A first course in continuum mechanics: for physical and - download pdf or read online

By Y. C. Fung

ISBN-10: 0130615242

ISBN-13: 9780130615244

ISBN-10: 0130615323

ISBN-13: 9780130615329

Revision of a vintage textual content by means of a exotic writer. Emphasis is on challenge formula and derivation of governing equations. new version good points elevated emphasis on functions. New bankruptcy covers long term adjustments in fabrics less than pressure.

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Additional info for A first course in continuum mechanics: for physical and biological engineers and scientists

Example text

The space of (q, q). ˙ 54 CHAPTER 2. LAGRANGE’S AND HAMILTON’S EQUATIONS The motion of the system sweeps out a path in the space (q, q, ˙ t) or a path in (q, P, t). Along this line, the variation of L is ∂L ∂L ∂L dt dq˙k + dqk + ∂ q˙k ∂qk ∂t ∂L dt Pk dq˙k + P˙k dqk + ∂t dL = k = k where for the first term we used the definition of the generalized momentum and in the second we have used the equations of motion P˙k = ∂L/∂qk . Then examining the change in the Hamiltonian H = k Pk q˙k − L along this actual motion, (Pk dq˙k + q˙k dPk ) − dL dH = k = k ∂L dt.

11) to get the third line. Plugging in the expressions we have found for the two terms in D’Alembert’s Principle, j d ∂T ∂T − − Qj δqj = 0. dt ∂ q˙j ∂qj We assumed we had a holonomic system and the q’s were all independent, so this equation holds for arbitrary virtual displacements δqj , and therefore d ∂T ∂T − − Qj = 0. 12) dt ∂ q˙j ∂qj Now let us restrict ourselves to forces given by a potential, with Fi = −∇i U({r}, t), or Qj = − i ∂ri ∂ U˜ ({q}, t) · ∇i U = − . ∂qj ∂qj t Notice that Qj depends only on the value of U on the constrained surface.

The phase point η(t) is also called the state of the system at time t. Many qualitative features of the motion can be stated in terms of the phase curve. Fixed Points There may be points ηk , known as fixed points, at which the velocity function vanishes, V (ηk ) = 0. This is a point of equilibrium for the system, for if the system is at a fixed point at one moment, η(t0 ) = ηk , it remains at that point. At other points, the system does not stay put, but there may be sets of states which flow into each other, such as the elliptical orbit for the undamped harmonic oscillator.

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A first course in continuum mechanics: for physical and biological engineers and scientists by Y. C. Fung

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