1. Analytical Mechanics Department of Physics Shanghai Jiaotong University 2004

2. Introduction

3. Newtonian Mechanics By Galileo and Newton Mathematical Principles of Natural Philosophy (1687). Three Laws of Mechanics The central physical quantity of Newtonian Mechanics is force F

4. Analytical Mechanics d’Alembert, Euler, Lagrange, Hamilton Analytical Mechanics by Lagrange (1788) extend to non-mechanical systems. The central physical quantity of Analytical Mechanics is energy

5. Difference between Newtonian Mechanics and Analytical Mechanics Newtonian: –Particles system –Force –Constraints Analytical: –Mechanics system –Energy (generalized coordinates) –Need not to consider constraints

6. Limitation v << cc= 2.997925 × 10 8 m/s speed of light Relative theory h h= 6.626 × 10 -34 J s Planck constant E × t ~ h or small size Quantum theory

7. Chapter One: Newtonian Mechanics 哲学推理规则 除去那些真实的而又足以说明自然界事物的表象的原因之外， 我们承认自然界事物没有更多的原因（简单性） 所以，对于同样的自然界的结果，我们必须尽可能地归之于同 样的原因 （因果性） 事物的属性，即不允许增强也不允许削弱，凡是在我们的实验 所能到达的范围内发现属于一切物体的属性，都应该视为一切 物体的普遍属性 （统一性） 在实验哲学上，我们把用一般归纳法从现象推导的命题，看作 准确的或很接近于真实的，虽然可以想象出任何相反的假设， 但是直到其他现象出现而使其变得更准确或出现例外之前，仍 应如此看待 （真理性）

8. Reference 郎道，《力学》 Goldstein ， Classical Mechanics 金尚年，《理论力学》 吴大猷，《古典动力学》 金桂林、潘孝仁、施善定 《理论物理学》

9. Chapter One The Principle of Least Action

10. Position of a particle

11. A system of N particles The number of independent quantities which must be specified in order to define uniquely the position of a system is called the number of degrees of freedom

12. Constrains 2 particles system 6 coordinates 4 constrains The number of degrees of freedom =2

13. Generalized coordinates A set of quantities q i i=1,2,…,s; which completely define the position of a system with s degrees of freedom are called generalized coordinates There is a great variety of coordinates may be employed The quantities are called generalized velocities

14. Mechanical state of a system is not determined when the values of the generalized coordinates are specified at an instant. From experience that the state of the system is completely determined if all the coordinates and velocities are simultaneously specified.

15. Hamilton’s principle Forget Newton’s Law now!

19. Lagrange’s equation Second-order differential equation

21. Frame of reference in which space is homogeneous and isotropic and time is homogenous Inertial Frame

22. A particle moving freely in an inertial frame Space is homogeneous: –System keeps the same under parallel movement x  x+dx Space is isotropic: –System keeps the same under rotation Time is homogeneous: –System keeps the same at any instant

23. In an inertial frame, any free motion takes place with a velocity which is constant in both magnitude and direction. Law of Inertia

24. Chapter One: Newtonian Mechanics Galileo’s principle of relativity The mechanical laws of physics are the same for every observer moving uniformly with constant speed in a straight line.

25. Chapter One: Newtonian Mechanics Galilean transformation Inertial frame S’ move with velocity v relative to inertial frame S Time t is absolute and concerns nothing with the objects and space.

26. S S’

27. m is called mass of the particle

28. Close system: system of particles interact with one another, but with on other bodies Potential energy which describes the interaction among particles

31. If A+B is close system Suppose the mechanical state of B is known, which means q B are given..

34. l l 11 22 m m

35. f m2m2 m1m1 x l