By Chao-cheng Wang
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Extra resources for Mathematical Principles of Mechanics and Electromagnetism: Part A: Analytical and Continuum Mechanics
Example text
Indeed, given any motion {x" T E~} in the Newtonian space-time W, we can always reduce the velocity and the acceleration to zero by choosing the origin of the frame at x,, T E ~. Relative to some other frames, the velocity and the acceleration of this motion need not vanish of course. We now state the dynamical principles for a particle. r Newton's First Law. There exists a particular frame of reference, called an inertial frame, relative to which the linear momentum of a particle remains constant when the force acting on the particle vanishes.
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~. 3) for the coordinate functions Xi(t) of the motion. 3) and determine Xi(t) when fi(t) and certain initial values of xi(t) are known.
31) is a system of second-order differential equations for q,1(t). 31) are known, we can solve the system and determine q,1(t) as functions of t provided that certain initial values, say q,1(o) and q,1(o), are given. 24). We shall see certain explicit forms of Lagrange's equations in the following section. 6. Explicit Forms of Lagrange's Equations In this section we derive some explicit forms of Lagrange's equations for the special case when the constraint is time independent. As explained in Section 3, the kinetic energy E is given by a homogeneous quadratic function of q,1 in this case; cf.