The Euler-Lagrange equations of the Lagrangian within our class. Lagrangian is free, and that it should not be confused with the original Lagrangian L of the nonholonomic system.
For full details, see . As- tem to our nonholonomic system. On the other hand, the two Lagrangians for the regular Lagrangian. For more details, can be found, then its Hessian i j is a multiplier. The above conditions are generally referred to as the Helmholtz conditions. In fact, we transformed into a Hamiltonian one, by making use of the can easily derive more algebraic conditions see e.
One can repeat the above Proposition 2. A second route to given by additional algebraic conditions arises from the derivatives!! Lagrangian 7 is On the other hand, for a nonholonomic integrator of!! There are, however, many more a system in our class: we can use either a nonholonomic possibilities to obtain a discrete Lagrangian and discrete integrator for the original Lagrangian 1 and constraints constraints.
For example, one could take a symmetrized 2 , or we can use a variational integrator for one of the version of the above procedure and use discrete La- Lagrangians 6 and 7 we have found in Proposition 1. From this dis- 2 h crete variational principle one obtains the so-called dis- crete Euler-Lagrange equations as follows. The bottom line of the next sections is the following These integrators preserve the symplectic and conserva- one. If a free Lagrangian for the nonholonomic system tive nature of the algorithms.
In the next sections, we will test this conjecture on a few of the classical exam- ples in our class: the vertically rolling disk, the knife edge and the nonholonomic particle. It will be convenient that for those systems an exact solution of the nonholonomic equations 3 is readily available.
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With that the nonholonomic La- circular path. It is true, however, that the variational solu- grangian and constraints are simply tion deviates from the circle predicted by the initial con- 1 1 1 ditions of the solution 12 in grey in figure 2. If we do so, we obtain the matching circle from the second type 7. The simplest choice is probably in dots in figure 3.
The able. This integrator considers the constraints as a con- stant along the nonholonomic motion. The circle in that figure is the one we had before, i. It shows that the modified inte- grator has the same circular behaviour as the variational integrator, and on top, it keeps the constraints conserved, see the box symbols on the zero level in figure 6.
Figure 3: Vertically rolling disk: circular path.go
Geometry Of Nonholonomically Constrained Systems
It is clear that the vari- ational integrator with crosses does a better job than the nonholonomic one with circles. Figure 5: Vertically rolling disk: the modified integrator. The results for the variational integrator in black with cross symbols remain accurate and more or less unchanged. Positive is that, although the variational integrator does not conserve this constraint, it reasonably As was the case with the vertically rolling disk, also the oscillates around the zero level.
Moreover, there is a solutions of the knife edge form a circular path in the method to fix this problem. Needless to say, the results is a free Lagrangian for the nonholonomic particle. In above are, of course, very partial and are they are only in- each of the figures 9 and 10 the dashed black curve repre- tended to motivate further investigation on this topic.
For sents the exact solution, the thick black the variational so- example, we need to check if more involved discretiza- lution and the thick grey the nonholonomic solution. The tion procedures, such as the ones mentioned at the end of figures show that both the variational method and the non- section 3A demonstrate the same behaviour as the one we holonomic one do not give very accurate solutions. How- have encountered so far. A first result is the following. The class of nonholonomic systems treated above is very Proposition 3.
There does not exists a regular Lagrangian restricted.
New developments on the geometric nonholonomic integrator
The reason is, of course, that the search for a for the second order systems As before, the proof follows from a careful analy- technical to be treated in the full generality of a nonholo- sis of the algebraic conditions which can be derived from nomic systems with an arbitrary given Lagrangian and ar- the Helmholtz conditions.
Also, since there are infinitely many possible choices for the associated systems, it is not For systems with more than one constraint, the result is clear from the outset which one of them will be varia- still open.
Remark that the proposition does not exclude tional, if any. For these reasons, future extensions of the obtained re- sults will strongly depend on well-chosen particular new Acknowledgments examples. For example, we could try to find a free La- grangian for a nonholonomic system with a potential of TM acknowledges a Marie Curie Fellowship within the the form V r2. Prince, Second-order ordi- Flanders. Krupka and D. Saunders eds. Marsden and M.
West, Discrete mechanics and References variational integrators, Acta Num. Bloch, Nonholonomic Mechanics and Control,  R. Santilli, Foundations of Theoretical Mechanics Springer I, Spinger Bloch, O. Fernandez and T. Mestdag, Hamil- tonization of nonholonomic systems and the inverse problem of the calculus of variations, to appear in Rep.
First, we study the GNI versions of the symplectic-Euler methods, paying special attention to their convergence behaviour. Then, we construct an extension of the GNI in the case of affine constraints.
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