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The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations.
Dynamics (mechanics) Classical mechanics; Isolated physical system. Lagrangian mechanics; Hamiltonian mechanics; Routhian mechanics; Hamilton-Jacobi theory; Appell's equation of motion; Udwadia–Kalaba equation; Celestial mechanics; Orbit; Lagrange point. Kolmogorov-Arnold-Moser theorem; N-body problem, many-body problem; Ballistics
Static problem For a set of N numbers find the maximal one. The problem may be solved in O(N) time. Dynamic problem For an initial set of N numbers, dynamically maintain the maximal one when insertion and deletions are allowed. A well-known solution for this problem is using a self-balancing binary search tree. It takes space O(N), may be ...
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems.Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does ...
Chapter fifteen, The General Theory of Orbits, describes two-dimensional mechanics of a particle subject to conservative forces and discusses special-case solutions of the Three-body problem. [9] The last chapter includes discussions of solutions of the problems of previous chapters by integration of series, particularly trigonometric series. [9]
One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H ( q , p , t ) {\displaystyle H({\boldsymbol {q}},{\boldsymbol ...