Chapter 1 Principles of Motion in Invariantive Mechanics 1.1. The Euler-Lagrange and Hamilton s equations obtained by means of exterior forms Let L = L(q 1,q 2,...,q n, q 1, q 2,..., q n,t) L(q, q,t) (1.1) be a function of class C 2, called Lagrangian, whereq (q 1,q 2,...,q n ) are the generalized coordinates, q j = dqi (j = 1,n) (1.2) are the generalized velocities, and t is the time. Let also S be the action, defined by the exterior 1-form ω = ds = L. (1.3) Theorem 1.1. The total derivative of the action with respect to time yields the Hamilton-Jacobi equation [1] S + H =0. (1.4) Indeed, if the functional dependence as well as the definition of generalized momenta S = S(q,t) (1.5) p j = S j (j = 1,n) (1.6) 1
2 Differentiability and Fractality in Dynamics of Physical Systems are accepted, the total derivative of S with respect to time is ds = S n + S j qj j=1 def = S + p q. (1.7) (N.B. Here and hereafter, the summation symbol and the summation indices shall be omitted). Since, according to (1.3) S = L, (1.8) relations (1.7) and (1.8) lead to (1.4), where the Hamiltonian H is defined as Theorem 1.2. Closeness of the 1-form (1.3) H = p q L. (1.9) Dω = dl =0 (1.10) reduces to Euler-Lagrange equations [1] ( ) d L L =0 (1.11) q and the integral of motion [1] = L. (1.12) Written explicitly, relation (1.10) becomes According to (1.1), we can write Dω = dlδt δl =0. (1.13) δl = L δq + L q L δ q + δt, (1.14) On the other hand, operators d and δ commute, so that L L δ q = q δ dq = L d δq. (1.15)
Principles of Motion in Invariantive Mechanics 3 Equation (1.3) can then be written as [ d ( ) L L ] [ ( d δq + L L ) q q q L ] δt =0. (1.16) Using the Hamiltonian defined by (1.9) and the fact that variations δq and δt are arbitrary, equation (1.16) yields (1.11) and (1.12). If, in particular, L does not explicitly depend on time, then (1.12) reduces to the energy conservation law =0. (1.17) As one can see, the 1-form (1.3) can also be written as ω =[p q H] = pdq H, (1.18) where the generalized momentum p is defined by Theorem 1.3. Closeness of the 1-form (1.18) p = L q. (1.19) Dω = dp dq =0 (1.20) reduces to Hamilton s canonical equations [1] q = H p ṗ = H (1.21) together with the integral of motion [1] Explicitating (1.20), we have = H. (1.22) dpδq δpdq δt + δh =0. (1.23)
4 Differentiability and Fractality in Dynamics of Physical Systems Since H = H(p, q, t), one can write δh = H δq + H p H δp + δt, (1.24) and (1.23), after some simple manipulation, becomes ( q + H ) ( δp + ṗ + H ) ( δq + p + H ) δt =0. (1.25) Equating to zero the coefficients of variations δq, δp, and δt,equations (1.21) and (1.22) follow immediately. In particular, if H does not explicitly depend on time, (1.22) reduces to the energy conservation law (1.17). 1.2. The Cartan motion principle The previous results allow us to operate with both Lagrange function L and Hamilton s function H. This means that, as long as the canonical formalism operates, Hamilton s canonical equations (1.16) and Euler-Lagrange equations (1.5) are equivalent. But, if the canonical formalism is not applicable, the two systems of equations are in general -different. When passing from the Lagrangian to canonical formalism, not all canonical momenta given by (1.19) are defined [2]. Between the two functions, Lagrangian and Hamiltonian, the invariantive mechanics chooses the last one. Here are some arguments in this respect: 1. In general, the Lagrangian does no have a direct physical signification (except for the case when it can be separated in two terms, of kinetic and potential energy significance). 2. The Hamiltonian represents not only energy, but also generates the motion (see paragraph concerning the Onicescu informational energy). This function is conserved in both previously discussed alternatives. 3. Using the Hamiltonian H and the exterior forms, one can construct the following principle (Cartan s principle):
Principles of Motion in Invariantive Mechanics 5 The law of motion for a discrete system with n components, characterized by the inertial 1-form ω = pdq H (1.26) is given by the cancellation of the exterior derivative of 1-form (1.26), that is Dω = dpδq δpdq δt δh =0. (1.27) The law of motion for a discrete system of n components, characterized by 1-form (1.26), is given by its closeness Dω = dp dq =0. (1.28) Observation 1.1. Euler-Lagrange and Hamilton s canonical equations keep their form in case of systems with an infinite number of degrees of freedom (continuous deformable media and fields), except for the facts that L and H are replaced by their densities L and H, while the usual derivatives are replaced by functional derivatives (see, for details, reference [4]).