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 describing the vectorial mechanics approach to problems involving multiple bodies, Lanczos remarks

This force-analysis [the standard approach of vectorial mechanics] sometimes becomes cumbersome. The unknown nature of the interaction forces makes it necessary to introduce additional postulates, and Newton thought that the principle "action equals reaction", stated as his third law of motion, would take care of all dynamical problems. This, however, is not the case, and even for the dynamics of a rigid body the additional hypothesis that the inner forces of the body are of the nature of central forces had to be made. In more complicated situations the Newtonian approach fails to give a unique answer to the problem. (p. 3-4)

Here are my related questions:

-What are examples of the failure of Newton's third law to solve problems involving multiple bodies and requiring the additional assumption of central forces?

-What are examples of 'more complicated situations' where Newton's third law fails to give a unique answer to multiple body problems?

-Assuming a variational mechanics approach is capable of solving a multiple body problem, is the motion that the variational mechanics approach predicts always unique? (Not asking whether the Lagrangian is always unique, but is the prediction for the motion of an object [the position of the object at each instant of time] always unique [of course allowing for the typical symmetries imposed on classical physics equations, e.g. position and time shifts]).

Note: I am aware that very similar questions were asked in this post in 2013, but the two answers there don't seem to be sufficiently explained or addressing to the questions.


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