Let \(T^*N\) be the cotangent bundle of an \(n\)-dimensional closed oriented manifold \(N\) with the natural symplectic structure and \({\mathcal G}\) the Hamiltonian group with compact support on \(T^*N\). In this paper two families of functions \(\mu_a:{\mathcal G}\to\mathbb{R}\) and \(\zeta_a: C^\infty_c(T^*N)\to \mathbb{R}\), \(a\in H^1(N;\mathbb{R})\), which possesses properties analogous to those of partial quasimorphisms and partial quasistates of \textit{M. Entov} and \textit{L. Polterovich} [Comment. Math. Helv. 81, No. 1, 75--99 (2006; Zbl 1096.53052)] are constructed (Theorems 1.3 and 1.8). It is shown that if \(N=\mathbb{T}^n\), then the family \(\mu_a\) is equivalent to Viterbo's symplectic homogenization (Theorem 1.1. in [\textit{C. Viterbo}, ``Symplectic homogenization'', Preprint, \url{arXiv: 0801.0206}]). Let \(\phi^t_H\) be the time-\(t\) map of the flow of \(H\) and \(\phi_H\) the time-\(1\) map, then \(\zeta_a\) is defined as \[ \zeta_a(H)= \mu_a(\phi_H). \] Using Lagrangian spectral invariants \(\ell_{\pm}(\phi)\) for Hamiltonian diffeomorphisms via Lagrangian Floer homology [\textit{Y.-G. Oh}, J. Differ. Geom. 46, No. 3, 499--577 (1997; Zbl 0926.53031); Commun. Anal. Geom. 7, No. 1, 1--55 (1999; Zbl 0966.53055)], \(\mu_a\) is defined as \[ \mu_0(\phi)= \lim_{k\to\infty} {\ell_+(\phi^k)\over k},\quad \mu_a(\phi)= \mu_0(T_\alpha \phi T_\alpha), \] where \(\alpha\in a\) and \(T_\alpha: T^* N\to T^* N\) is defined as \(T_\alpha(q, p)= (q,p+ \alpha(q))\) (\S3.1). By definition, restriction of \(\mu_a\) to \({\mathcal G}_0\), the subgroup of \({\mathcal G}\) fixing the zero section \(N\) of \(T^*N\) as a set, coincides with the action homomorphism \({\mathcal A}:{\mathcal A}(\phi)={\mathcal A}_H(\gamma_q, \gamma_q(t))= \phi^t_H(q)\), where \[ {\mathcal A}_H(\gamma)= \int^1_0 H_t(\gamma(t))\,dt- \int\gamma^*\lambda \] is the action functional on the space of paths \(\Omega= \{\gamma: [0,1]\to T^* N\mid\gamma(0)\in N\}\). Lagrangian and Hamiltonian spectral invariants for Hamiltonian diffeomorphisms are reviewed in \S2. Then coincidence of the spectral invariants of \(H\) and \(H'\) when \(\phi_H= \phi_{H'}\) (Lemma 2.7) and the inequalities \[ c_-(\phi)\leq\ell_-(\phi)\leq \ell_+(\phi)\leq c_+(\phi), \] where \(c_{\pm}(\phi)\) means Hamiltonian spectral invariants (Proposition 2.17) are proved. Existence of \(\mu_a(\phi)\) follows from these results (\S3.1). \(\mu_a\) is invariant under conjugation by elements of \({\mathcal G}\) and Lipschitz with respect to the Hofer metric. The function \(H^1(N; \mathbb{R})\to\mathbb{R}\) defined by \(a\to\mu_a(\phi)\) for a fixed \(\phi\) is also Lipschitz. These are proved in \S3.1 together with several other statements. \(\zeta_a\) is also invariant under the \({\mathcal G}\)-action and \(\zeta_a(F+ G)=\zeta_a(F)\) if \(F\), \(G\) commute and the support \(S\) of \(G\) is displaceable, that is there is \(\phi\in{\mathcal G}\) such that \(\overline S\cap\phi(S)=\emptyset\). As a consequence, if \(N= N_1\times N_2\), \(F_i\in C^\infty(T^*N)\), and \(a= (a_1, a_2)\in H^1(N_1; \mathbb{R})\times H^2(N_2;\mathbb{R})\subset H^1(N;\mathbb{R})\), then \[ \zeta_a(F_1\oplus F_2)_a= \zeta_{a_1}(F_1)+ \zeta_{a_2}(F_2), \] (Proposition 1.9). These are also proved in \S3.1. \(\mu_0(\phi)\) gives lower bounds of the fragmentation norm; see Corollary 1.10 in [\textit{A. Banyaga}, Comment. Math. Helv. 53, 174--227 (1978; Zbl 0393.58007)]. If \(H\) is a time-periodic Tonnelli Hamiltonian, then \[ \alpha_H(a)= \mu_a(\phi_H), \] where \(\alpha_H\) is Mather's alpha function; see Theorem 1.11 in [\textit{J. N. Mather}, Math. Z. 207, No. 2, 169--207 (1991; Zbl 0696.58027)]. The spectral norm \(\Gamma(\psi)\) is defined as \(c_+(\psi)- c_-(\psi)\) and the displacement energy of a displacement subset by \[ e(S)= \text{inf}\{\Gamma(\psi)\mid\psi(S) \cap(\overline S)= \emptyset\}. \] Then, if the support of \(G\) is displaceable, \(\zeta_a\) satisfies \[ |\zeta_a(F+ G)- \zeta_a(F)- \zeta_a(G)|\leq \sqrt{2e(U)\|\{F, G\}\|_{C^0}}. \] From this inequality, a lower bound for the norm of the Poisson bracket is derived. These results together with other applications of \(\mu_a\) and \(\zeta_a\), are proved in \S3, the last section. In the appendix, generating functions for Tonelli flows are studied. The results are used in the proof of Theorem 1.11 (given in \S3.3).