Mathematical Principles of Mechanics and Electromagnetism: by Chao-cheng Wang

By Chao-cheng Wang

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Extra info for Mathematical Principles of Mechanics and Electromagnetism: Part A: Analytical and Continuum Mechanics

Sample text

6) where the first term on the right-hand side denotes the moment of the resultant force relative to the origin, and the second term denotes the moment (a couple) relative to the center of mass. 7) G. This equation is also called the moment of momentum equation (relative to the center of mass), and it can be stated as Euler's second law. Euler's Second Law (center-of-mass version). 4. 7), we obtain Iw + wxIw = G. 8) 26 Cbapter 1 Sec. 4 In component form relative to an imbedded basis {f;} at the center of mass this equation is given by i = 1,2,3.

24), is a manifold of dimension 2n. 9K)2. 1(q)}. p) at each point (q, p) E ~*(1). 1 = I, ... 1 = I, ... 1 coordinate function. 1}. 1' H r > = 0 for all. 1, = I, ... ,n. p). In Sections 51 and 55, IVT-2, we have defined the concepts of differential forms and exterior derivative. 34), IVT-2. We now apply these concepts to the cotangent bundle ~*(1). First, ~*(1) is endowed with a canonical I-form e, which has the component form r + (8. 1} Hamiltonian Systems in Phase Space Sec. A), the components of 8 in H-1 being P-1 and the components in K-1 being O.

Newton's Second Law. Relative to an inertial frame the motion of a particle satisfies the following equation of motion: d (it I(t) = f(t), tE~. 1 ) 24 Sec. 1) f = 0 implies I = const. This remark does not mean that the first principle is a consequence of the second principle, however, since without the existence of an inertial frame, the equation of motion is meaningless. 1) cannot possibly be equal in all frames, since the right-hand side is frame indifferent, while the left-hand side is not. 1) can be rewritten as ma(t)=m d 2x(t) dt 2 =f(t), tE~.