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On complete solutions and complete singular solutions of second order ordinary differential equations

Masatomo Takahashi (2007)

Colloquium Mathematicae

A complete solution of an implicit second order ordinary differential equation is defined by an immersive two-parameter family of geometric solutions on the equation hypersurface. We show that a completely integrable equation is either of Clairaut type or of first order type. Moreover, we define a complete singular solution, an immersive one-parameter family of singular solutions on the contact singular set. We give conditions for existence of a complete solution and a complete singular solution...

On Galilean connections and the first jet bundle

James Grant, Bradley Lackey (2012)

Open Mathematics

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order...

On implicit Lagrangian differential systems

S. Janeczko (2000)

Annales Polonici Mathematici

Let (P,ω) be a symplectic manifold. We find an integrability condition for an implicit differential system D' which is formed by a Lagrangian submanifold in the canonical symplectic tangent bundle (TP,ὡ).

On the envelope of a vector field

Bernard Malgrange (2011)

Banach Center Publications

Given a vector field X on an algebraic variety V over ℂ, I compare the following two objects: (i) the envelope of X, the smallest algebraic pseudogroup over V whose Lie algebra contains X, and (ii) the Galois pseudogroup of the foliation defined by the vector field X + d/dt (restricted to one fibre t = constant). I show that either they are equal, or the second has codimension one in the first.

On the Mayer problem. I: General principles

Veronika Chrastinová, Václav Tryhuk (2002)

Mathematica Slovaca

On the Mayer problem. II: Examples

Veronika Chrastinová, Václav Tryhuk (2002)

Mathematica Slovaca

On the tangential velocity arising in a crystalline approximation of evolving plane curves

Shigetoshi Yazaki (2007)

Kybernetika

In a crystalline algorithm, a tangential velocity is used implicitly. In this short note, it is specified for the case of evolving plane curves, and is characterized by using the intrinsic heat equation.