Lagrangian mechanics

Lagrangian mechanics is a formulation of classical mechanics that uses the principle of least action to describe the motion of objects. Developed by the mathematician Joseph-Louis Lagrange in the 18th century, this approach reformulates Newtonian mechanics, providing a powerful and elegant framework for analyzing mechanical systems.

Satellites orbiting Lagrange points

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Satellites orbiting Lagrange points refer to spacecraft that are positioned at or near one of the five specific points in a two-body system where the gravitational forces and the orbital motion of the bodies create a stable or semi-stable location for smaller objects. These points are known as Lagrange points, named after the French mathematician Joseph-Louis Lagrange.

AQUAL

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AQUAL can refer to different things depending on the context. Here are a few possibilities: 1. **AQUAL (Assured Quality of Life)**: It's used in various contexts related to ecological or social aspects, focusing on the quality of life concerning water resources and environmental sustainability. 2. **Aqual - Related to Water**: The term "aqual" is derived from the Latin word for water ("aqua") and is sometimes used in branding or product names that emphasize hydration or purity.

Averaged Lagrangian

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The **Averaged Lagrangian** is a concept often used in the context of dynamical systems, particularly in the fields of mechanics and control theory. It is associated with the method of averaging, which is a mathematical technique used to simplify the analysis of systems with periodic or oscillatory behavior.

Conformal gravity

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Conformal gravity is a theoretical framework in gravity research that extends the principles of general relativity by focusing on conformal invariance, which is a symmetry involving the scaling of the metric tensor without altering the underlying physics. In simpler terms, conformal gravity posits that physical phenomena should remain unchanged under transformations that scale distances uniformly, which is a more generalized symmetry than the Lorentz invariance of general relativity.

D'Alembert's principle

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D'Alembert's principle is a fundamental concept in classical mechanics that provides a powerful tool for analyzing the motion of dynamic systems. Named after the French mathematician Jean le Rond d'Alembert, the principle can be seen as a reformulation of Newton's second law of motion. In essence, D'Alembert's principle states that the sum of the differences between the applied forces and the inertial forces (which are proportional to the mass and acceleration) acting on a system is zero.

FLEXPART

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FLEXPART is a numerical model designed for simulating the transport and dispersion of atmospheric pollutants and tracers. It stands for "FLEXible PARTicle dispersion model," and it is often used in atmospheric science to study how substances such as gases, aerosols, or other particles move through the atmosphere under the influence of various meteorological conditions.

Generalized coordinates

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Generalized coordinates are a set of parameters used in the field of classical mechanics and theoretical physics to describe the configuration of a mechanical system. They provide a way to express the degrees of freedom of a system, which correspond to the number of independent parameters needed to uniquely specify its position or configuration.

Generalized forces

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Generalized forces are a concept from classical mechanics used in the context of Lagrangian and Hamiltonian mechanics. They extend the idea of force beyond merely the conventional forces acting on a system (like gravity, friction, etc.) to include other types of influences that can affect the motion of a system.

Gibbons–Hawking–York boundary term

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The Gibbons–Hawking–York (GHY) boundary term is an important concept in the context of general relativity and gravitational action principles, particularly when dealing with the Einstein-Hilbert action, which describes the dynamics of gravity.

Halo orbit

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A Halo orbit is a type of orbital path that an object can take around a point in space, specifically around a Lagrangian point in the Earth-Moon system or any other two-body system. Lagrangian points are positions in space where the gravitational forces of two large bodies, like the Earth and the Moon, balance out the centrifugal force felt by a smaller object. There are five such points, denoted as L1, L2, L3, L4, and L5.

Joseph-Louis Lagrange

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Joseph-Louis Lagrange (1736–1813) was an influential mathematician and astronomer of Italian origin who later became a naturalized French citizen. He made significant contributions to many areas of mathematics, including calculus, number theory, and mechanics. Lagrange is known for several key achievements: 1. **Lagrange's Theorem**: In group theory, he established that the order of a subgroup divides the order of the group.

Lagrange point

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A Lagrange point is a position in space where the gravitational forces of two large bodies, such as a planet and a moon or a planet and the sun, balance out the centripetal force experienced by a smaller body. This results in a stable or semi-stable location where the smaller body can maintain a position relative to the two larger bodies, effectively "parking" in that location.

Lissajous orbit

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A Lissajous orbit refers to a specific type of trajectory that a body can follow in a dynamical system, especially within the context of celestial mechanics. These orbits are characterized by the interplay of two oscillatory motions that combine to form a complex, looping pattern, much like the Lissajous figures seen in mathematics and physics when plotting parametric equations.

Palatini variation

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The Palatini variation, often discussed in the context of the Einstein-Hilbert action in general relativity, refers to a particular formulation of the variational principle from which the equations of motion for a gravitational field can be derived. In general relativity, one can employ different approaches to derive the field equations, and one such approach is the Palatini formalism, which differs from the more common metric formulation.

Rayleigh dissipation function

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The Rayleigh dissipation function is a concept used in classical mechanics, particularly in the analysis of systems that experience non-conservative forces, such as friction or air resistance. It is a mathematical tool that helps to describe the energy lost in a system due to these non-conservative forces. In Lagrangian mechanics, the equations of motion for a system can be derived using the Lagrangian function, which is defined as the difference between the kinetic and potential energies of the system.

Relativistic Lagrangian mechanics

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Relativistic Lagrangian mechanics is an extension of classical Lagrangian mechanics that incorporates the principles of special relativity into the framework of theoretical mechanics. While classical Lagrangian mechanics is effective for describing the motion of objects at non-relativistic speeds (much less than the speed of light), it requires modification to properly address situations where speeds approach the speed of light.

Rheonomous

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Rheonomous is a term that could refer to a variety of concepts depending on the context, but it is not widely recognized in common use or scientific literature. It may be a specialized term within a niche field or a newly coined term that has not gained widespread acceptance.

Scleronomous

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"Scleronomous" typically refers to a class of structures in mathematics, specifically in the field of differential geometry and the study of manifolds. However, the term may not be widely recognized in common mathematical literature, and its specific definition can vary depending on the context in which it is used. In general terms, "scleronomous" is often contrasted with "holonomous.

Total derivative

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The total derivative is a concept from calculus that extends the idea of a derivative to functions of multiple variables. It takes into account how a function changes as all of its input variables change simultaneously.

Virtual displacement

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Virtual displacement is a concept used in the fields of mechanics and physics, particularly in the study of classical mechanics and systems in equilibrium. It refers to a hypothetical or imagined small change in the configuration of a system that occurs without the passage of time. In other words, it is a conceptual tool used to analyze the equilibrium of a system by considering small variations in position of the particles or bodies constituting the system.

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Lagrangian mechanics by

Ciro Santilli 37  Updated 2025-06-17  +Created 1970-01-01

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Originally it was likely created to study constrained mechanical systems where you want to use some "custom convenient" variables to parametrize things instead of global x, y, z. Classical examples that you must have in mind include:

compound Atwood machine. Here, we can use the coordinates as the heights of masses relative to the axles rather than absolute heights relative to the ground
double pendulum, using two angles. The Lagrangian approach is simpler than using Newton's laws
- pendulum, use angle instead of x/y
two-body problem, use the distance between the bodies

lagrangian mechanics lectures by Michel van Biezen (2017) is a good starting point.

When doing lagrangian mechanics, we just lump together all generalized coordinates into a single vector

q (t) : R \to R^{n}

that maps time to the full state:

q (t) = [q_{1} (t), q_{2} (t), q_{3} (t), ..., q_{n} (t)]

(1)

where each component can be anything, either the x/y/z coordinates relative to the ground of different particles, or angles, or nay other crazy thing we want.

The Lagrangian is a function

R^{n} \times R^{n} \times R \to R

that maps:

L (q (t), \dot{q} (t), t)

(2)

to a real number.

Then, the stationary action principle says that the actual path taken obeys the Euler-Lagrange equation:

\frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i}} - \frac{d}{d t} \frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i} ˙} = 0

(3)

This produces a system of partial differential equations with:

$n$ equations
$n$ unknown functions $q_{i}$
at most second order derivatives of $q_{i}$ . Those appear because of the chain rule on the second term.

The mixture of so many derivatives is a bit mind mending, so we can clarify them a bit further. At:

\frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i}}

(4)

the

q_{i}

is just identifying which argument of the Lagrangian we are differentiating by: the i-th according to the order of our definition of the Lagrangian. It is not the actual function, just a mnemonic.

Then at:

\frac{d}{d t} \frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i} ˙}

(5)

the $\frac{\partial}{\partial q _{i} ˙}$ part is just like the previous term, $\overset{q_{i}}{˙}$ just identifies the argument with index $n + i$ ( $n$ because we have the $n$ non derivative arguments)
after the partial derivative is taken and returns a new function $\frac{\partial L ( q ( t ) , q ˙ ( t ) , t )}{\partial q _{i} ˙}$ , then the multivariable chain rule comes in and expands everything into $2 n + 1$ terms

However, people later noticed that the Lagrangian had some nice properties related to Lie group continuous symmetries.

Basically it seems that the easiest way to come up with new quantum field theory models is to first find the Lagrangian, and then derive the equations of motion from them.

For every continuous symmetry in the system (modelled by a Lie group), there is a corresponding conservation law: local symmetries of the Lagrangian imply conserved currents.
Genius: Richard Feynman and Modern Physics by James Gleick (1994) chapter "The Best Path" mentions that Richard Feynman didn't like the Lagrangian mechanics approach when he started university at MIT, because he felt it was too magical. The reason is that the Lagrangian approach basically starts from the principle that "nature minimizes the action across time globally". This implies that things that will happen in the future are also taken into consideration when deciding what has to happen before them! Much like the lifeguard in the lifegard problem making global decisions about the future. However, chapter "Least Action in Quantum Mechanics" comments that Feynman later notice that this was indeed necessary while developping Wheeler-Feynman absorber theory into quantum electrodynamics, because they felt that it would make more sense to consider things that way while playing with ideas such as positrons are electrons travelling back in time. This is in contrast with Hamiltonian mechanics, where the idea of time moving foward is more directly present, e.g. as in the Schrödinger equation.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

And partly due to the above observations, it was noticed that the easiest way to describe the fundamental laws of particle physics and make calculations with them is to first formulate their Lagrangian somehow: S.

TODO advantages:

Bibliography:

www.physics.usu.edu/torre/6010_Fall_2010/Lectures.html Physics 6010 Classical Mechanics lecture notes by Charles Torre from Utah State University published on 2010,
- Classical physics only. The last lecture: www.physics.usu.edu/torre/6010_Fall_2010/Lectures/12.pdf mentions Lie algebra more or less briefly.
www.damtp.cam.ac.uk/user/tong/dynamics/two.pdf by David Tong

Video 1.

Euler-Lagrange equation explained intuitively - Lagrangian Mechanics by Physics Videos by Eugene Khutoryansky (2018)

Source. Well, unsurprisingly, it is exactly what you can expect from an Eugene Khutoryansky video.

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