• Most text books give a derivation of Euler Lagrange equations from Newton’s laws using d Alembert’s principle. See for example [1, 3, 6]
  • If a transformation of generalized coordinates is made from \( q_k \) to another set \( Q_k \), the Lagrangian, \( L^′ \), in terms of new coordinates \( Q_k \) can be obtained by expressing the coordinates \( q_k \) in terms of new coordinates. Thus

$$ L^′ (Q, \dot Q, t) = L (q(Q), \dot q (Q, \dot Q), t) \hspace{7em} (37) $$

7.1 Want to learn related skills? This will be critical later for problem-solving.

Optional now, but recommended for problem-solving later.

  • Simple Examples of constraints
  • Counting the number of independent variables to be determined
  • Examples of choosing generalized coordinates.

7.2 Want to dig deeper? or What you missed here?

Most of this will not be required later.
Here is a short list of what you can explore.

  • Different types of constraints:
    The constraints are classified according to whether or not they are scleronomic or rheonomic, holonomic or nonholonomic, and conservative or nonconservative.
  • Principle of virtual work illustrated with examples
  • d’ Alembert’s principle and generalized coordinates with an application.

References

  1. Goldstein Herebert et al, Classical Mechanics
  2. Landau, L. D. and Lifshitz, E. M., Mechanics, Volume 1 of Course of Theoretical Physics, Butterworth-Heinenann Linacre House, Jordan Hill, Oxford 3rd Ed.(1976)
  3. Calkin, M.G. Lagrangian and Hamiltonian Mechanics, World Scientific Publishing Co. Pte. Ltd. (1996);
    This book has several interesting problems on d’ Alembert’s principle.
  4. Whitaker E. T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Dover Publications (1944)
  5. Rana, N. C. and Jog P. C., Mechanics Tata McGraw Hill Publishing Co, New Delhi (1991)
  6. Greiner, Walter, Clasical Mechanics – Systems of Particles and Hamiltonian Dynamics Springer New York (2003);
    Lot of details about types of constraints can be found in this book. It has examples of applications of principles of virtual work and d’ Alembert’s principle.