Question

The dynamics of a system with \(n\) degrees of freedom are governed by a Lagrangian \(L(q, v, t)\). Show that if \(f(q, t)\) is any function on \(C T\), then $$ L^{\prime}=L+\frac{\partial f}{\partial q_{a}} v_{a}+\frac{\partial f}{\partial t} $$ generates the same dynamics.

Short answer

Question: Show that if we have a Lagrangian \(L(q, v, t)\) for a system with \(n\) degrees of freedom, and \(f(q, t)\) is any function on \(C T\), then the modified Lagrangian \(L^{\prime}=L+\frac{\partial f}{\partial q_{a}} v_{a}+\frac{\partial f}{\partial t}\) will generate the same dynamics. Answer: By applying the Euler-Lagrange equation to both the original and modified Lagrangians and showing that they result in the same equations of motion, we can conclude that the modified Lagrangian \(L^{\prime}\) generates the same dynamics as the original Lagrangian \(L\).

Step-by-step solution

Write down the Euler-Lagrange equation for the original Lagrangian

To find the equations of motion for the original Lagrangian, we first apply the Euler-Lagrange equation: $$ \frac{d}{dt}\frac{\partial L}{\partial v_a} - \frac{\partial L}{\partial q_a} = 0 $$ This equation represents the dynamics of the system for each degree of freedom \(a\).

Write down the modified Lagrangian

Now we have the modified Lagrangian \(L^{\prime}\) defined as follows: $$ L^{\prime}=L+\frac{\partial f}{\partial q_{a}} v_{a}+\frac{\partial f}{\partial t} $$

Write down the Euler-Lagrange equation for the modified Lagrangian

First, we will calculate \(\frac{\partial L^{\prime}}{\partial q_a}\), \(\frac{\partial L^{\prime}}{\partial v_a}\), and \(\frac{d}{dt}\frac{\partial L^{\prime}}{\partial v_a}\) and then, we will apply the Euler-Lagrange equation for the modified Lagrangian \(L^{\prime}\). $$ \frac{d}{dt}\frac{\partial L^{\prime}}{\partial v_a} - \frac{\partial L^{\prime}}{\partial q_a} = 0 $$

Calculate necessary derivatives of \(L^{\prime}\)

First, we find the derivatives of \(L^{\prime}\) with respect to \(q_a\) and \(v_a\). $$ \frac{\partial L^{\prime}}{\partial q_a} = \frac{\partial L}{\partial q_a} + \frac{\partial^2 f}{\partial q_a^2}v_a + \frac{\partial^2 f}{\partial q_a \partial t} $$ $$ \frac{\partial L^{\prime}}{\partial v_a} = \frac{\partial L}{\partial v_a} + \frac{\partial^2 f}{\partial q_a^2} $$ Now, we find the time derivative of \(\frac{\partial L^{\prime}}{\partial v_a}\). $$ \frac{d}{dt}\frac{\partial L^{\prime}}{\partial v_a} = \frac{d}{dt}(\frac{\partial L}{\partial v_a} + \frac{\partial^2 f}{\partial q_a^2}) $$ $$ \frac{d}{dt}\frac{\partial L^{\prime}}{\partial v_a} = \frac{d}{dt}\frac{\partial L}{\partial v_a} + \frac{d}{dt}(\frac{\partial^2 f}{\partial q_a^2}) $$

Apply the Euler-Lagrange equation for the modified Lagrangian

Now, plug the derivatives we found into the Euler-Lagrange equation for \(L^{\prime}\). $$ \frac{d}{dt}\frac{\partial L^{\prime}}{\partial v_a} - \frac{\partial L^{\prime}}{\partial q_a} = \frac{d}{dt}\frac{\partial L}{\partial v_a} + \frac{d}{dt}(\frac{\partial^2 f}{\partial q_a^2}) - \frac{\partial L}{\partial q_a} - \frac{\partial^2 f}{\partial q_a^2}v_a - \frac{\partial^2 f}{\partial q_a \partial t} = 0 $$ Notice that the terms \(\frac{d}{dt}\frac{\partial L}{\partial v_a} - \frac{\partial L}{\partial q_a}\) represent the Euler-Lagrange equation for the original Lagrangian, which is equal to zero.

Simplify and show that the dynamics are the same

Since the Euler-Lagrange equation for the original Lagrangian is equal to zero, we have: $$ \frac{d}{dt}(\frac{\partial^2 f}{\partial q_a^2}) - \frac{\partial^2 f}{\partial q_a^2}v_a - \frac{\partial^2 f}{\partial q_a \partial t} = 0 $$ This equation does not depend on the original Lagrangian \(L(q, v, t)\), which implies that the dynamics generated by the modified Lagrangian \(L^{\prime}\) are the same as the dynamics generated by the original Lagrangian \(L\).

Key concepts

Euler-Lagrange equation

The Euler-Lagrange equation is vital in Lagrangian dynamics, serving as the foundation to derive the equations of motion for a system. It is represented as: \[\frac{d}{dt}\frac{\partial L}{\partial v_a} - \frac{\partial L}{\partial q_a} = 0\] This formula tells us how the position \( q_a \) and velocity \( v_a \) of a system change over time, where \( L \) is the Lagrangian of the system. The Lagrangian itself is usually a function representing the difference between kinetic and potential energy.

• **Key role:** Enables the calculation of how a system evolves by specifying its motion through the principle of least action.
• **Application:** Extends to systems with multiple degrees of freedom; one equation for each degree of freedom.

The Lagrangian dynamics differ from the Newtonian approach by utilizing scalar quantities, making it easier to handle complex systems. Its elegance lies in working with generalized coordinates \( q \), functioning well even in non-Cartesian systems. This broadens the flexibility of analysis, particularly for systems that aren't straightforward to observe directly.

Degrees of freedom

Degrees of freedom (DoF) are an essential concept in physics, especially when discussing dynamic systems like those in Lagrangian mechanics. In simple terms, it represents the number of independent variables needed to define the state of a system. For example, a rigid body in three-dimensional space has six degrees of freedom: three translational and three rotational movements.

• **Importance:** Provides a quantitative measure of the possible independent motions of a system.
• **Complexity:** Increasing degrees of freedom often lead to more complicated dynamics and require more equations to solve.

In the context of Lagrangian dynamics, each degree of freedom corresponds to an independent coordinate, leading to a distinct Euler-Lagrange equation. These equations together define the complete behavior of the system.

Analytical mechanics

Analytical mechanics is a broad field in classical mechanics that includes techniques like Lagrangian and Hamiltonian mechanics. This approach differs from the Newtonian framework by focusing on energy rather than force. Analytical mechanics deals with the effectiveness of calculating mechanics problems using functions rather than forces, as seen in Newton's laws.

• **Lagrangian mechanics:** Utilizes the principle of least action, where the path taken by the system minimizes the action, which is the integral of the Lagrangian.
• **Hamiltonian mechanics:** Often introduced as a reformulation of Lagrangian mechanics, it uses canonical coordinates and has applications in quantum mechanics.

These methods offer powerful tools for handling a wide variety of physical systems. They are particularly advantageous when dealing with constraints or non-Cartesian coordinates, providing a more general and sometimes simpler framework for solving mechanics problems. Lagrangian mechanics, for example, bypasses the need to consider forces directly and instead works with energy, which can simplify the analysis of complex systems.