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While we discuss this in the next section, it may be instructive to carry out an intuitive computation that illustrates a relationship between the quantity L and equations of motion. Take the harmonic oscillator with L D 12 mPx2 12 kx2 . t/, and with the same initial and final points. t2 / D 0. Let us compute the variation of the function L for the two curves in a linear order on ıx (that is, the differential) at a fixed t; see Fig. 6 on page 30. 115) Note that mRx C kx D 0 is equation of motion of the harmonic oscillator.

The last equality represents a necessary condition of minimum of the Lagrangian action; see the next section. 1 Variational Analysis Definition of a Functional Consider a set of functions C D fqa . /; q W R ! Rn g. The functional S is a rule that associates a real number with any function of the set, that is, S is the map S W C ! R; or S W qa . / ! SŒqa . 117) The straight brackets are used to distinguish functionals from functions. In Lagrangian mechanics we are interested in a functional of a special form called the Lagrangian action functional.

B) ... i i+1 ... 135) by qP a , obtain the law of conservation of total energy. Lagrangian theories are classified according to the properties of the Hessian matrix M which appears in front of the second-derivative terms. The theory is called nonQ Otherwise, it is singular if det M ¤ 0 (then there is an inverse matrix denoted M/. called a singular theory. 137) As has been mentioned above, this implies causal dynamics, as should be the case for a classical-mechanics system. So, in classical mechanics we usually deal with non-singular systems (while a system with holonomic constraints can be formulated in terms of a singular Lagrangian as well, see Sect.

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