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An important class of systems in classical mechanics are those of *Newton type*:
d^{2}*q* / d*t*^{2} = *M*(*q*),
where *q* = (*q*_{1},…,*q*_{n})
is a point in **R**^{n}.
Equation of this form are obtained from Newton's second law
(acceleration equals force divided by mass)
if the force field depends only on positions and not on velocities.

Newton systems appearing in physics often take the form
d^{2}*q* / d*t*^{2} = −∇*V*(*q*)
for some scalar function *V*(*q*)
called the *potential*.
This reflects the conservation of the energy (kinetic + potential);
in mathematical terms,
*E* = *T* + *V* = (1/2) ∑ (d*q*_{i} / d*t*)^{2} + *V*(*q*)
is a constant of motion for the system.
There is a lot of mathematical machinery for dealing with such
systems: Lagrangian mechanics, Hamiltonian mechanics, etc.

In order to have any hope of integrating the system (that is, computing
the solution explicitly),
one must find sufficiently many other constants of motion besides the energy.
Such extra constants of motion may or may not exist,
depending on what the potential *V* is.

For those rare systems where it is possible to perform the integration
explicitly, this is often accomplished through the powerful
Hamilton–Jacobi method, which uses separation of variables in a clever
way in order to compute the solution. A potential for which this
method is applicable is called a
*separable potential*.
(There is a remarkable algorithm, due to Stefan Rauch-Wojciechowski and Claes
Waksjö, for determining whether a given potential is separable or not.)

*Quasi-potential systems* are more general;
they are Newton systems having a constant of motion
which is energy-like in the sense that it depends quadratically
on the the velocity components:
*F* = (1/2) ∑∑ *A*_{ij}(*q*) (d*q*_{i} / d*t*) (d*q*_{j} / d*t*) + *W*(*q*).
(Such constants of motions appear besides *E* for systems given by
a separable potential, but here we are not assuming that the system
has a potential at all.)

By simply computing d*F*/d*t* and setting it to zero,
one finds that the right-hand side *M* of such a Newton system
(written as a column vector) must be given by
*M*(*q*) = −*A*(*q*)^{−1}∇*W*(*q*)
for some scalar function *W*(*q*), called the *quasi-potential*,
and some *n*×*n* symmetric
matrix function *A*(*q*) satisfying the condition
∂_{i} *A*_{jk} + ∂_{j} *A*_{ki} + ∂_{k} *A*_{ij} = 0
for all indices *i*,*j*,*k* = 1,…,*n*.
(Such a matrix is called a *Killing tensor for the Euclidean metric*.)

When I started out as a PhD student, my first project was to study quasi-potential systems in two dimensions together with Stefan Rauch-Wojciechowski and Krzysztof Marciniak.

Later, when I tried to generalize our results to *n* dimensions,
it turned out that one has to restrict the class of quasi-potential systems
a bit in order for the theory to work.

A *cofactor system* is a quasi-potential system where the matrix
*A*(*q*) has the following special form:
*A*(*q*) = cof *J*(*q*),
where *J*_{ij}(*q*) = α*q*_{i}*q*_{j} + β_{i}*q*_{j} + β_{j}*q*_{i}+γ_{ij}.
Here, "cof" denotes taking the cofactor matrix (defined by the formula
*X* cof *X* = (det *X*)*I* ;
also called the adjoint matrix),
and α, β_{i} and
γ_{ij} = γ_{ji}
are real parameters.
The corresponding quadratic constant of motion *F*
is said to be of *cofactor type*.

(Notice that if *n* > 2, then
*A*(*q*) depends on the parameters α, β_{i},
γ_{ij}
in a fairly complicated way,
so the set of matrices of the form
*A*(*q*) = cof *J*(*q*) is
a nonlinear variety in the (linear) solution space to
∂_{i} *A*_{jk} + ∂_{j} *A*_{ki} + ∂_{k} *A*_{ij} = 0.
When *n* = 2, these matrices fill out the whole solution space.)

I called matrices *J*(*q*) of this particular type
*elliptic coordinates matrices* because of their relation to
Jacobi's generalized elliptic coordinates:
in fact,
the eigenvalues *u*_{1},…,*u*_{n}
of such a matrix are functions of *q*,
and this relationship *u* = *u*(*q*) is
precisely the change of variables from Cartesian coordinates
(*q*_{1},…,*q*_{n})
to elliptic coordinates
(*u*_{1},…,*u*_{n}).
(In three dimensions, the level surfaces of elliptic coordinates are
ellipsoids,
one-sheeted hyperboloids,
and
two-sheeted hyperboloids,
intersecting at right angles as illustrated below.)

As I learned later, such matrices *J*(*q*) had appeared earlier in the
literature under various names; for example, Sergio Benenti had
referred to a special case as *inertia tensors*. Mike Crampin and
Willy Sarlet (who generalized the concept of cofactor systems from
Euclidean space to Riemannian manifolds) called them **special
conformal Killing (SCK) tensors**, which is the name that seems to
be most used nowadays.

Some of the classical machinery for potential systems can still be
used for cofactor systems. In particular (although I will not try to
explain here exactly what it means), a cofactor system can be embedded
in a (2*n*+1)-dimensional phase space in such a way that it is
Hamiltonian with respect to a certain noncanonical Poisson structure
which depends on the parameters appearing in the matrix *J*(*q*).
Maciej Błaszak and Krzysztof Marciniak have shown that cofactor systems
are essentially potential systems up to reparametrization of time.
Work in progress by Alain Albouy and me shows that cofactor systems
are precisely those Newton systems which admit a potential
"in the sense of projective dynamics".

A *bi-cofactor system* (or a *cofactor pair system*)
is a system which can be written as
a cofactor system in two independent ways.
Thus, by definition, a bi-cofactor system is a system which has
**two** independent constants of motion **of cofactor type**,
with SCK tensors *J*_{1} and *J*_{2}, respectively.

It is a remarkable and quite nontrivial fact that a bi-cofactor system
automatically has ** n** quadratic constants of
motion! My original proof of this used that each of the two
cofactor-type constants of motion gives rise to a Poisson structure,
so that the system is bi-Hamiltionan. Crampin and Sarlet gave a more
direct proof which is very elegant, but uses a bit of machinery about
differential operators of a certain kind ("Frölicher–Nijenhuis
derivations of type

There is a simple recipe (which I skip here) for finding
the *n*−2 extra constants of motion from the two given ones.
Provided that all these constants of motion are functionally independent,
the system can be considered as completely integrable in the
Liouville–Arnold sense.
In particular, when one of the two given constants of motion is
the usual energy *E*, the system is given by a separable potential, and
this theorem explains a lot of the structure of the constants of motion
for such a system.

Now suppose that that we have two cofactor pair systems,
both associated to the same pair
(*J*_{1}, *J*_{2})
of SCK tensors.
The multiplication theorem gives a formula for producing a third
cofactor system associated to (*J*_{1}, *J*_{2}).
Briefly, it works like this: for each of the two systems,
compute the full set of *n* constants of motion
(as guaranteed by the "2 implies *n*" theorem),
and put the *n* quasipotentials as coefficients in a polynomial
of degree *n*−1 in a parameter μ.
Then multiply the two polynomials, and reduce modulo the degree *n*
polynomial det(*J*_{1} + μ *J*_{2}).
The resulting polynomial contains the quasipotentials for the
resulting "product" system.

A special case of this formula, when *J*_{1} = [1,0; 0,−1]
and *J*_{2} = [0,1; 1,0],
corresponds to the fact that the product of two holomorphic functions
is again holomorphic. (The quasipotentials in this case are the real
and imaginary parts of the holomorphic functions.)
Another way of saying this is that there is a multiplicative structure
on the solution set of the Cauchy–Riemann equations. Jens Jonasson has
developed this point of view further, and studied linear systems of
PDE with a similar multiplicative structure on their solution set.

Starting from the trivial system
d^{2}*q* / d*t*^{2} = 0,
which is a bi-cofactor system with constant quasipotentials
for any pair of SCK tensors,
and taking "powers" with respect to this multiplication,
one can produce infinite sequences of nontrivial bi-cofactor systems.

Last modified 2015-01-07. Hans Lundmark (hans.lundmark@liu.se)