faizan2722 Posted February 6, 2015 Share Posted February 6, 2015 (edited) Hello Everyone, What are the most important unsolved problems in classical mecahics (mechanics of rigid-bodies, mechanics of deofrmable bodies, fluid mechanics) Edited February 6, 2015 by faizan2722 Link to comment Share on other sites More sharing options...
ajb Posted February 6, 2015 Share Posted February 6, 2015 A big question that spring to mind is Arnold diffusion. For n ≥ 3 degrees of freedom Arnold conjectured that a general small perturbation of a completely integrable system leads to instability. There are nice examples that show Arnold diffusion but the conjecture is not properly proved. In fluid dynamics there is the question of the existence of smooth solutions to the NavierStokes equation. A shameless plug for my own work now, my collaborators and I were the first to write down the higher order Euler-Lagrange equations on a Lie algebroid [1]. We did this fully geometrically using the ideas of Tulczyjew and some techniques from systems subject to affine constraints. We have now found out that independently, Martínez [2] using variational calculus also derived the same equations. References [1] Andrew James Bruce, Katarzyna Grabowska and Janusz Grabowski, Higher order mechanics on graded bundles, arXiv:1412.2719 [math-ph] [2] Eduardo Martínez, Higher-order Variational Calculus on Lie algebroids, arXiv:1501.06520 [math-ph]. 1 Link to comment Share on other sites More sharing options...
studiot Posted February 6, 2015 Share Posted February 6, 2015 faizan What are the most important unsolved problems in classical mecahics Well it partly depends what you mean by important and also what you include in classical mechanics. Are you thinking in purely theoretical terms or do applications count? There is much work yet to be undertaken int the field of non linear mechanics. You have listed several application areas which are the most heavily studied. but how about The mechanics of granular assemblies The mechanics of fracture and fatigue The mechanics of fibre bundles and other composites High and low termperature mechanics Near vacuum statistical mechanics the mechanics of non linear wavelike motion Link to comment Share on other sites More sharing options...
ajb Posted February 7, 2015 Share Posted February 7, 2015 The mechanics of fibre bundles and other composites Not to be confused with mechanics on fibre bundles! Link to comment Share on other sites More sharing options...
studiot Posted February 7, 2015 Share Posted February 7, 2015 Not to be confused with mechanics on fibre bundles! Pun intended. Link to comment Share on other sites More sharing options...
John Cuthber Posted February 7, 2015 Share Posted February 7, 2015 The weather. Link to comment Share on other sites More sharing options...
MigL Posted February 7, 2015 Share Posted February 7, 2015 Darn weather ! Snow up to my knees from the last snowstorm, and its snowing again. Where is Global Warming when you need it ! ( just kidding ) Link to comment Share on other sites More sharing options...
faizan2722 Posted February 8, 2015 Author Share Posted February 8, 2015 Well it partly depends what you mean by important and also what you include in classical mechanics. Are you thinking in purely theoretical terms or do applications count? There is much work yet to be undertaken int the field of non linear mechanics. You have listed several application areas which are the most heavily studied. but how about The mechanics of granular assemblies The mechanics of fracture and fatigue The mechanics of fibre bundles and other composites High and low termperature mechanics Near vacuum statistical mechanics the mechanics of non linear wavelike motion Please mention theoratical terms as well as application problems. Link to comment Share on other sites More sharing options...
studiot Posted February 8, 2015 Share Posted February 8, 2015 I thought ajb did a pretty good rundown of theoretical. I am an applied man myself, but isn't non linear stuff pretty theoretical? Link to comment Share on other sites More sharing options...
ajb Posted February 9, 2015 Share Posted February 9, 2015 I thought ajb did a pretty good rundown of theoretical. I only gave two well-known problems. Because of this thread I was thinking about collating 5 to 10 open problems in classical mechanics, I will have to ask around and see if people will send me problems. I will be at a conference soon which would give me a great chance to ask about problem. Anyway, another problem is the general n-body problem. There is no known general analytic solution for the n-bodies interacting via Newtonian gravity (1/r^2 force). The two body problem is okay and there are lots of results for small n when we have some restrictions on the motion. People have developed all kinds of tools here but the problem remains open. Another very interesting question is 'what is the covariant Hamiltonian formalism for field theories?'. There are several similar approaches here of multisymplectic and polysymplectic and refinements of these two notions. None of this is a clear of the standard Hamiltonian approach to mechanics and there are conceptional problems with brackets and then the quantisation. The literature is is quite large and the ideas are not new, just it does not fit together as well as the classical situation. Maybe related to the above is the formulation of classical field theories on Lie algebroids. There are again some different approaches on would like to find 'minimal approach' in which one does not have to add too much by hand. Higher order field theories in this context have yet to be formulated, but I know of one person in Spain who is probability looking into this now. Link to comment Share on other sites More sharing options...
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