# Homogeneity of space and the form of the Lagrangian

• ian_dsouza
In summary: So, in summary, the homogeneity of space leads to the conclusion that the Lagrangian of a free particle is not explicitly dependent on its position.
ian_dsouza
I was reading that the homogeneity of space can lead to the conclusion that the lagrangian of a free particle is not explicitly dependent on its position. At the moment, this does not come very intuitively to me. By homogeneity, I understand that if you displace the initial position of a particle by a vector $$\vec{c}$$

, then all points on the trajectory of the particle are displaced by the same vector 'c'.

I am trying to work with the Euler lagrange equation. For now, consider the 1-dimensional case: $$L(x,\dot{x},t)$$ If x(t) = x1(t) is a solution to the Euler-Lagrange equation corresponding to initial conditions x(t1) = X1 and v(t1) = V1 as in:
$$\frac{\partial L(x_1(t),\dot{x}_1(t),t)}{\partial x} - \frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial L(x_1(t),\dot{x}_1(t),t)}{\partial \dot{x}}=0 \tag{1}$$then I'd like to prove that if x(t) = x1(t)+c is also a solution to the equation corresponding to the initial conditions x(t1) = X1 +c and v(t1) = V1, as in:$$\frac{\partial L(x_1(t)+c,\dot{x}_1(t),t)}{\partial x} - \frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial L(x_1(t)+c,\dot{x}_1(t),t)}{\partial \dot{x}}=0 \tag{2}$$then this must mean that $$\frac{\partial L(x,\dot{x},t)}{\partial x}=0 \tag{3}$$
I tried expanding the term $$L(x_1(t)+c,\dot{x}_1(t),t)$$ with Taylor's theorem. Assuming 'c' is not necessarily small, the series has an infinite number of terms. The LHS in eqn. (2) evaluated to zero using the identity in eqn. (1). But I am not sure how to prove eqn (3).

Any help is much appreciated.

The homogeneity of space implies that the Lagrangian of a free particle is not explicitly dependent on its position. To see this, we can consider the Euler-Lagrange equation for a single particle:$$\frac{\partial L(x, \dot{x}, t)}{\partial x} - \frac{d}{dt}\left(\frac{\partial L(x, \dot{x}, t)}{\partial \dot{x}}\right) = 0$$If we assume that the Lagrangian is homogeneous of degree n in the position variable, then it can be written as$$L(x, \dot{x}, t) = x^n \tilde{L}(\dot{x}, t)$$where $\tilde{L}$ does not depend on x explicitly. Substituting this into the Euler-Lagrange equation yields$$x^n\frac{\partial \tilde{L}(\dot{x}, t)}{\partial x} - x^n \frac{d}{dt}\left(\frac{\partial \tilde{L}(\dot{x}, t)}{\partial \dot{x}}\right) = 0$$But since $x^n \neq 0$, we must have that $$\frac{\partial \tilde{L}(\dot{x}, t)}{\partial x} = 0.$$This implies that the Lagrangian does not depend on the position explicitly.

## 1. What is homogeneity of space?

Homogeneity of space refers to the property of space being uniform and the same at every point. This means that there is no preferred location or direction in space, and all points are equivalent. This concept is important in the study of physics and is a fundamental assumption in the formulation of many physical theories.

## 2. How does homogeneity of space relate to the Lagrangian?

The Lagrangian is a mathematical function that describes the dynamics of a physical system. It is based on the principle of least action, which states that the actual path taken by a system between two points in space and time is the one that minimizes the action, a measure of the system's energy. The Lagrangian is homogeneous in space, meaning it is unchanged by translations in space, which reflects the homogeneity of space itself.

## 3. What is the significance of the form of the Lagrangian?

The form of the Lagrangian is crucial in determining the equations of motion for a physical system. It contains all the information about the system's potential and kinetic energies, as well as any external forces acting on the system. By varying the Lagrangian with respect to the system's variables, one can obtain the equations of motion that govern the system's behavior.

## 4. How does the form of the Lagrangian affect the behavior of a system?

The form of the Lagrangian determines the equations of motion for a system, which in turn dictate the system's behavior. By varying the Lagrangian, one can derive the equations of motion and study how the system will evolve over time. The form of the Lagrangian can also reveal important properties of the system, such as symmetries and conservation laws.

## 5. Can the form of the Lagrangian be modified?

Yes, the form of the Lagrangian can be modified to include additional terms or factors to account for various physical phenomena. In fact, the process of modifying the Lagrangian to incorporate new information or theories is an active area of research in physics. However, the form of the Lagrangian must still adhere to certain principles, such as homogeneity of space, in order to accurately describe the dynamics of a system.

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