In The Variational Principles of Mechanics Lanczos describes what he calls 'vectorial mechanics': the process of solving mechanical problems by recourse to the immediate consequences of Newton's laws, exemplified by decomposing the forces that act on particles and applying Newton's differential equation $(F=\frac{dp}{dt})$ to derive solutions. Lanczos contrasts such a process with 'variational mechanics', where one focuses on the analysis and synthesis of work functions and kinetic energies rather than forces and momenta. When comparing the two, Lanczos describes the vectorial mechanics approach to the analysis of constraints:

The vectorial treatment has to take account of the forces which maintain the constraints and has to make definite hypotheses concerning them. Newton's third law of motion, "actions equals reaction", does not embrace all cases. It suffices only for the dynamics of rigid bodies. (2nd Ed, p. xx-xxi)

Here are my related questions:

-When considering constraints imposed on the motions of individual particles, why is Newton's third law within vectorial mechanics sufficient for problems involving rigid bodies but not other bodies?

-What would be examples of the failure of the third law for other bodies?

Edit: removed a question that wasn't sufficiently well-defined ("Is variational mechanics able to solve all constrained motion problems?")


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.