Determining equations for cinf-structures

The determining equations for ${\mathcal{C}}^{\mathrm{\infty }}$-structures are the system of partial differential equations (PDEs) that the components of the vector fields must satisfy to constitute a valid cinf-structure.

General Definition

For a set of vector fields $\{X_1,\dots ,X_{m-1}\}$ and the vector field $A$ associated with the ODE, the determining equations arise from the involutivity conditions:

1. Compatibility with the associated vector field$$[X_i,A]\in \mathcal{S}(\{A,X_1,\dots ,X_i\})$$
2. Internal compatibility:$$[X_i,X_j]\in \mathcal{S}(\{A,X_1,\dots ,X_i\})\text{for }j<i$$

These conditions ensure that each distribution in the chain is involutive and of the correct rank.

For Canonical Structures

In the specific case of a canonical cinf-structure, where the vector fields are constructed using the recursive prolongation formula, the system simplifies significantly.

That is, we can start with lambda-generated vector fields, $X_i$ entirely determined by the functions ${\lambda }_1,\dots ,{\lambda }_{m-1}$, and the determining equations become a system of PDEs such that:

• Structure functions: The unknowns are the functions ${\lambda }_i(x,u,\dots ,u_{m-1})$.
• Less PDEs: Due to the particular form of this vector fields some of the equations are trivially satisfied.
• Sequential solution: The triangular nature of the canonical structure often allows the equations to be solved sequentially. One can determine ${\lambda }_1$, then use it to set up the equations for ${\lambda }_2$, and so on.
• Ansatz: To solve these complex PDEs, one often applies an ansatz for cinf-structures (e.g., assuming ${\lambda }_i$ depends on fewer variables) to reduce the system to a tractable form.

See also: recursive prolongation formula, canonical cinf-structure, ansatz for cinf-structures