## Blogger

# How Lagrangian and Hamiltonian Dynamics Differ –

# A Brief Annotation

## Dr. Jonathan Kenigson, FRSA

Lagrangian and Hamiltonian are two powerful tools used by physicists to describe the behavior of physical systems. While they share similarities, they are two distinct formulations. The Lagrangian formulation is based on the principle of least action and is used to describe the motion of objects in a system. It is written as the sum of the kinetic energy and the potential energy of the system. On the other hand, the Hamiltonian formulation is based on the principle of least constraint and is used to describe the evolution of a system over time. This formulation is written as the sum of the kinetic energy and the potential energy of the system, plus an additional term that represents the interactions between the objects in the system. To summarize, the Lagrangian formulation is used to describe the motion of objects in a system, while the Hamiltonian formulation is used to describe the evolution of a system over time. The Euler-Lagrange equations are a set of partial differential equations that play an important role in the calculus of variations. They are used to find the extremum of a functional, which is a function of several variables and their derivatives. In other words, they help us find the minimum or maximum of a function. The equations are named after the mathematicians Leonhard Euler and Joseph-Louis Lagrange, who developed them independently in the 18th century. The equations are most used in physics and engineering, where they are used to determine the behavior of physical systems.

Lagrangian Dynamics is a powerful tool used in physics to investigate the motion of a system. It is based on the principle of least action, which states that for a system to move from one point to another, the sum of all its actions must be at a minimum. This principle is the basis for Lagrangian Dynamics, which states that the motion of a system is determined by its Lagrangian, or kinetic minus potential energy. With this, we can use Lagrangian Dynamics to determine the equations of motion for a system and to uncover the hidden relationships between the variables. It can also give us insights into the stability of a system, allowing us to predict how it will behave in different conditions. Lagrangian Dynamics is the cornerstone of modern physics and can be applied to a wide variety of systems, from particles to galaxies. Hamiltonian Dynamics is a field of mathematics and physics that studies the behavior of mechanical systems. It is based on the principles of analytical mechanics, which was devised by William Hamilton in the early 19th century. In Hamiltonian Dynamics, the behavior of a system is described using the Hamiltonian, which is derived from the system's Lagrangian. The Hamiltonian is a function of the system's position and momentum, and it provides a mathematical framework for studying the dynamics of the system. With it, one can calculate the system's energy, momentum, and trajectories in space and time. Hamiltonian Dynamics can be used to study many different types of systems, including mechanical systems, electrical circuits, and even quantum systems.

# Works Consulted and Further Study.

Bell, John Stewart. "Hamiltonian mechanics." (1987).

Calkin, Melvin G. *Lagrangian and Hamiltonian mechanics*. 1996.

Crampin, M. "Tangent bundle geometry Lagrangian dynamics." *Journal of Physics A: Mathematical and General* 16.16 (1983): 3755.

Kozlov, Valery V. "Integrability and non-integrability in Hamiltonian mechanics." *Russian Mathematical Surveys* 38.1 (1983): 1.

Mané, Ricardo. "Lagrangian flows: the dynamics of globally minimizing orbits." *Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathematical Society* 28.2 (1997): 141-153.

Meneveau, Charles. "Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows." *Annual Review of Fluid Mechanics* 43.1 (2011): 219-245.

Puta, Mircea. *Hamiltonian mechanical systems and geometric quantization*. Vol. 260. Springer Science & Business Media, 2012.

Salmon, Rick. "Hamiltonian fluid mechanics." *Annual review of fluid mechanics* 20.1 (1988): 225-256.

Sarlet, W. "The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics." *Journal of Physics A: Mathematical and General* 15.5 (1982): 1503.

Vinogradov, Alexandre Michailovich, and Boris Abramovich Kupershmidt. "The structures of Hamiltonian mechanics." *Russian Mathematical Surveys* 32.4 (1977): 177.