DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 553–561

VARIATIONAL COMPATIBILITY OF FORCE LAWS

MARTIN MRÁZ AND JANA MUSILOVÁ

Abstract. The paper presents a simplified version of conditions for the variational compatibility of general forces in equations of motions of physical systems with the finite number of degrees of freedom. The forces are classified with respect to their transformation properties, and the classification is discussed from the physical point-of-view.

1. Introduction

The variational principle is the fundamental element of most physical theories, particularly mechanics, classical electrodynamics as well as the theory of gravitation. However, the interpretation of phenomena which can be explained within the framework of these disciplines need not be restricted to the use of the variational principle. Alternative approaches are often admitted including the wider region of phenomena than those based on the variational principle. As the typical example, the classical Newtonian mechanics can be mentioned: it enables us to take into account the more extensive class of force laws compared with the Lagrangean (variational) mechanics. Thus, the aim can be formulated: what are the conditions for the variational compatibility of force laws in classical physics? The answer is, of course, contained in the solution of the inverse variational problem formulated and solved by Helmholtz [2] and then treated by many authors from various points-of-view. In the presented contribution, this problem is solved by the use of a very simple integration procedure, and its physical aspects are pointed out. Especially, the variationally compatible force laws in classical physics are characterized, and the role of coordinate transformations is discussed in detail. The force laws are classified into basic classes containing the so-called real, inertial, geometrical and metric forces. The invariance aspects are not studied here.

1991 Mathematics Subject Classification. 49H05, 70H35.

Key words and phrases. Inverse variational problem, variational force laws.

Research supported by the Grant No. 201/98/0853 of the Czech Grant Agency, and Project No. VS 96003 (“Global Analysis”) of the Ministry of Education, Youth and Sports of the Czech Republic.

This paper is in final form and no version of it will be submitted for publication elesewhere.

2. Basic notations

We briefly summarize the basic concepts and notations which are used in our considerations. Throughout the paper, we adopt the commonly used notation as presented e. g. in [3,5,8].

Consider a mechanical system with m degrees of freedom. We can describe all reasonable cases of the time development of its configuration by means of the fibered manifold (Y, π, X), with the one-dimensional base X, total space Y, dim Y = m + 1, and the projection π. For the dynamical description of the system, we introduce the r−jet prolongation of the fibered manifold denoted by (J^{r}Y, π^{r}, X), r ≥1, and the spaces Ω_{k}^{r} of differential k−forms, k ≥1, functions for k = 0, defined on J^{r}Y. In the standard notation, π^{r,s} : J^{r}Y → J^{s}Y, r ≥s≥0, are the canonically defined projections. Let V ⊂ Y be an open set. We denote (V, ψ = (t, q^{σ}))_{σ=1,... ,m} a fibered chart on Y, (U, φ = (t)), U = π(V), the associated chart on X and (V^{r}, ψ^{r} = (t, q^{σ}, q_{1}^{σ}, …, q_{r}^{σ})), V^{r} = (π^{r,0})^{−1}(V), the associated fibered chart on J^{r}Y. By Ω_{k}^{r}V, the space of differential k−forms defined on V^{r} is denoted. Let λ = L dt ∈ Ω_{1}^{1}V be a Lagrangian with the Lagrange function L ∈ Ω_{0}^{1}V, (π, λ) the corresponding Lagrange structure and E_{λ} = E_{σ} dq^{σ} ∧ dt ∈ Ω_{2}^{s}V (s ≥ r is the appropriate integer), the associated Euler-Lagrange form. Functions E_{σ} of the order s are called Euler-Lagrange expressions. Equations E_{σ} = 0 define the system of ordinary differential equations of the order s. The solutions of this system are sections of the underlying fibered manifold, so-called extremals of the given Lagrange structure (π, λ). They represent the real motion of the studied physical system. For r = 1, it holds s = 2, and in the fibered chart (V, ψ = (t, q^{σ}, q_{1}^{σ}, q_{2}^{σ})), we have

E_{σ} = \frac{∂L}{∂q^{σ}} − \frac{d}{dt} (\frac{∂L}{∂q_{1}^{σ}} ) . (1)

The quantities q^{σ}, q_{1}^{σ}, q_{2}^{σ} are the generalized coordinates, velocities and accelerations, respectively.

3. Variational equations

Let E = E_{σ} dq^{σ} ∧ dt ∈ Ω_{2}^{2}V be a so-called dynamical form. The form E and, equivalently, the system of equations E_{σ} = 0 is variational, i. e. arising from a Lagrangian, if and only if it satisfies the well-known Helmholtz conditions (Helmholtz [2]) formulated for the expressions E_{σ} themselves. Variational expressions E_{σ} are of the form E_{σ} = A_{σ} + B_{σν}q_{2}^{ν} with A_{σ}, B_{σν} ∈ Ω_{0}^{1}V satisfying certain additional conditions ( Havas [1]). The alternative approach to the variationality conditions leading to the explicite formulae for tensor of mass (−B_{σν}) and general forces (A_{σ}), based on geometrical concepts, was advised by Novotný [9]. The entirely correct and complete procedure for obtaining these formulae was first given by Krupková [6] and then still treated in more detailed form by the same author [7,8]. The problem was also treated by Sarlet [10]. Our following results are equivalent to those of Krupková. However, our calculation procedure as well as the results themselves are simpler.

Let V ⊂ R^{m} be an open convex set containing the origin 0_{R^{m}}. For the purposes of two following lemmas, we define the mapping

η : [0, 1] × V ∋ (s, y^{k}) ⟶ η(s, y^{k}) = (sy^{k}) ∈ V , 1≤k ≤m .

Lemma 1. Let (B_{ij})_{i,j=1,... ,m}, B_{ij} : V ∋ (y^{k}) ⟶ B_{ij}(y^{k}) ∈ R, 1 ≤ k ≤ m, be a given system of differentiable functions. Then the system of equations

B_{ij} = \frac{∂A_{i}}{∂y^{j}} − \frac{∂A_{j}}{∂y^{i}} , 1≤i, j ≤m (2)

is integrable with respect to the unknown functions A_{i} : V → R if and only if it holds

\frac{∂B_{ij}}{∂y^{k}} + \frac{∂B_{jk}}{∂y^{i}} + \frac{∂B_{ki}}{∂y^{j}} , B_{ij} + B_{ji} = 0 . (3)

The general solution is

A_{i} = y^{j} ∫_{0}^{1}(B_{ij} ∘ η) s ds + \frac{∂F}{∂y^{i}} , (4)

with the arbitrarily chosen function F : V → R.

Proof. Lemma 1 is the direct consequence of the well-known Poincaré lemma applied to the form ω = B_{ij} dy^{i} ∧ dy^{j}. □

Lemma 2. Let p be an integer and let (B_{i_{1}...i_{p}}), 1 ≤ i_{1}, …, i_{p} ≤ m, B_{i_{1}...i_{p}} : V ∋ (y^{k}) ⟶ B_{i_{1}...i_{p}}(y^{k}) ∈ R, 1≤k ≤m, be a given system of differentiable functions. Then the system of equations

B_{i_{1}...i_{p}} = \frac{∂^{p}A}{∂y^{i_{1}} … ∂y^{i_{p}}} (5)

is integrable with respect to the unknown function A : V ⟶ R if and only if the set of quantites B_{i_{1}...i_{p}} is completely symmetrical and the following conditions are satisfied:

\frac{∂B_{i_{1}...i_{p−1}i_{p}}}{∂y^{i_{p+1}}} = \frac{∂B_{i_{1}...i_{p−1}i_{p+1}}}{∂y^{i_{p}}} . (6)

Then the general solution of the system (5) is A = A^{H} + A^{P} where

A^{H} =∑_{l=0}^{p−1} C_{i_{1}...i_{l}}y^{i_{1}} … y^{i_{l}} , A^{P} =y^{i_{1}}...y^{i_{p}} ∫_{0}^{1} ∑_{l=1}^{p} \frac{(−1)^{p−l}}{(p−l)!(l−1)!}s^{p−l}(B_{i_{1}...i_{p}} ∘ η) ds, (7)

with arbitrary constants C_{i_{1}...i_{l}}, 0≤l ≤p − 1.

Proof. A function A : V ⟶ R is the solution of the system (5) if and only if it satisfies the following system of first order equations:

\frac{∂A}{∂y^{i_{1}}} = B_{i_{1}} , \frac{∂B_{i_{1}}}{∂y^{i_{2}}} = B_{i_{1}i_{2}} , …, \frac{∂B_{i_{1}...i_{p−1}}}{∂y^{i_{p}}} = B_{i_{1}...i_{p}} . (8)

The symmetry conditions for B_{i_{1}...i_{p}} follow directly from (5) and the conditions (6) are immediately connected with the Poincaré lemma applied to the last set of equations (8). A^{P} is the particular integral of the system (5). This can be proved by the use of Poincaré lemma for p = 1 and then by induction with respect to p. A^{H} is the general solution of the homogenized system (5). □

The following theorem gives the possibility to recognize the variationally compatible general force laws.

Theorem. Let E = E_{σ} dq^{σ} ∧ dt ∈ Ω_{2}^{2}V be a dynamical form. Then E is variational if and only if its components are of the form

E_{σ}(t, q^{λ}, q_{1}^{λ}, q_{2}^{λ}) = A_{σ}(t, q^{λ}, q_{1}^{λ}) + B_{σν}(t, q^{λ}, q_{1}^{λ})q_{2}^{ν} (9)

in which functions A_{σ}, B_{σν} ∈ Ω_{0}^{1}V obey the following conditions:

B_{σν} = B_{νσ} , \frac{∂B_{σν}}{∂q_{1}^{λ}} = \frac{∂B_{σλ}}{∂q_{1}^{ν}}

A_{σ} =q_{1}^{ν}q_{1}^{λ} ∫_{0}^{1} (\frac{∂B_{σν}}{∂q^{λ}} ∘η− \frac{∂B_{λν}}{∂q^{σ}} ∘η(1−s)) ds+q_{1}^{ν} ∫_{0}^{1} \frac{∂B_{σν}}{∂t} ∘η ds+N_{σν}q_{1}^{ν} +M_{σ} (10)

N_{σν}, M_{σ} ∈ Ω_{0}^{0}V, N_{σν} =−N_{νσ}, \frac{∂N_{σν}}{∂q^{λ}} +\frac{∂N_{νλ}}{∂q^{σ}} +\frac{∂N_{λσ}}{∂q^{ν}} = 0, \frac{∂M_{σ}}{∂q^{ν}} −\frac{∂M_{ν}}{∂q^{σ}} −\frac{∂N_{σν}}{∂t} = 0 .

The corresponding Lagrange function is

L = −q_{1}^{σ}q_{1}^{ν} ∫_{0}^{1}(B_{σν} ∘ η)(1 − s)ds + q^{σ}q_{1}^{ν} ∫_{0}^{1}(N_{σν} ∘ \overline{η}) s ds + q^{σ} ∫_{0}^{1}(M_{σ} ∘ \overline{η}) ds + \frac{dF}{dt} (11)

with the arbitrary function F ∈ Ω_{0}^{0}V. The mappings η and \overline{η} in equations (10,11) are defined by

\overline{η} : [0, 1] × ψ(V) ∋ (s, t, q^{λ}) ⟶ \overline{η}(s, t, q^{λ}) = (t, sq^{λ}) ∈ ψ(V) ,

η : [0, 1] × ψ^{1}(V^{1}) ∋ (s, t, q^{λ}, q_{1}^{λ}) ⟶ η(s, t, q^{λ}, q_{1}^{λ}) = (t, q^{λ}, sq_{1}^{λ}) ∈ ψ^{1}(V^{1}) .

Proof. The variationality of the dynamical form E = E_{σ} dq^{σ} ∧ dt is equivalent to the integrability of equations

E_{σ} = \frac{∂L}{∂q^{σ}} − \frac{d}{dt} (\frac{∂L}{∂q_{1}^{σ}} )=(\frac{∂L}{∂q^{σ}} − \frac{∂^{2}L}{∂t∂q_{1}^{σ}} −q_{1}^{ν} \frac{∂^{2}L}{∂q^{ν}∂q_{1}^{σ}} )−q_{2}^{ν} \frac{∂^{2}L}{∂q_{1}^{ν}∂q_{1}^{σ}} (12)

with respect to the unknown function L ∈ Ω_{0}^{1}V. The conditions for functions B_{σν} = −\frac{∂^{2}L}{∂q_{1}^{ν}∂q_{1}^{σ}} and the function L are immediately given by the use of lemma 2 for p = 2:

L = −q_{1}^{σ}q_{1}^{ν} ∫_{0}^{1}(B_{σν} ∘ η)(1 − s) ds + C_{σ}q_{1}^{σ} + D ; C_{σ}, D ∈ Ω_{0}^{0}V .

Functions C_{σ} and D can be easily obtained from (12). Denoting M_{σ} = \frac{∂D}{∂q^{σ}} − \frac{∂C_{σ}}{∂t} and N_{σν} = \frac{∂C_{ν}}{∂q^{σ}} − \frac{∂C_{σ}}{∂q^{ν}} and using lemma 1, we can finish the proof. □

Note that the Lagrange function L and the variationally compatible general force laws (A_{σ}) are expressed through the variationally compatible tensor of mass (B_{σν}) by the single integral only, i.e. by the simpler way than in [6,7,8]. By the relation (10), the forces (A_{σ}) are decomposed into two terms: the term containing B_{σν} corresponds to the so-called canonical connection on R×T M, and the contribution N_{σν}q_{1}^{ν}+M_{σ} is connected with the soldering form, both introduced by Krupková [7]. The function L given by the relation (11) defines the so-called minimal Lagrangian (see e.g. [8]) which is of the first order for the second order equations of motion.

4. Variational forces

Let us suppose that \frac{∂B_{σν}}{∂q_{1}^{λ}} = 0, 1≤σ, ν, λ≤m and denote

g_{σν} = −B_{σν} , Γ_{σνμ} = \frac{1}{2} (\frac{∂g_{σν}}{∂q^{λ}} + \frac{∂g_{σλ}}{∂q^{ν}} − \frac{∂g_{λν}}{∂q^{σ}} ) , Λ_{σν} = \frac{∂g_{σν}}{∂t} . (13)

Then the variational expressions can be written as E_{σ} = −g_{σν}q_{2}^{ν} + A_{σ} where

A_{σ} = −q_{1}^{ν}q_{1}^{λ}Γ_{σνλ} − q_{1}^{ν}Λ_{σν} + q_{1}^{ν}N_{σν} + M_{σ}. (14)

Let (V, t, q^{σ}) and (\overline{V} , \overline{t}, \overline{q}^{σ}) be two fibered charts on Y and q^{σ} = q^{σ}(\overline{t}, \overline{q}^{ν}), t = t(\overline{t}), the corresponding coordinate transformation. (These relations represent the most general coordinate transitions preserving fibres of the underlying fibered manifold. They do not contain e.g. Lorentzian transformations.) We denote

T = \frac{dt}{d\overline{t}}, \overline{T} = \frac{d^{2}t}{d\overline{t}^{2}} ,

C_{ν}^{σ} = \frac{∂q^{σ}}{∂\overline{q}^{ν}} , C_{νλ}^{σ} = \frac{∂^{2}q^{σ}}{∂\overline{q}^{ν}∂\overline{q}^{λ}} , C_{T}^{σ} = \frac{∂q^{σ}}{∂\overline{t}} , C_{T T}^{σ} = \frac{∂^{2}q^{σ}}{∂\overline{t}^{2}} , C_{T ν}^{σ} = \frac{∂^{2}q^{σ}}{∂\overline{t}∂\overline{q}^{ν}} .

For the coordinate transformation it holds, of course, det(C_{ν}^{σ}) ≠ 0 and T ≠ 0 on W = V ∪ \overline{V}. The quantities Γ_{σνλ}, Λ_{σν}, N_{σν} and M_{σ} in the relation (14) for variational general forces (A_{σ}) obey the following transformation rules:

\overline{g}_{σν} = g_{λμ}C_{σ}^{λ}C_{ν}^{μ}T^{−1} ,
\overline{Γ}_{σνλ} = (Γ_{αβγ}C_{σ}^{α}C_{ν}^{β}C_{λ}^{γ} + g_{αβ}C_{σ}^{α}C_{νλ}^{β}) T^{−1} ,

\overline{Λ}_{σν} = Λ_{αβ}C_{σ}^{α}C_{ν}^{β} + g_{αβ}(C_{σ}^{α}C_{T ν}^{β} + C_{ν}^{α}C_{T σ}^{β})T^{−1} + g_{αβ}C_{σ}^{α}C_{ν}^{β}\overline{T} , (15)

\overline{N}_{σν} = N_{αβ}C_{σ}^{α}C_{ν}^{β} − Γ_{αβγ}C_{σ}^{α}(C_{T}^{γ} C_{ν}^{β} + C_{T}^{β}C_{ν}^{γ}) + g_{αβ}(C_{T σ}^{β}C_{ν}^{α} − C_{σ}^{α}C_{T ν}^{β})T^{−1} ,

\overline{M}_{σ} = M_{α}C_{σ}^{α}T + N_{αβ}C_{σ}^{α}C_{T}^{β} − Γ_{αβγ}C_{σ}^{α}C_{T}^{β}C_{T}^{γ} T^{−1}−
−Λ_{αβ}C_{σ}^{α}C_{T}^{β} − g_{αβ}C_{σ}^{α}C_{T}^{β} \overline{T} − g_{αβ}C_{σ}^{α}C_{T T}^{β} T^{−1} .

Note that the equation (13) for Γ_{σνλ} is similar to the definition of the Christoffel symbols of the well-known Levi-Civita connection. Thus, we can identify (g_{σν}) with the metric tensor of the fibre π^{−1}{t}. As the equations of motion have the form g_{σν}q_{2}^{ν} = A_{σ}, we can see, by analogy with the second Newton law, that the expressions (A_{σ}) are related to the forces. However, the family of functions (A_{σ}) is not in general the geometrical object, in contradiction to (g_{σν}).

If we denote A_{σ}^{M} = −q_{1}^{ν}q_{1}^{λ}Γ_{σνλ} − q_{1}^{ν}Λ_{σν} and A_{σ}^{D} = q_{1}^{ν}N_{σν} + M_{σ}, we can again decompose the general forces (A_{σ}) into two terms. The first of them is related to the canonical connection, and from the physical point-of-view, it corresponds to the free particle; the second one is given by the soldering form, after [7],

A_{σ} = A_{σ}^{M} + A_{σ}^{D}.

Let M be the set of all expressions (A_{σ}). We say that two families of expressions from this set are equivalent if it is possible to transform one to another by a change of coordinates. We factorize the set M by this equivalence. We can classify the equivalence classes into four types, according to the transformation behaviour of sets of functions (Γ_{σνλ}), (Λ_{σν}), (N_{σν}) and (M_{σ}).

• Type I. is given by the only class containing zero expressions (A_{σ}) on V.

• Type II. is generated by classes containing the expressions (A_{σ}) with all Γ_{σνλ} = 0, Λ_{σν} = 0 on V (but some N_{σν}q_{1}^{ν} + M_{σ} ≠ 0).

Next two types are given by classes which contain no expressions (A_{σ}) with all Γ_{σνλ} = 0, Λ_{σν} = 0.

• Type III. classes contain expressions (A_{σ}) with all N_{σν} = 0, M_{σ} = 0 (but some Γ_{σνλ} ≠ 0 or Λ_{σν} ≠ 0)

• Type IV. contains all other classes.

The forces (A_{σ}^{M}) and (A_{σ}^{D}) are called, in the agreement with the presented classification and in the physical context, as follows:

Type  A_{σ}^{M}  A_{σ}^{D}
I. geometrical inertial
II. geometrical real
III.  metric  inertial
IV.  metric  real

The set of functions (A_{σ}^{D}) is, in physical terminology, considered as representing the so-called inertial and real forces, whereas the family (A_{σ}^{M}) corresponds to geometrical and metric forces. The expressions (M_{σ}) represent the electric-type forces, and the quantities (N_{σν}) define by the expressions (N_{σν}q_{1}^{ν}) the magnetictype forces. Geometrical forces have no actual physical basis, because they can be removed by the appropriate change of coordinates on fibres, t = \overline{t}, q^{σ} = q^{σ}(\overline{q}^{ν}). Metric forces can be made zero along extremals. On the other hand, the difference between inertial and real forces is not so outstanding. For eliminating the inertial forces, a more general transformation t = \overline{t}, q^{σ} = q^{σ}(\overline{q}^{ν}, \overline{t}) must be performed, containing \overline{t} explicitely. The sum of the real and inertial forces is again real. Any real forces can be always decomposed into the sum of inertial and other real forces. Let us discuss this problem somewhat in detail.

Now let us take into account only such coordinate systems in which (A_{σ}^{M}) = 0 on V for the I and II type classes of expressions (A_{σ}), or the coordinate systems in which this condition is satisfied along some extremals, for the types III and IV. These coordinate systems are preferred in the certain sense. We choose one of them and declare the corresponding (in general nonzero) expressions (A_{σ}^{D}) as true forces. The choice if such a coordinate system is, of course, arbitrary from the mathematical point-of-view. However, in real physical situations it can be supported by physical arguments, e.g. a usual inertial system of classical mechanics can be chosen. So, in the physical analogy, we call such an arbitrarily chosen system the inertial-like coordinate system. In all coordinate systems related to the inertial-like one by Galilean transformations, t = \overline{t}, q^{σ} = \overline{q}^{σ} + V^{σ}\overline{t}, V^{σ} = const., the expressions (A_{σ}^{D}) remain unchanged and the expressions (A_{σ}) = (A_{σ}^{D}) are again the true forces. Such systems can be called also inertial-like and remaining ones are considered as non-inertial ones. In non-inertial systems the expressions (A_{σ}) = (A_{σ}^{D}) include non only the true forces, but the inertial forces as well. In the following example we illustrate the concept of true, real and inertial forces. The example shows that there are the physical situations in which the true forces can be compensated by the inertial ones in an appropriately chosen non-inertial system.

5. Example

Let us take into account two particles with masses and charges m_{1}, e_{1} and m_{2}, e_{2}, respectively, in the electric field with the intensity E⃗ and the gravitational field with the intensity G⃗. Let time be the parameter t, X = R. The configuration of the system is given by cartesian coordinates of radius-vectors of particles r⃗_{1}, r⃗_{2}, i.e. m = 6, Y = R × R^{6}, (q^{σ})_{σ=1,... ,6}, q^{σ} = x_{1}^{σ} for 1 ≤ σ ≤ 3 and q^{σ} = x_{2}^{σ−3} for 4 ≤ σ ≤ 6, g_{σν} = δ_{σν}. Denote as v⃗_{1}, v⃗_{2} and a⃗_{1}, a⃗_{2} velocities and accelerations of particles, respectively. The equations of motion in the reference frame which is inertial in the usual sense accepted in Newtonian mechanics are

m_{1}G⃗ + e_{1}E⃗ − m_{1}a⃗_{1} = 0 , m_{2}G⃗ + e_{2}E⃗ − m_{2}a⃗_{2} = 0 .

These equations are variational. It is clear, that A_{σ}^{M} = 0 and then A_{σ} = A_{σ}^{D}. We choose this system as the inertial-like one and thus we declare the expressions

A_{σ} = m_{1}G^{σ} + e_{1}E^{σ} for 1≤σ ≤3 , A_{σ} = m_{2}G^{σ−3} + e_{2}E^{σ−3} for 4≤σ ≤6 as the true forces. We discuss two cases:

• \frac{e_{1}}{m_{1}} = \frac{e_{2}}{m_{2}} . The true forces (A_{σ})_{σ=1,... ,6} are of the type I and they vanish in the new (non-inertial) coordinate system defined by the transformation t = \overline{t} and q^{σ} =\overline{q}^{σ} + \frac{1}{2}(G^{σ} + \frac{e_{1}}{m_{1}} E^{σ}) t^{2}, 1≤σ ≤3 ; q^{σ} =\overline{q}^{σ} + \frac{1}{2}(G^{σ−3}+ \frac{e_{2}}{m_{2}} E^{σ−3}) t^{2}, 4≤σ ≤6. (The true forces occurring in the inertial system were compensated by the inertial forces.)

• \frac{e_{1}}{m_{1}} ≠ \frac{e_{2}}{m_{2}} . The true forces e_{2}E⃗, e_{2}E⃗ cannot be entirely removed by any coordinate transition (type II) and thus they are the real forces. On the other hand, the tranformation

q^{σ} = \overline{q}^{σ} + \frac{1}{2}G^{σ}t^{2} for 1≤σ ≤3 , q^{σ} = \overline{q}^{σ} + \frac{1}{2}G^{σ−3}t^{2} , for 4≤σ ≤6 , t = \overline{t} ,

leads to the absence of forces m_{1}G⃗, m_{2}G⃗ which are compensated by the inertial forces −m_{1}G⃗, −m_{2}G⃗ occurring in the new non-inertial system.

Summarizing the results of the example, we can conclude: The forces (A_{σ}^{D}) are declared as true by the choice of some of coordinate systems with (A_{σ}^{M}) = 0 which is then taken as inertial-like. The forces of various physically specified types (electrical, magnetical, gravitational, centrifugal, etc.) can contribute to (A_{σ}^{D}). These contributions or their appropriately chosen sums which can be removed by a coordinate transformations and thus compensated by inertial forces, are purely inertial, the remaining ones are real. Only purely inertial forces can be recognized in the complete expressions (A_{σ}^{D}) by finding the coordinate transitions eliminating them. This is e.g. the case of forces m_{1}G⃗, m_{2}G⃗ in both alternatives of the presented example and the case of forces e_{1}E⃗ + m_{1}G⃗, e_{2}E⃗ + m_{2}G⃗ for \frac{e_{1}}{m_{1}} = \frac{e_{2}}{m_{2}} . For different specific charges of particles the forces e_{1}E⃗, e_{2}E⃗ are real because there is no coordinate transition removing them entirely. Acknowledgement. The authors are indebted to Prof. Olga Krupková for her kind interest in their work and for helpful consultations.

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[10] Sarlet, W., Geometrical structures related to second-order equations, in: Differential Geometry and its Applications, Proc. Conf. August 1986, Masaryk Univ. Brno, 1986, 279-288.

Masaryk University Brno, Kotlářská 2 611 37 Brno, CZECH REPUBLIC E-mail: mraz@physics.muni.cz janam@physics.muni.cz