Herglotz's Variational Principle

In mathematics and physics, Herglotz's variational principle, named after German mathematician and physicist Gustav Herglotz, is an extension of the Hamilton's principle, where the Lagrangian L explicitly involves the action $S$ as an independent variable, and $S$ itself is represented as the solution of an ordinary differential equation (ODE) whose right hand side is the Lagrangian $L$, instead of an integration of $L$. Herglotz's variational principle is known as the variational principle for nonconservative Lagrange equations and Hamilton equations.

Mathematical formulation

Suppose there is a Lagrangian $L = L(t,\boldsymbol, \boldsymbol,S)$ of $2n+2$ variables, where $\boldsymbol q = (q_1, q_2, \dots, q_n)$ and $\boldsymbol u = (u_1, u_2, \dots, u_n)$ are $n$ dimensional vectors, and $t, S$ are scalar values. A time interval _0, t_1 is fixed. Given a time-parameterized curve $\boldsymbol q = \boldsymbol q (t)$, consider the ODE \begin
\dot S (t) = L(t, \boldsymbol q (t), \boldsymbol \dot q (t), S(t)), & t \in _0, t_1\\
S(t_0) = S_0
\endWhen $L(t, \boldsymbol q, \boldsymbol u, S), \boldsymbol q (t), \boldsymbol u(t)$ are all well-behaved functions, this equation allows a unique solution, and thus $S_1 := S(t_1)$ is a well defined number which is determined by the curve $\boldsymbol q (t)$. Herglotz's variation problem aims to minimize $S_1$ over the family of curves $\boldsymbol q (t)$ with fixed value $\boldsymbol q_0$ at $t=t_0$ and fixed value $\boldsymbol q_1$ at $t = t_1$, i.e. the problem \underset S_1 boldsymbol q/math>Note that, when $L$ does not explicitly depend on $S$, i.e. $L = L(t,\boldsymbol, \boldsymbol)$, the above ODE system gives exactly $S(t) = \int_^t L(t, \boldsymbol q(\tau ), \boldsymbol q(\tau )) \tau$, and thus $S_1 = S(t_1) = \int_^ L(t, \boldsymbol q(t), \boldsymbol q(t )) t$, which degenerates to the classical Hamiltonian action. The resulting Euler-Lagrange-Herglotz equation is $\frac \left(\frac\right) - \frac = \frac \frac$which involves an extra term $\frac \frac$ that can describe the dissipation of the system.

Derivation

In order to solve this minimization problem, we impose a variation $\delta \boldsymbol q$ on $\boldsymbol q$, and suppose $S(t)$ undergoes a variation $\delta S(t)$ correspondingly, then$\begin \delta \dot S(t) & = L (t, \boldsymbol q (t) + \delta \boldsymbol q (t), \dot \boldsymbol q(t) + \delta \dot \boldsymbol q (t), S(t) + \delta S(t)) - L (t, \boldsymbol q (t), \dot \boldsymbol q(t), S(t)) \\ & = \frac \delta \boldsymbol q(t) + \frac \delta \dot \boldsymbol q(t) + \frac \delta S(t) \end$and since the initial condition is not changed, $\delta S_0 = 0$. The above equation a linear ODE for the function $\delta S(t)$, and it can be solved by introducing an integrating factor $\mu (t) = \mathrm^$, which is uniquely determined by the ODE $\dot( t) =-\mu ( t)\frac, \quad u(t_0) = 1.$By multiplying $\mu(t)$ on both sides of the equation of $\delta \dot S$ and moving the term $\mu (t) \frac \delta S(t)$ to the left hand side, we get $\mu ( t) \delta \dot( t) -\mu ( t)\frac \delta S( t) =\mu ( t)\left(\frac \delta \boldsymbol( t) +\frac \delta \dot( t)\right).$Note that, since $\dot( t) =-\mu ( t)\frac$, the left hand side equals to $\mu ( t) \delta \dot( t) +\dot( t) \delta S( t) =\frac$and therefore we can do an integration of the equation above from $t=t_0$ to $t=t_1$, yielding $\mu ( t_) \delta S_ -\mu ( t_) \delta S_ =\int _^ \mu ( t)\left(\frac \delta \boldsymbol( t) +\frac \delta \dot( t)\right)\mathrm t$where the $\delta S_0 = 0$ so the left hand side actually only contains one term $\mu(t_1) \delta S_1$, and for the right hand side, we can perform the integration-by-part on the $\frac \delta \dot( t)$ term to remove the time derivative on $\delta$:$\begin & \int _^ \mu ( t)\left(\frac \delta \boldsymbol( t) +\frac \delta \dot( t)\right)\mathrm t\\ = & \int _^ \mu ( t)\frac \delta \boldsymbol( t)\mathrm t+\int _^ \mu ( t)\frac \delta \dot( t)\mathrm t\\ = & \int _^ \mu ( t)\frac \delta \boldsymbol( t)\mathrm t+\mu ( t_)\frac\underbrace_ -\mu ( t_)\frac\underbrace_ -\int _^\frac\left( \mu ( t)\frac\right) \delta \boldsymbol( t)\mathrm t\\ = & \int _^ \mu ( t)\frac \delta \boldsymbol( t)\mathrm t-\int _^\frac\left( \mu ( t)\frac\right) \delta \boldsymbol( t)\mathrm t\\ = & \int _^ \mu ( t)\frac \delta \boldsymbol( t)\mathrm t-\int _^\left(\dot( t)\frac +\mu ( t)\frac\frac\right) \delta \boldsymbol( t)\mathrm t\\ = & \int _^ \mu ( t)\frac \delta \boldsymbol( t)\mathrm t-\int _^\left( -\mu ( t)\frac\frac +\mu ( t)\frac\frac\right) \delta \boldsymbol( t)\mathrm t\\ = & \int _^ \mu ( t)\underline \delta \boldsymbol( t)\mathrm t, \end$and when $S_1$ is minimized, $\delta S_1 = 0$ for all $\delta \boldsymbol q$, which indicates that the underlined term in the last line of the equation above has to be zero on the entire interval
_0, t_1/math>, this gives rise to the Euler-Lagrange-Herglotz equation.

Examples

One simple one-dimensional ($n=1$) example is given by the Lagrangian $L(t,x,\dot x,S)=\frac m \dot x^2 - V(x) -\gamma S$The corresponding Euler-Lagrange-Herglotz equation is given as $\frac( m\dot) +V'( x) =-\gamma \dot,$which simplifies into $m \ddot x = -V'(x) - \gamma \dot x.$This equation describes the damping motion of a particle in a potential field $V$, where $\gamma$ is the damping coefficient.

References

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Physics theorems
Mathematical theorems