This chapter offers a pedagogical introduction to classical analytical dynamics including Lagrangian mechanics, symplectic and contact structures, and Hamiltonian dynamics. In Sect. 35.1 we sketch the classical Lagrangian “analytical dynamics” for systems of point particles. Originally based on the Newtonian paradigm, such systems are described in terms of a finite number n of “degrees of freedom” and evolve as particular solutions to a system of n second-order ordinary differential equations. One advantage of the Lagrangian formulation is that there is no pre-requisite that the “degrees of freedom” be associated with any particular Cartesian coordinate system or should evolve with respect to Newtonian time. The n degrees of freedom constitute the coordinates of a point in an n-dimensional configuration space \(\mathcal {Q}\) and a particular evolution of the system is a particular \(\tau \) -parametrised curve in \(\mathcal {Q}\) obtained by solving a second-order ODE system subject to 2n initial conditions given at some value of \(\tau \) . Based on the notion of a Lagrangian function on the extended tangent bundle \(T\mathcal {Q}\times \mathbb {R}\) , the classical Euler-Lagrange equations are derived from extremal curves of an action functional. This section concludes by extending the domain of the Lagrangian function to accommodate extrema that yield higher-order coupled non-autonomous ordinary differential equations. These are needed to describe physical phenomena that involve electromagnetic radiation from accelerating charges and the associated “backreaction” and “radiation damping”. In Sect. 35.2, we turn to a dual formulation of classical analytical dynamics based on a Hamiltonian function derived from a Lagrangian function \(\widehat {\mathcal {L}}\) . Classical Lagrangian functions play an important role in constructing many classical variational “action principles” while classical Hamiltonian functions are fundamental ingredients in the formulation of quantum mechanics, in the development of quantum electrodynamics and in the “canonical” description of classical gravitation. A general explicitly time-dependent Hamiltonian function is obtained from a real-valued map \(\widehat {\mathcal {H}}:T^{*}\mathcal {Q}\times \mathbb {R}\rightarrow \mathbb {R}\) and offers certain computational advantages over “Lagrangian dynamics” by exploiting the calculus of differential forms, the exterior derivative on forms and the facility to pull back forms with maps that need not be diffeomorphisms. These advantages are best appreciated in the framework of “symplectic manifolds”, i.e. differentiable manifolds endowed with a “symplectic structure”. The chapter concludes with the definition of a dynamical system associated with the contact form for a contact structure and the definitions of 0-form Poisson brackets. These were used by Dirac to motivate the structure of canonical quantum commutation brackets (see Appendix M).

An Introduction to Classical Analytical Dynamics

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发表日期：

2025年

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Robin W. Tucker,
Timothy J. Walton

摘要

This chapter offers a pedagogical introduction to classical analytical dynamics including Lagrangian mechanics, symplectic and contact structures, and Hamiltonian dynamics. In Sect. 35.1 we sketch the classical Lagrangian “analytical dynamics” for systems of point particles. Originally based on the Newtonian paradigm, such systems are described in terms of a finite number n of “degrees of freedom” and evolve as particular solutions to a system of n second-order ordinary differential equations. One advantage of the Lagrangian formulation is that there is no pre-requisite that the “degrees of freedom” be associated with any particular Cartesian coordinate system or should evolve with respect to Newtonian time. The n degrees of freedom constitute the coordinates of a point in an n-dimensional configuration space \(\mathcal {Q}\) and a particular evolution of the system is a particular \(\tau \) -parametrised curve in \(\mathcal {Q}\) obtained by solving a second-order ODE system subject to 2n initial conditions given at some value of \(\tau \) . Based on the notion of a Lagrangian function on the extended tangent bundle \(T\mathcal {Q}\times \mathbb {R}\) , the classical Euler-Lagrange equations are derived from extremal curves of an action functional. This section concludes by extending the domain of the Lagrangian function to accommodate extrema that yield higher-order coupled non-autonomous ordinary differential equations. These are needed to describe physical phenomena that involve electromagnetic radiation from accelerating charges and the associated “backreaction” and “radiation damping”. In Sect. 35.2, we turn to a dual formulation of classical analytical dynamics based on a Hamiltonian function derived from a Lagrangian function \(\widehat {\mathcal {L}}\) . Classical Lagrangian functions play an important role in constructing many classical variational “action principles” while classical Hamiltonian functions are fundamental ingredients in the formulation of quantum mechanics, in the development of quantum electrodynamics and in the “canonical” description of classical gravitation. A general explicitly time-dependent Hamiltonian function is obtained from a real-valued map \(\widehat {\mathcal {H}}:T^{*}\mathcal {Q}\times \mathbb {R}\rightarrow \mathbb {R}\) and offers certain computational advantages over “Lagrangian dynamics” by exploiting the calculus of differential forms, the exterior derivative on forms and the facility to pull back forms with maps that need not be diffeomorphisms. These advantages are best appreciated in the framework of “symplectic manifolds”, i.e. differentiable manifolds endowed with a “symplectic structure”. The chapter concludes with the definition of a dynamical system associated with the contact form for a contact structure and the definitions of 0-form Poisson brackets. These were used by Dirac to motivate the structure of canonical quantum commutation brackets (see Appendix M).