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14.1: Prelude to Lie Analysis

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  • In Chapter 11, the ideas of calculus of variations were applied to energy conversion processes. We began with two forms of energy and studied how those forms of energy varied with variation in some generalized path and some generalized potential. The result was an equation of motion that described the variation of the generalized path. The equation of motion had the form of conservation of generalized potential. In Chapter 12, conservation laws were listed in the last row of the tables. Knowing how forms of energy vary with path and with potential provide significant information about energy conversion processes. The purpose of this chapter is to show that we can find symmetries, invariants, and other information about the energy conversion process by applying Lie analysis techniques to this equation of motion. If continuous symmetries of an equation can be identified, it is often possible to extract quite a bit of information by starting only with the equation.

    The equations of motion that result from calculus of variations are not always linear. It may or may not be possible to solve a nonlinear equation of motion for the path. Even in the cases where it is possible, it is often quite difficult because techniques for solving nonlinear differential equations are much less developed than techniques for linear equations. Furthermore, many nonlinear differential equations do not have closed form solutions. In this chapter, we will see a systematic technique for getting information out of nonlinear differential equations that comes from calculus of variation. The technique is known as Lie analysis based on the work of Sophus Lie in the late part of the nineteenth century. Additionally, this chapter introduces Noether's theorem. Using this theorem and an equation of motion, we may be able to derive conserved quantities. The techniques discussed in this chapter apply even for nonlinear equations.

    Lie analysis is a systematic procedure for identifying continuous symmetries of an equation. If the equation possesses continuous symmetries, we may be able to find related conservation laws. Some equations possess multiple symmetries and conservation laws while other equations do not contain any symmetries or conservation laws. Using this procedure with a known generalized path, we may be able to derive conserved quantities even if we do not know how to choose the generalized potential at first. Some systems might even contain multiple conserved quantities, and this procedure will give us a complete set of conserved quantities.

    Lie analysis has been used to find continuous symmetries of many fundamental equations of physics, and it has been applied to both classical and quantum mechanical equations. References [164, p. 117] and [181] apply the procedure to the heat equation

    \[\frac{dy}{dt} = \frac{d^2y}{dx^2}\]

    describing the function \(y(t, x)\). It has been applied to both the two dimensional wave equation [164, p. 123] and the three dimensional wave equation [181]. Other equations analyzed by this procedure include Schrödinger's equation [182] [183], Maxwell's equations [184] [185], and equations of nonlinear optics [186].

    A tremendous amount of information can be gained by looking at the symmetries of equations. Knowledge of continuous symmetries may allow us to solve equations or at least reduce the order of differential equations [164]. If we can identify symmetries, we may be able to simplify or speed up numerical calculations by using known repetition in the form of the solution. If multiple equations contain the same symmetry elements, we can draw comparisons between the equations [164]. We may be able to find invariant quantities of the system from known continuous symmetries of equations. Hopefully this chapter will provide an appreciation for the amount of information that can be gained from applying symmetry analysis to equations of motion describing energy conversion processes.

    14.1.1: Assumptions and Notation

    The techniques of this chapter are applied to equations of motion that results from describing an energy conversion processes by calculus of variations. All starting equations of motion are assumed to have only one independent and one dependent variable. These equations may or may not be linear. Furthermore, all independent and dependent variables are assumed to be purely real. We made the same assumptions in Chapter 11. Most of the examples in this chapter involve second order differential equations because many of the energy conversion processes studied in Chapter 11 led to equations of motion which were second order differential equations. However, these techniques apply to algebraic equations and to differential equations of other orders.

    In this chapter, total derivatives will be denoted as either \(\frac{dy}{dt}\) or \(\dot{y}\). Partial derivatives will be denoted as either \(\frac{\partial y}{\partial t}\) or \(\partial_ty\) for shorthand. If the quantity \(y\) is just a function of a single independent variable, there is no reason to distinguish between total and partial derivatives, \(\frac{dy}{dt} = \frac{\partial y}{\partial t}\). Equations of motion in this chapter will involve one independent and one dependent variable, \(y(t)\). However, we will encounter functionals of multiple independent variables such as the Lagrangian \(\mathcal{L} = \mathcal{L}(t, y, \frac{dy}{dt})\). For such quantities, we will have to distinguish between total and partial derivatives carefully.

    The analysis here is in no way mathematically rigorous. Furthermore, the examples in this chapter are not original. References to the literature are included below.

    These techniques generalize to more complicated equations. They apply to equations with multiple independent and multiple dependent variables, and they apply when these variables are complex [164]. Also, these techniques apply to partial differential equations as well as ordinary differential equations, and they even apply to systems of equations [164]. See references [164] for how to generalize the methods introduced in this chapter to the other situations.