An application of geodesic modeling of second-order differential equations
It is proved that for the second-order ordinary differential equation (generalizing the electromagnetic field equation \[ (\nabla /dt)(m(x)dx/dt)=(2e/c)F^{\#}(x,dx/dt)-(e^ 2/2c^ 2m(x))\text{grad}(1/f) \] for any points \(p,q\in M\) not conjugate, in a Riemannian manifold M, in every homotopy class there exists a continuum of distinct solutions joining p with q. Here \(F^{\#}\) is defined by \(F(X,Y)=g(X,F^{\#}Y)\) for g a complete Riemannian metric (\(\nabla\) is its affine Levi-Civita connection), F a closed 2-form on M, m,f function on M, \(f>0\). The idea is to lift the equation to a geodesic one in a higher dimensional manifold (a geodesic modeling). The reader will find a lot of references to applications in physics.
second-order ordinary differential equation
scientific article; zbMATH DE number 4013571