# Exploring the Homogeneity of Space & Time in Lagrangian Mechanics

• Andrea Vironda
In summary, the conversation discusses the homogeneity of space and time and its implications on the Lagrangian, which must only be a function of velocity and not position or time. However, the definition of the Lagrangian is not limited to just homogeneous space and time, as in a uniform gravitational field, it does depend on position through potential energy. The conversation also touches on the difference between the general case of ##L(q,\dot{q},t)## and the specific restriction of ##L(v^2)##, with the latter being a specific case of the former.
Andrea Vironda
Hi,
i know that The homogeneity of space and time implies that the Lagrangian cannot contain
explicitly either the radius vector r of the particle or the time t, i.e. L must be a function of v only

but the lagrangian definition is ##L=\int L(\dot q,q,t)##, so velocity appears in the definition and it's in contrast with ##L=L(v^2)##
why?

The definition is general, and not just for homogeneous space and time. For example, in a uniform gravitational field, the Lagrangian of a particle does depend on position, through the potential energy (mgy). The Lagrangian of a free particle does not depend on position. It also does not depend on direction (isotropy) and depends only on the square of the speed.

Why should ##L(q,\dot{q},t)## and ##L(v^2)## be in contrast at all? The latter sais, that ##L## in that specific case depends on ##v=\dot{q}## only via ##v^2## and that the dependence on ##q## and ##t## drops, so ##L(v^2)## is a specific restriction of the most general case ##L(q,\dot{q},t)##, but not in contrast with it.

vanhees71

## 1) What is Lagrangian Mechanics?

Lagrangian mechanics is a mathematical framework used to study the motion of particles and systems in classical mechanics. It was developed by Italian mathematician and astronomer Joseph-Louis Lagrange in the late 18th century.

## 2) How is Lagrangian Mechanics different from Newtonian Mechanics?

While Newtonian mechanics uses the concept of force to describe the motion of objects, Lagrangian mechanics uses the concept of energy. This approach often simplifies the equations of motion and allows for a more elegant and intuitive description of the system.

## 3) What is the significance of exploring the homogeneity of space and time in Lagrangian Mechanics?

The homogeneity of space and time refers to the idea that the laws of physics should be the same for all observers, regardless of their location or the time at which they are making observations. By exploring this concept in Lagrangian mechanics, we can better understand the fundamental principles of classical mechanics and how they apply to different reference frames.

## 4) How is the homogeneity of space and time related to symmetries in Lagrangian Mechanics?

In Lagrangian mechanics, symmetries refer to the invariance of the equations of motion under certain transformations. The homogeneity of space and time is a symmetry in this framework, meaning that the laws of physics are unchanged when we shift our reference frame in space or time.

## 5) What are some practical applications of studying the homogeneity of space and time in Lagrangian Mechanics?

Understanding the homogeneity of space and time is essential for accurately predicting the behavior of physical systems, such as satellites in orbit or objects moving in a non-inertial reference frame. It also has important applications in fields such as astrophysics, where the laws of physics must be consistent across vast distances and over billions of years.

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