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14.5: Non-Standard Lagrangians

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  • The definition of the standard Lagrangian was based on d’Alembert’s differential variational principle. The flexibility and power of Lagrangian mechanics can be extended to a broader range of dynamical systems by employing an extended definition of the Lagrangian that is based on Hamilton’s Principle, equation 131. Hamilton’s Principle was introduced 46 years after the standard formulation of Lagrangian mechanics. Hamilton’s Principle provides a general definition of the Lagrangian that applies to standard Lagrangians, which are expressed as the difference between the kinetic and potential energies, as well as to non-standard Lagrangians where there may be no clear separation into kinetic and potential energy terms. These non- standard Lagrangians can be used with the Euler-Lagrange equations to generate the correct equations of motion even though they may have no relation to the kinetic and potential energies. The extended definition of the Lagrangian based on Hamilton’s action functional 131 can be exploited for developing non-standard definitions of the Lagrangian that may be applied to dynamical systems where use of the standard definition is inapplicable. Non-standard Lagrangians can be equally as useful as the standard Lagrangian for deriving equations of motion for a system. Secondly, non-standard Lagrangians, that have no energy interpretation, are available for deriving the equations of motion for many nonconservative systems. Thirdly, Lagrangians are useful irrespective of how they were derived. For example, they can be used to derive conservation laws or the equations of motion. Coordinate transformations of the Lagrangian is much simpler than that required when using the equations of motion. The relativistic Lagrangian defined in chapter 166 is a well-known example of a non-standard Lagrangian.