformation to
vector x can be represented symbolically as the “matrix product” y q = Mx
of “intrinsic” quantities, or spelled out explicitly in components yi = Mi
jxj.
Frequently both formswill be given. This leads to a notational difficulty in distinguishing
between the “active” and “passive” interpretations of the transformation.
The new components yi can belong to a new arrow in the old frame
(active interpretation) or to the old arrow in a new frame (passive interpretation).
On the other hand, the intrinsic form y q = Mx seems to support only an
active interpretation according to which M “operates” on vector x to yield a
different vector y. To avoid this problem, when we wish to express a passive
interpretation we will ordinarily use the form x q = Mx and will insist that x
and x stand for the same arrow. The significance of the overhead bar then is that
x is simply an abbreviation for an array of barred-frame coordinates xi. When
the active interpretation is intended the notation will usually be expanded to
clarify the situation. For example, consider a moving point located initially at
r(0) and at r(t) at later time t. These vectors can be related by r(t) = O(t) r(0)
where O(t) is a time-dependent operator. This is an active transformation.
established coordinate system. From the point of view of elementary physics
these three are equivalent quantities, differing only in the ways they are to
be manipulated; “addition” of arrows is by ruler and compass, addition of
intrinsic vectors is by vector algebra, and addition of coordinate vectors is
component wise. Because of this multiplicity of meanings, the word “vector”
is ambiguous in some contexts. For this reason, we will often use the word
arrow in situations where independence of choice of coordinates is being emphasized
(even in dimensionality higher than 3.) According to its definition
above, the phrase intrinsic vector could usually replace arrow, but some would
complain of the redundancy, and the word arrow more succinctly conveys the
intended geometric sense. Comments similar to these could bemade concerning
higher order tensors but they would be largely repetitive.
Avirtue of arrows is that they can be plotted in figures. This goes a longway
toward *** their meaning unambiguous but the conditions defining the
figure must still be made clear. In classical mechanics “inertial frames” have
a fundamental significance and we will almost always suppose that there is a
“preferred” reference system, its rectangular axes fixed in an inertial system.
Unless otherwise stated, figures in this text are to be regarded as “snapshots”
taken in that frame. In particular, a plotted arrow connects two points fixed in
the inertial frame at the instant illustrated. As mentioned previously, such an
arrow is symbolized by a true vector such as x.
It is, of course, essential that these vectors satisfy the algebraic properties
defining a vector space. In such spaces “transformations” are important; a
“linear” transformation can be represented by a matrix symbolized, for example,
by M, with elements Mi
j. The result of applying this transformation to
vector x can be represented symbolically as the “matrix product” y q = Mx
of “intrinsic” quantities, or spelled out explicitly in components yi = Mi
jxj.
Frequently both formswill be given. This leads to a notational difficulty in distinguishing
between the “active” and “passive” interpretations of the transformation.
The new components yi can belong to a new arrow in the old frame
(active interpretation) or to the old arrow in a new frame (passive interpretation).
On the other hand, the intrinsic form y q = Mx seems to support only an
active interpretation according to which M “operates” on vector x to yield a
different vector y. To avoid this problem, when we wish to express a passive
interpretation we will ordinarily use the form x q = Mx and will insist that x
and x stand for the same arrow. The significance of the overhead bar then is that
x is simply an abbreviation for an array of barred-frame coordinates xi. When
the active interpretation is intended the notation will usually be expanded to
clarify the situation. For example, consider a moving point located initially at
r(0) and at r(t) at later time t. These vectors can be related by r(t) = O(t) r(0)
where O(t) is a time-dependent operator. This is an active transformation.
The beauty and power of vector analysis as it is applied to physics is that
a bold face symbol such as V indicates that the quantity is intrinsic and also
abbreviates its multiple components Vi into one symbol. Though these are
both valuable purposes, they are not the same. The abbreviation works in
vector analysis only because vectors are the only multiple component objects
occurring. That this will no longer be the case in this book will cause considerable
notational difficulty because the reader, based on experience with
vector analysis, is likely to jump to unjustified conclusions concerning bold
face quantities.1 We will not be able to avoid this problem however since we
wish to retain familiar notation. Sometimes we will be using bold face symbols
to indicate intrinsically, sometimes as abbreviation, and sometimes both.
Sometimes the (redundant) notation v will be used to emphasize the intrinsic
aspect. Though it may not be obvious at this point, notational insufficiency
was the source of the above-mentioned need to differentiate verbally between
active and passive transformations. In stressing this distinction the text differs
from a text such as Goldstein that, perhaps wisely, de-emphasizes the issue.
According to Arnold “it is impossible to understand mechanics without the
use of differential forms.” Accepting the validity of this statement only grudgingly
(and trying to corroborate it) but knowing from experience that typical
physics students are innocent of any such knowledge, a considerable portion
of the text is devoted to this subject. Briefly, the symbol dx will stand for an
old-fashioned differential displacement of the sort familiar to every student of
physics. But a new quantity dx to be known as a differential form, will also
be used. This symbol is distinguished from dx both by being bold face and
having an overhead tilde. Displacements dx1, dx2, . . . in spaces of higher dimension
will have matching forms dx1,dx2, . . . . This notation is mentioned
at this point only because it is unconventional. In most treatments one or the
other form of differential is used, but not both at the same time. I have found it
impossible to cause classical formulations tomorph into modern formulations
without this distinction (and others to be faced when the time comes.)
It is hard to avoid using termswhose meanings are vague. (See the previous
paragraph, for example.) I have attempted to acknowledge such vagueness, at
least in extreme cases, by placing such terms in quotationmarks when they are
first used. Since quotationmarks are also usedwhen the term is actually being
defined, a certain amount of hunting through the surrounding sentences may
be necessary to find if a definition is actually present. (If it is not clear whether
or not there is a definition then the term is without any doubt vague.) Italics
are used to emphasize key phrases, or pairs of phrases in opposition, that are
central to the discussion. Parenthesized sentences or sentence fragments are
supposedly clear only if they are included right there but they should not be
allowed to interrupt the logical flow of the surrounding sentences. Footnotes,
though sometimes similar in intent, are likely to be real digressions, or technical
qualifications or clarifications.
The text contains at least enough material for a full year course and far more
than can be covered in any single term course. At Cornell the material has
been the basis for several distinct courses: (a) Junior/senior level classical mechanics
(as that subject is traditionally, and narrowly, defined.) (b) First year
graduate classical mechanics with geometric emphasis. (c) Perturbative and
adiabatic methods of solution and, most recently, (d) “Geometric Concepts in
Physics.” Course (d) was responsible for the math/physics reformulation of
this edition. The text is best matched, therefore, to filling a curricular slot that
allows variation term-by-term or year-by-year.
Organization of the book: Chapter 1, containing review/examples, provides
appropriate preparation for any of the above courses; it contains a brief
overview of elementary methods that the reader may wish (or need) to review.
Since the formalism (primarily Lagrangian) is assumed to be familiar,
this review consists primarily of examples,manyworked out partially or completely.
Chapter 2 and the first half of Chapter 3 contain the geometric concepts
likely to be both “new” and needed. The rest of Chapter 3 as well as
Chapter 4 contain geometry that can be skipped until needed. Chapters 5, 6,
7, and 8 contain, respectively, the Lagrangian, Newtonian, Hamiltonian and
Hamilton–Jacobi, backbone of course labeled (a) above. The first half of Chapter
10, on conservation laws, is also appropriate for such a course, and methods
of solution should be drawn from Chapters 14, 15, and 16.
The need for relativistic mechanics is what characterizes Chapters 9, 11, 12,
and 13. These chapters can provide the “physics” content for a course such
as (d) above. The rest of the book does not depend on the material in these
chapters. A course should therefore include either none of this material or all
of it, though perhaps emphasizing either, but not both, of general relativity
and string theory.
Methods of solution are studied in Chapters 14, 15, and 16. These chapters
would form an appreciable fraction of a course such as (c) above.
Chapter 17 is concerned mainly with the formal structure of mechanics in
Hamiltonian form. As such it is most likely to be of interest to students planning
to take a subsequent courses in dynamical systems, chaos, plasma or accelerator
physics. Somehow the most important result of classical mechanics
– Liouville’s theorem – has found its way to the last section of the book.
The total number of problems has been almost doubled compared to the
first edition. However, in the chapters covering areas of physics not traditionally
regarded as classical mechanics, the problems are intended to require no
special knowledge of those subjects beyond what is covered in this text.
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