# Time-Dependent Lagrangians: Reference Work & Euler-Lagrange Equations

• StuckOnZero
In summary, a time-dependent Lagrangian is a mathematical function used to describe the behavior of a physical system over time, taking into account position, velocity, and acceleration. It differs from a regular Lagrangian by also considering acceleration, making it more accurate for systems with changing dynamics. The Euler-Lagrange equation is a mathematical tool used to find the equations of motion for a system described by a Lagrangian. It is derived from the principle of least action and has practical applications in physics and engineering, including classical mechanics, quantum mechanics, and general relativity.
StuckOnZero
I'm looking for a good reference work on time-dependent Lagrangians. For example, the Lagrangian and resultant Euler-Lagrange equations for a forced harmonic oscillator. All the classical textbooks just skip over this subject area. Obviously the system is non-energy conserving. In deriving the E-L equations one has to add a partial wrt T. I am looking to solve some electron trajectory problems when there is a weak oscillating external E-field.
-Thanks

Your best bet will be a book on applied mechanics, either celestial mechanics, tokamak or particle accelerators.

## 1. What is a time-dependent Lagrangian?

A time-dependent Lagrangian is a mathematical function that describes the behavior of a physical system over time. It takes into account the position, velocity, and acceleration of the system at any given time and can be used to calculate the system's dynamics and motion.

## 2. How is a time-dependent Lagrangian different from a regular Lagrangian?

A regular Lagrangian is a function that only depends on the position and velocity of a system, whereas a time-dependent Lagrangian also takes into account the acceleration. This makes it more accurate for describing systems with changing dynamics over time.

## 3. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical equation used to find the equations of motion for a system described by a Lagrangian. It takes into account the partial derivatives of the Lagrangian with respect to the variables of the system, and sets them equal to zero to find the stationary points of the system.

## 4. How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived from the principle of least action, which states that the actual path a system takes between two points is the one that minimizes the action, or the integral of the Lagrangian over time. By applying the calculus of variations to this principle, the Euler-Lagrange equation can be derived.

## 5. What are some practical applications of time-dependent Lagrangians and the Euler-Lagrange equation?

Time-dependent Lagrangians and the Euler-Lagrange equation have a wide range of practical applications in physics and engineering, such as in classical mechanics, quantum mechanics, and general relativity. They are used to model and analyze the behavior of complex systems, including the motion of particles, the dynamics of fluids, and the behavior of electromagnetic fields.

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