- #1
Storm Butler
- 78
- 0
I was wondering why when we derive the euler lagrange equations and when we use them we treat x and x dot as independent quantities?
Hmm, that's not how I usually see it done. In the derivations I've seen, they say something like "Since the integral of [itex]\eta[/itex] times blah is zero and [itex]\eta[/itex] was arbitrary, it follows that blah is zero." Is that not fully rigorous?vanhees71 said:After taking the derivative you take as another limit the support of [itex]\eta[/itex] to the single point [itex]t[/itex], i.e., in a sense [itex]\eta(t') \propto \delta(t'-t)[/itex].
The Lagrangian, which is used in classical mechanics to describe the motion of a system, is based on the principle of least action. This principle states that the path a system takes between two points in time is the one that minimizes the action, which is defined as the integral of the Lagrangian over time. In order for this principle to work, the Lagrangian must be a function of independent variables. In this case, the independent variables are position and velocity.
Treating velocity and position as independent variables allows us to describe the motion of a system in a more general and elegant way. It also allows us to easily incorporate constraints and external forces into the Lagrangian, which would not be possible if we treated velocity and position as dependent variables.
No, it is not possible to treat velocity and position as dependent variables in a Lagrangian. This is because the Lagrangian is based on the principle of least action, which requires the Lagrangian to be a function of independent variables. If velocity and position were treated as dependent, the Lagrangian would not be able to accurately describe the motion of a system.
Treating velocity and position as independent variables in the Lagrangian leads to a set of second-order differential equations known as the Euler-Lagrange equations. These equations are derived from the principle of least action and provide a concise and efficient way to describe the motion of a system.
While treating velocity and position as independent variables in the Lagrangian is a powerful tool, it does have its limitations. For example, the Lagrangian cannot accurately describe systems that exhibit chaotic behavior or systems that involve quantum mechanics. In these cases, alternative methods must be used to describe the motion of the system.