Abstract Motivated by filling the gap we felt after years of teaching analytical mechanics, a non-relativistic, classical introduction to Lagrangian mechanics has accordingly been provided here, which covers all possible forms of Euler–Lagrange equation, derived through dealing with different kinds of forces including conservative forces, forces of constraint, velocity-dependent forces, and non-conservative/dissipative forces. Hamiltonian mechanics has also been concluded as a reformulation of Lagrangian mechanics via applying Legendre transformation. Ignorable coordinates have finally been introduced, leading to Hamilton–Jacobi formalism, from which an equivalence between dynamics of a classical point particle and that of a plane wave has been inferred. We have showed that such an equivalence had long laid the required theoretical ground for the advent of wave mechanics; therefore, a number of landmark advancements in theoretical physics, including Hamiltonian mechanics, canonical transformations, and formulations of quantum mechanics have roots in Lagrangian mechanics.

The problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one‐dimensional systems. The relationship between the quantum theory and multiply‐periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.

Homogenous Lagrangian systems The application of the Legendre transformation to a hyperregular Lagrangian system results in a Hamiltonian vector field generated by a Hamiltonian defined on the phase space of the mechanical system. The Legendre transformation in its usual interpretation can not be applied to homogeneous Lagrangians found in relativistic mechanics. The dynamics of relativistic systems must be formulated in terms of implicit differential equations in the phase space and not in terms of Hamiltonian vector fields. The constrained Hamiltonian systems introduced by Dirac [1] are not general enough to cover some important cases. We formulate a geometric framework which permits Lagrangian and Hamiltonian descriptions of the dynamics of a wide class of mechanical systems. Lagrangians and Hamiltonians are presented as families of functions. The Legendre transformation and the inverse Legendre transformation are described as transitions between these families. Two examples, the dynamics of a relativistic particle and a space-time formulation of geometric optics (relativistic massless particle), are given.

Geometrical Mechanics on algebroids A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler-Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem. Besides, the constructions seem to be more natural and simpler. Published as: Int. J. Geom. Meth. Mod. Phys. 3 (2006), 559-575. arXiv categories: math-ph math.DG math.MP

Abstract Chapter 1 begins laying a foundation for classical statistical mechanics with a discussion of relevant topics in classical mechanics. The chapter begins with a discussion of Newton’s laws of motion and the concept of a phase space and builds up to the Lagrangian and Hamiltonian formulations of classical mechanics and the Legendre transform relation that connects them. The action integral is introduced, and its stationarity is shown to lead to the Euler-Lagrange equations of motion. It is shown how mechanical constraints can be incorporated into the action integral and Euler-Lagrange equations via Lagrange multipliers, and analytical forms of Lagrange multipliers are derived using Gauss’ principle of least constraint. The chapter concludes with a discusion of rigid-body motion in terms of Euler angles and quaternions followed by a brief discussion of non-Hamiltonian equations of motion and the types of problems they describe. Throughout the chapter, concepts are illustrated with analytically solvable examples.

and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). Both formulations are equivalent by a Legendre transformation on In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses scalar properties of motion representing the system as a whole—usually its kinetic energy and potential energy. The equations of motion are derived from the scalar quantity by some underlying principle abou In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses scalar properties of motion representing the system as a whole—usually its kinetic energy and potential energy. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation. Analytical mechanics was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system; it can also be called vectorial mechanics. A scalar is a quantity, whereas a vector is represented by quantity and direction. The results of these two different approaches are equivalent, but the analytical mechanics approach has many advantages for complex problems. Analytical mechanics takes advantage of a system's constraints to solve problems. The constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics. Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of…

In physics, Lagrangian mechanics is an alternate formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced In physics, Lagrangian mechanics is an alternate formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécanique analytique. Lagrange's approach greatly simplifies the analysis of a…